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On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

Abstract

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

Introduction

Fractional differential equations are considered as prolongation of the concept of derivative operator from integer order to any real or complex order. Fractional differential equations usually describe the nonlocal effects. Over the last two decades, there has been a blistering growth in the field of fractional calculus. Owing to the vast amount of applications, many mathematicians focused their engrossment on fractional calculus.

There exist several definitions for fractional derivatives and fractional integrals in the literature like Riemann–Liouville, Caputo, Hadamard, Riesz, Grunwald–Letnikov, Marchaud, Erdelyi–Kober, etc. The process of developing these operators began with a series of stages ranging from exponential functions to different classes of functions. Having lately come into holocene Udita N. Katugampola [1] generalized the above mentioned integral and differential operators. Meanwhile the well-developed theory and many more applications of the said operators are still a spotlight area of research in applied sciences.

As we know, the existence theory is meat-and-potatoes in every field of science, as it is very applicative to comprehend whether there is a solution to a given differential equation beforehand; otherwise, all the attempts to find a numerical or analytic solution will become valueless. The analysis of fractional differential equations has been carried out by various authors (see, for example, [218]).

As most fractional derivatives are computed using the corresponding integrals, researchers describe the nonlocal effects in terms of left and the right derivative. Thus, many mathematicians are in a hunt to generalize the notions further. In this context, Riesz [19] demonstrated the two-sided fractional operators using both left and right Riemann–Liouville’s fractional differential and integral operators.

Due to the two-sided nature of Riesz’s differential operator, the interesting differential is specifically used for fractional modeling on a finite domain. Some optimality conditions are discussed by Almeida for fractional variational problems with Riesz–Caputo derivative [20]. Frederico et al. derived Noether’s theorem for variational problems having Riesz–Caputo derivatives. In [21], Mandelbrot demonstrated that there is a close connection between Brownian motion and fractional calculus.

In [22], the authors solved the fractional Poisson equation having Riesz derivative using Fourier transform. Due to the validity of Riesz derivative operator on the whole domain, it appears in the fractional turbulent diffusion model. In [23], the authors numerically solved the advection-diffusion equation having Riesz derivative. For further applications of Riesz derivative on the anomalous diffusion, see [2429].

In this work, we define the generalized Riesz–Caputo type derivative operator by using the generalized operators. We present basic perspectives on the existence and uniqueness of solutions of fractional differential equations. Motivated by [30, 31], we provide the analysis on existence of solutions for the following nonlinear fractional differential equation involving generalized Riesz–Caputo type derivative operator with general boundary conditions:

$$\begin{aligned} \textstyle\begin{cases} {}_{0}^{RC}D_{T} ^{\alpha ,\rho } \phi (\mu ) = g(\mu ,\phi (\mu ), _{0}^{RC}D_{T} ^{{\alpha {^{*},\rho }}}\phi (\mu )), \quad \mu \in [0,T ], \\ \phi (0) = {\phi _{0}}, \qquad \phi (T ) = {\phi _{T} }, \end{cases}\displaystyle \end{aligned}$$
(1)

where \({\phi _{0}}\) and \({\phi _{T} }\) are constants, while \(g:[0,T] \times \mathbb{R}^{2} \to \mathbb{R} \) is continuous with \(1 < \alpha \le 2 \), \(0 < \alpha ^{*} \le 1 \), and \(1<\rho <\infty \).

The rest of the paper is organized as follows: Sect. 2 presents some basic definitions and lemmas from literature. In Sect. 3 we introduce the generalized Riesz–Caputo’s fractional operators and derived some useful results, while in Sect. 4 we establish some equivalence results for boundary value problem (1) and establish the results for the existence and uniqueness of solutions for BVP (1). The last section of this paper presents the stability of solutions for BVP (1) by means of continuous dependence on parameters.

Preliminaries

In this section we demonstrate some useful results including definitions and lemmas related to Riesz–Caputo derivatives and integrals that will help us in our later discussions. Following the same traditional definitions of Riesz–Caputo derivative and integral [19, 30, 32], we can generalize these definitions using a generalized Caputo type derivative operator. Some preliminary structural properties, which we will frequently use in our later discussion, are also introduced in this section. In 2010, Om Prakash Agrawal defined the generalized fractional in the following way.

Definition 2.1

([33])

Let \(\alpha >0 \). Then the generalized fractional integral operator \(A_{(a,T;r,s)}^{\alpha }\) is defined as

$$\begin{aligned} A_{(a,T;r,s)}^{\alpha }\phi (\mu ): = {r \int _{a}^{\mu }{{K_{\alpha }}( \mu ,\eta )\phi ( \eta )\,d\eta } + s \int _{\mu }^{T} {{K_{\alpha }}(\mu , \eta )\phi ( \eta )\,d\eta } }, \end{aligned}$$

where the kernel function \({K_{\alpha }}(\mu ,\eta ) \) may depend on α and \(a<\mu <T \) and \(r,s\in \mathbb{R} \).

This is the generalized fractional integral operator which, by using the specific kernel function, leads to the specific operator. For example, if \({K_{\alpha }}(\mu ,\eta ) = \frac{{{{(\mu - \eta )}^{\alpha - 1}}}}{{\Gamma (\alpha )}} \) and by taking \(T=0 \) leads to the left sided R-L integral operator and by taking \({K_{\alpha }}(\mu ,\eta ) = \frac{{{\eta ^{\rho - 1}}{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{\alpha - 1}}}}{{{\rho ^{\alpha - 1}}\Gamma (\alpha )}} \) with \(T=0 \) gives the left generalized integral defined below. Furthermore, the limits of integration a and T can be extended to −∞ and ∞ respectively.

Definition 2.2

([34])

Let \(\alpha \in \mathbb{R}_{+} ,c\in \mathbb{R} \), and \(g \in X^{p}_{c}(a,b)\), where \(X^{p}_{c}(a,b) \) is the space of Lebesgue measurable functions. Then corresponding generalized left- and right-sided fractional integrals \(({}^{\rho }I^{\alpha }_{a+}g)(\mu ) \) and \({({}^{\rho }}I_{{b^{-} }}^{\alpha }g)(\mu ) \) of order \(\alpha \in \mathbb{C} (\operatorname{Re}(\alpha ))>0\) are defined by

$$\begin{aligned} \begin{aligned} &\bigl({}^{\rho }I^{\alpha }_{a+}g \bigr) (\mu ) = \frac{\rho ^{1-\alpha }}{\Gamma (\alpha )} \int _{a}^{\mu } \frac{\eta ^{\rho -1}g(\eta )}{({\mu ^{\rho }-\eta ^{\rho }})^{1-\alpha }}\,d \eta ,\quad \mu >a, \rho >0, \\ &{\bigl({}^{\rho }}I_{{b^{-} }}^{\alpha }g\bigr) (\mu ) = \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{b} { \frac{{{\eta ^{\rho - 1}}g(\eta )}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta ,\quad \mu < b,\rho > 0, \end{aligned} \end{aligned}$$

respectively, where \(\Gamma (\cdot) \) is Euler’s gamma function.

Theorem 2.3

([35])

Let \(\alpha , \rho \in \mathbb{R}\) and \(\rho , a>0 \). Then, for \(\phi \in X^{p}_{c}(a,b)\), the following relation holds:

$$\begin{aligned} \bigl(_{*}^{\rho }D_{{a^{+} }}^{\alpha }{}^{\rho }I_{{a^{+} }}^{\alpha } \phi \bigr) ( \mu ) = \phi (\mu ). \end{aligned}$$

Similarly, the inverse property holds for a right-hand-sided integral and a derivative operator as well.

Lemma 2.4

([35])

Let \(0<\alpha <\beta <1\) and \(\rho , a>0 \). Then, for \(\phi \in X^{p}_{c}(a,b)\), the following relation holds:

$$\begin{aligned} _{*}^{\rho }D_{{a^{+} }}^{\alpha }{}^{\rho }I_{{a^{+} }}^{\beta } \phi (\mu ) = {}^{\rho }I_{{a^{+} }}^{\beta - \alpha }\phi (\mu ). \end{aligned}$$

Lemma 2.5

([35])

Let \(\alpha ,\rho \in \mathbb{R}_{+}\) and \(g \in AC_{\delta }^{n}[0,T]\): the space of complex-valued functions g which have continuous derivatives up to order \((n -1)\) on \([a, b]\) such that \(\delta _{\rho }^{(n-1)}g(\mu ) \in AC[0, T]\) is absolutely continuous on \([0, T]\), where \(\delta _{\rho }(g(\mu ))= {{\mu ^{1 - \rho }}\frac{d}{{\mu }}}(g(\mu )) \). Then, for \(0 \le \mu \le T \), the following relations hold:

  1. (i)

    \(( {_{0}^{\rho }I_{\mu }^{\alpha }{}{_{*}^{\rho }D_{0,\mu }^{\alpha }} g} )(\mu ) = g(\mu ) - \sum_{j = 0}^{n} { \frac{{\delta _{\rho }^{j}g(0)}}{{j!}}{{ ( { \frac{{{\mu ^{\rho }} }}{\rho }} )}^{j}}}\),

  2. (ii)

    \(( {_{\mu }^{\rho }I_{T} ^{\alpha }{}{_{*}^{\rho }D_{\mu ,T }^{\alpha }} g} )(T ) = \{ {g(\mu ) - \sum_{j = 0}^{n} { \frac{{( - 1)^{j}\delta _{\rho }^{j}g(T )}}{{j!}}{{ ( { \frac{{{T ^{\rho }} - {\mu ^{\rho }}}}{\rho }} )}^{j}}} } \} \),

where \(n = \lceil \alpha \rceil \) and \(\delta _{\rho }^{j} = { ( {{\eta ^{1 - \rho }}\frac{d}{{d\eta }}} )^{j} } \).

Generalized Riesz–Caputo fractional operators

In this section we introduce the generalized Riesz–Caputo fractional integrals and derivative operators.

Definition 3.1

([19])

For \(g (\mu ) \in C(0,T ) \), the classical Riesz–Caputo derivative is defined by

$$\begin{aligned} {}_{*}^{R}D_{0,T }^{\alpha }g (\mu ) &= \frac{1}{{\Gamma (n - \alpha )}} \int _{0}^{T} {{{ \vert {\mu - \eta } \vert }^{n - \alpha - 1}}} {g ^{(n)}}(\eta )\,d\eta \\ &= \frac{1}{2}\bigl({}_{*}D_{0,\mu }^{\alpha }+ {( - 1)^{n}} {}_{*}D_{\mu ,T }^{\alpha }\bigr)g (\mu ), \end{aligned}$$

where \({}_{*}D_{0,\mu }^{\alpha }\) and \({}_{*}D_{\mu ,T }^{\alpha }\) are left and right Caputo derivatives [36], respectively.

Following the same mechanism, we generalize the Riesz fractional integral by means of Definition 2.2 as follows.

Definition 3.2

Let \(g(\mu ) \in X^{p}_{c}(a,b)\) and \(\alpha ,\rho > 0\). Then, for \(0 \le \mu \le T \), the generalized Riesz type integral is defined as

$$\begin{aligned} \bigl(_{0}^{\rho }I_{T} ^{\alpha }g\bigr) (\mu ) &= \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{T} {{ \eta ^{\rho - 1}} {{ \bigl\vert {\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)} \bigr\vert }^{ \alpha - 1}}} g(\eta )\,d\eta \\ &= {}_{0}^{\rho }I_{\mu }^{\alpha }g(\mu ) + {}_{\mu }^{\rho }I_{T} ^{\alpha }g( \mu ). \end{aligned}$$

Accordingly, the Riesz–Caputo derivative [19] can be generalized by means of generalized Caputo type derivative operators [1] as follows.

Definition 3.3

Let \(\alpha , \rho \in \mathbb{C} \) with \(\operatorname{Re}({\alpha }), \operatorname{Re}({\rho })>0 \) and \(g(\mu ) \in X_{c}^{\rho }(a,b) \) for \(0 \le \mu \le T \). Then the generalized Riesz–Caputo type derivative operator is defined as

$$\begin{aligned} {}_{0}^{RC}D_{T} ^{\alpha ,\rho } g (\mu ) &= \frac{{{\rho ^{\alpha - n + 1}}}}{{\Gamma (n - \alpha )}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{ \vert {({\mu ^{\rho }} - {\eta ^{\rho }})} \vert }^{\alpha - n + 1}}}}{{\biggl({ \eta ^{1 - \rho }}\frac{d}{{d\eta }}\biggr)}^{n}}g (\eta )} \,d\eta \\ &= \frac{1}{2} \bigl( {_{*}^{\rho }D_{0,\mu }^{\alpha }+ {{( - 1)}^{n}}_{*}^{\rho }D_{\mu ,T }^{\alpha }} \bigr)g(\mu ), \end{aligned}$$

where \({_{*}^{\rho }D_{0,\mu }^{\alpha }} \) and \({_{*}^{\rho }D_{\mu ,T }^{\alpha }} \) are left and right generalized Caputo type derivatives [37] as follows:

$$\begin{aligned} {_{*}^{\rho }D_{0,\mu }^{\alpha }} = \frac{{{\rho ^{\alpha - n + 1}}}}{{\Gamma (n - \alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{\alpha - n + 1}}}}{{\biggl({ \eta ^{1 - \rho }}\frac{d}{{d\eta }}\biggr)}^{n}}g (\eta )} \,d\eta \end{aligned}$$

and

$$\begin{aligned} {_{*}^{\rho }D_{\mu ,T }^{\alpha }} = \frac{{{\rho ^{\alpha - n + 1}}}}{{\Gamma (n - \alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{\alpha - n + 1}}}}{{\biggl( - {\eta ^{1 - \rho }}\frac{d}{{d\eta }}\biggr)}^{n}}g (\eta )} \,d\eta , \end{aligned}$$

where \(n = \lceil \alpha \rceil \).

Since for \(\alpha =1 \) the right generalized derivative is the negative of the left generalized derivative, so for integer values of α, the generalized Riesz–Caputo type derivative defined above comes to term with the conventional definitions of derivative.

Lemma 3.4

Let \(g \in AC_{\delta }^{n}[0,T]\) with \(0 \le \mu \le T \). Then the following relation is true:

$$\begin{aligned} _{0}^{\rho }I_{T} ^{\alpha }{}_{0}^{RC}D_{T} ^{\alpha ,\rho } g(\mu ) = \frac{1}{2} \bigl(_{0}^{\rho }I_{\mu }^{\alpha } {}_{*}^{\rho }D_{0, \mu }^{\alpha }+ {{( - 1)}^{n}} {}_{\mu }^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{ \mu ,T }^{\alpha } \bigr)g(\mu ). \end{aligned}$$
(⁎)

Proof

Using the above definitions, we can write

$$\begin{aligned} {} _{0}^{\rho }I_{T} ^{\alpha }{}_{0}^{RC}D_{T} ^{\alpha ,\rho } g(\mu )&= \frac{1}{2}{}_{0}^{\rho }I_{T} ^{\alpha } \bigl( {_{*}^{\rho }D_{0,\mu }^{\alpha }+ {{( - 1)}^{n}}_{*}^{\rho }D_{\mu ,T }^{\alpha }} \bigr)g(\mu ) \\ &= \frac{1}{2}{}_{0}^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{0,\mu }^{\alpha }g(\mu ) + \frac{{{{( - 1)}^{n}}}}{2}{}_{0}^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{\mu ,T }^{\alpha }g( \mu ) \\ &= \frac{1}{2} \bigl( {_{0}^{\rho }I_{\mu }^{\alpha }{}_{*}^{\rho }D_{0,\mu }^{\alpha }+ _{\mu }^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{0,\mu }^{\alpha }} \bigr)g(\mu ) + \frac{{{{( - 1)}^{n}}}}{2} \bigl( {_{0}^{\rho }I_{\mu }^{\alpha }{}_{*}^{\rho }D_{\mu ,T }^{\alpha }+ _{\mu }^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{\mu ,T }^{\alpha }} \bigr)g(\mu ) \\ &= \frac{1}{2} \bigl(_{0}^{\rho }I_{\mu }^{\alpha } {}_{*}^{\rho }D_{0,\mu }^{\alpha }+ {{( - 1)}^{n}} {}_{\mu }^{\rho }I_{T} ^{\alpha }{}_{*}^{\rho }D_{\mu ,T }^{\alpha } \bigr)g(\mu ), \end{aligned}$$

and the proof is finished. □

Remark 3.5

If \(0<\alpha \le 1 \), then for \(g (\mu ) \in C[0,T]\) the relation illustrated in () becomes

$$\begin{aligned} _{0}^{\rho }I_{T} ^{\alpha }{}_{0}^{RC}D_{T} ^{\alpha ,\rho } g(\mu ) = g(0) - \frac{1}{2} \bigl( {g(0) + g(T )} \bigr). \end{aligned}$$

Proof

The proof simply follows by using \(n=1 \) in Lemma 3.4 and Lemma 2.5, which yields the required result. □

Theorem 3.6

Let \(\alpha >0 \) and \(\{\phi _{j}\}_{j = 1}^{\infty }\) be a uniformly convergent sequence of continuous functions on \([a,b]\). Then we can interchange the generalized fractional integral operator and the limit, i.e.,

$$\begin{aligned} {\bigl({}^{\rho }}I_{a + }^{\alpha }\mathop{\lim } _{j \to 0} {\phi _{j}}\bigr) ( \mu ) = \bigl(\mathop{\lim } _{j \to \infty } {}^{\rho }I_{a + }^{\alpha }{\phi _{j}}\bigr) (\mu ). \end{aligned}$$

Proof

Let ϕ be the limit of the sequence \(\{\phi _{j}\} \). Since \(\{\phi _{j}\}\) is the convergent sequence of continuous functions, so ϕ is also continuous. To prove that under the given conditions we can interchange fractional integral and limit, it is enough to show that the sequence \(\{{}^{\rho }I_{a + }^{\alpha }{\phi _{j}}\}_{j = 1}^{\infty }\) is also uniformly convergent. That is, \(\vert {{}^{\rho }I_{a + }^{\alpha }{\phi _{j}}(\mu ) - {}^{\rho }I_{a + }^{\alpha }\phi (\mu )} \vert \to 0\) as \(j \to \infty \). For this, consider

$$\begin{aligned} & \bigl\vert {{}^{\rho }I_{a + }^{\alpha }{ \phi _{j}}(\mu ) - {}^{\rho }I_{a + }^{\alpha }\phi (\mu )} \bigr\vert \\ &\quad = \biggl\vert {\frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{a}^{\mu }{ \frac{{{\phi _{j}}(\eta ){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}\,d \eta - } \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{a}^{\mu }{ \frac{{{\phi _{j}}(\eta ){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta } \biggr\vert \\ &\quad\leq \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{a}^{\mu }{ \biggl\vert { \frac{{({\phi _{j}}(\eta ) - \phi (\eta )){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \biggr\vert } \,d\eta \\ &\quad\leq \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}}{ \Vert {{ \phi _{j}} - \phi } \Vert _{\infty }} \int _{a}^{\eta }{{\eta ^{\rho - 1}}} {\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)^{\alpha - 1}}\,d\eta. \end{aligned}$$
(2)

Now, we first shall evaluate the integral

$$\begin{aligned} \int _{a}^{\eta }{{\eta ^{\rho - 1}}} {\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)^{ \alpha - 1}}\,d\eta = {\mu ^{\alpha \rho - \rho }} \int _{a}^{\eta }{{ \eta ^{\rho - 1}}} {\biggl(1 - {\frac{\eta ^{\rho }}{\mu ^{\rho }}\biggr)}^{\alpha - 1}}\,d \eta. \end{aligned}$$

Substituting \(\frac{{{\eta ^{\rho }}}}{{{\mu ^{\rho }}}}=u \), we have

$$\begin{aligned} & \int _{a}^{\eta }{{\eta ^{\rho - 1}}} {\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)^{ \alpha - 1}}\,d\eta \\ &\quad=\frac{{{\mu ^{\alpha \rho }}}}{\rho } \int _{ \frac{{{\eta ^{\rho }}}}{{{\mu ^{\rho }}}}}^{1} {{u^{0}}} {(1 - u)^{ \alpha - 1}}\,du =\frac{{{\mu ^{\alpha \rho }}}}{\rho } \int _{ \frac{{{\eta ^{\rho }}}}{{{\mu ^{\rho }}}}}^{1} {{\biggl(u-{ \frac{{{\eta ^{\rho }}}}{{{\mu ^{\rho }}}}} \biggr)^{1-1}}} {(1 - u)^{\alpha - 1}}\,du . \end{aligned}$$

Now, using the result \(\int _{{\xi _{1}}}^{{\xi _{2}}} {{{(u - {\xi _{1}})}^{ \alpha - 1}}} {({\xi _{2}} - u)^{\beta - 1}}\,du = {({\xi _{2}} - {\xi _{1}})^{ \alpha + \beta - 1}}B(\alpha ,\beta ) \), the above equation leads to

$$\begin{aligned} & \int _{a}^{\eta }{{\eta ^{\rho - 1}}} {\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)^{ \alpha - 1}}\,d\eta \\ &\quad=\frac{{{\mu ^{\alpha \rho }}}}{\rho } \biggl\{ { \frac{{{{({\mu ^{\rho }} - {a^{\rho }})}^{\alpha }}B(1,\alpha )}}{{{\mu ^{\alpha \rho }}}}} \biggr\} = \frac{{{{({\mu ^{\rho }} - {a^{\rho }})}^{\alpha }}B(1,\alpha )}}{\rho } = \frac{{{{({\mu ^{\rho }} - {a^{\rho }})}^{\alpha }}}}{{\alpha \rho }}. \end{aligned}$$

Consequently, from equation (2), we arrive at

$$\begin{aligned} \bigl\vert {{}^{\rho }I_{a + }^{\alpha }{\phi _{j}}(\mu ) - {}^{\rho }I_{a + }^{\alpha }\phi ( \mu )} \bigr\vert \le \frac{{{{({\mu ^{\rho }} - {a^{\rho }})}^{\alpha }}}}{{\rho \Gamma (\alpha + 1)}}{ \Vert {{\phi _{j}} - \phi } \Vert _{\infty }}. \end{aligned}$$

Since \((\phi _{j})\) is a uniformly convergent sequence, thus

$$\begin{aligned} \bigl\vert {^{\rho }I_{a + }^{\alpha }{\phi _{j}}(\mu ){ - ^{\rho }}I_{a + }^{\alpha }\phi ( \mu )} \bigr\vert \to 0\quad \text{as }j \to \infty. \end{aligned}$$

Therefore, the sequence \(\{{}^{\rho }I_{a + }^{\alpha }{\phi _{j}}\}_{j = 1}^{\infty }\) is also uniformly convergent, and hence the result follows. □

The similar result holds true for the right-sided generalized fractional integral as well.

Lemma 3.7

If \(\phi (\mu ) \) is an analytic function in \((a_{0}-\xi , a_{0}+\xi ) \), where \(t>0 \) and \(\alpha ,a_{0} >0 \), then

$$\begin{aligned} {\bigl({}^{\rho }}I_{{a_{0}}}^{\alpha }\phi \bigr) (\xi ) = \sum _{j = 0}^{\infty }{ \frac{{\Gamma (\frac{j}{\rho } + 1){\xi ^{\alpha (\rho - 1)}}{{(\xi - {a_{0}})}^{j + \alpha }}}}{{j!\Gamma (\frac{j}{\rho } + \alpha + 1){\rho ^{\alpha }}}}} {\phi ^{j}}({a_{0}}). \end{aligned}$$

In particular, \({^{\rho }}I_{{a_{0}}}^{\alpha }\phi \) is also analytic.

Proof

Since ϕ is an analytic function, thus it can be written in the form of convergent power series, i.e.,

$$\begin{aligned} \phi (\xi ) = \sum_{j = 0}^{\infty }{ \frac{{{{(\xi - {a_{0}})}^{j}}}}{{j!}}} {\phi ^{j}}({a_{0}}). \end{aligned}$$

Using Definition 2.2, we get

$$\begin{aligned} {\bigl({}^{\rho }}I_{{a_{0}}}^{\alpha }\phi \bigr) (\xi ) = \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{{a_{0}}}^{\xi }{{\eta ^{\rho - 1}} {{\bigl({\xi ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}} \sum _{j = 0}^{\infty }{\frac{{{{(\eta - {a_{0}})}^{j}}}}{{j!}}} {\phi ^{j}}({a_{0}})} \,d\eta. \end{aligned}$$

Using Theorem 3.6, the summation and integral sign are interchanged as follows:

$$\begin{aligned} {\bigl({}^{\rho }}I_{{a_{0}}}^{\alpha }\phi \bigr) (\xi ) &= \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}}\sum_{j = 0}^{\infty }{ \frac{{{\phi ^{j}}({a_{0}})}}{{j!}}} \int _{{a_{0}}}^{\xi }{{ \eta ^{\rho - 1}} {{(\eta - {a_{0}})}^{j}} {{\bigl({\xi ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} \,d\eta , {\bigl({}^{\rho }}I_{{a_{0}}}^{\alpha } \phi \bigr) ( \xi ) \\ &= \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}}\sum_{j = 0}^{\infty }{ \frac{{{\phi ^{j}}({a_{0}}){\xi ^{\alpha (\rho - 1)}}{{(\xi - {a_{0}})}^{j + \alpha }}}}{{\rho j!}}} B\biggl(\frac{j}{\rho } + 1,\alpha \biggr) \\ &=\sum_{j = 0}^{\infty }{ \frac{{\Gamma (\frac{j}{\rho } + 1){\xi ^{\alpha (\rho - 1)}}{{(\xi - {a_{0}})}^{j + \alpha }}}}{{j!\Gamma (\frac{j}{\rho } + \alpha + 1){\rho ^{\alpha }}}}} { \phi ^{j}}({a_{0}}). \end{aligned}$$

 □

Theorem 3.8

Let \(\phi \in X_{c}^{p}{(a,b)} \) and \(\{ {{\lambda _{j}}} \}_{j = 1}^{\infty }\) be a convergent sequence of nonnegative real numbers with limit λ. Then

$$\begin{aligned} \lim_{j \to 0} \bigl({{}^{\rho }}I_{a + }^{{\lambda _{j}}} \phi \bigr) (\mu ) = {\bigl({}^{\rho }}I_{a + }^{\lambda }\phi \bigr) (\mu ), \end{aligned}$$

where convergence of the sequence \({\{^{\rho }}I_{a + }^{{\lambda _{j}}}\phi \}_{j = 1}^{\infty }\) is signified in terms of \(X_{c}^{p}({a,b}) \) norm with \(1\le p \le \infty \), \(p,c\in \mathbb{R}\), \(\rho >0 \), and \(c\leq \rho + 1 \).

Proof

Let the sequence \(\{ {{\lambda _{j}}} \}_{j = 1}^{\infty }\) converge to the limit λ. Then, by definition,

$$\begin{aligned} {\bigl({}^{\rho }}I_{a + }^{{\lambda _{j}}}\phi \bigr) (\mu ) = \frac{{{\rho ^{1 - {\lambda _{j}}}}}}{{\Gamma ({\lambda _{j}})}} \int _{a}^{\mu }{ \frac{{{\eta ^{\rho - 1}}\phi (\eta )}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - {\lambda _{j}}}}}}} \,d\eta ,\quad \mu > a,\rho > 0, \end{aligned}$$

and by taking limit on both sides and by using Theorem 3.6, we have

$$\begin{aligned} \mathop{\lim } _{j \to 0} {\bigl({}^{\rho }}I_{a + }^{{\lambda _{j}}} \phi \bigr) (\mu ) &= \frac{{{\rho ^{\mathop{\lim } _{j \to 0} (1 - {\lambda _{j}})}}}}{{\mathop{\lim } _{j \to 0} \{ {({\lambda _{j}} - 1)!} \} }} \int _{a}^{\mu }{{\eta ^{\rho - 1}}\phi (\eta ) \bigl\{ {\mathop{\lim } _{j \to 0} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{{\lambda _{j}} - 1}}} \bigr\} } \,d\eta \\ &= \frac{{{\rho ^{(1 - \lambda )}}}}{{(\lambda - 1)!}} \int _{a}^{\mu }{{ \eta ^{\rho - 1}}\phi (\eta ){{ \bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \lambda - 1}}} \,d \eta \\ &= {\bigl({}^{\rho }}I_{a + }^{\lambda }\phi \bigr) (\mu ), \end{aligned}$$

and this ends the proof. □

Theorem 3.9

Let \(\alpha >0 \) and \(\{\phi _{j}\}_{j = 1}^{\infty }\) be a uniformly convergent sequence of continuous functions on \([a,b]\). Then we can interchange the generalized Riesz fractional integral operator and the limit, i.e.,

$$\begin{aligned} \bigl({}_{0}^{\rho }I_{T} ^{\alpha }\mathop{\lim } _{j \to 0} {\phi _{j}}\bigr) ( \mu ) = \bigl(\mathop{\lim } _{j \to \infty } {}_{0}^{\rho }I_{T} ^{\alpha }{ \phi _{j}}\bigr) (\mu ). \end{aligned}$$

Proof

The result follows taking into account Definition 3.2, Theorem 3.6, and the fact that sum of two convergent sequence is convergent. □

Lemma 3.10

([38])

Let \(\alpha >0 \), \(g(\mu ) \) and \(u_{1}(\mu ) \) be locally integrable, nonnegative, and nondecreasing functions with \(\mu \in [0,T] \). Also, assume that \(v_{1}(\mu ) \) is a nondecreasing continuous function such that \(0\le v_{1}(\mu )< L \), where L is a constant. Furthermore, if

$$\begin{aligned} g(\mu ) \le {u_{1}}(\mu ) + {\rho ^{1 - \alpha }} {v_{1}}(\mu ) \int _{0}^{\mu }{{\eta ^{\rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}}g( \eta ) \,d \eta\quad (0 \le \mu \le T), \end{aligned}$$

then the following inequality is true:

$$\begin{aligned} g(\mu ) \le {u_{1}}(\mu ) + \int _{0}^{\mu } \Biggl[ {\sum _{j = 1}^{\infty }{ \frac{{{\rho ^{1 - n\alpha }}{{({v_{1}}(T )\Gamma (\alpha ))}^{n}}}}{{\Gamma (n\alpha )}}} {\eta ^{\rho - 1}} {u_{1}}(\eta ){{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{n \alpha - 1}}} \Biggr] \,d\eta. \end{aligned}$$

Corollary 3.11

([38])

Let \(\alpha >0 \) and assume that \(g(\mu ) \), \(u_{1}(\mu ) \), and \(v_{1}(\mu ) \) are defined in the same way as in Lemma 3.10. Furthermore, if g satisfies

$$\begin{aligned} g(\mu ) \le {u_{1}}(\mu ) + {\rho ^{1 - \alpha }} {v_{1}}(\mu ) \int _{0}^{\mu }{{\eta ^{\rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}}g( \eta ) \,d \eta , \quad 0 \le \eta \le \mu , \end{aligned}$$

on \(\mu \in [0,T]\), then

$$\begin{aligned} g(\mu ) \le {{u_{1}}(\mu )} {E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{1}}( \mu )\Gamma (\alpha ){\mu ^{\alpha \rho }}} \bigr), \end{aligned}$$

where \({E_{\alpha ,1}} ( \cdot ) \) is a Mittag-Leffler function [12].

Likewise, the Gronwall inequality for generalized right-sided generalized fractional operator is expressed as follows.

Lemma 3.12

([38])

Let \(\alpha >0 \), \(\mu \in [0,T] \) and assume that \(g(\mu ) \), \(u_{2}(\mu ) \), and \(v_{2}(\mu ) \) are defined in the same way as in Lemma 3.10. Furthermore, if

$$\begin{aligned} g(\mu ) \le u_{2}(\mu ) + {\rho ^{1 - \alpha }}v_{2}(\mu ) \int _{\mu }^{T} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}}g( \eta ) \,d\eta , \quad 0 \le \mu \le \eta , \end{aligned}$$

then the following inequality holds true:

$$\begin{aligned} g(\mu ) \le u_{2}(\mu ) + \int _{\mu }^{T} { \Biggl[ {\sum _{j = 1}^{\infty }{ \frac{{{\rho ^{1 - n\alpha }}{{(v_{2}(T )\Gamma (\alpha ))}^{n}}}}{{\Gamma (n\alpha )}}} {\eta ^{\rho - 1}}u_{2}(\eta ){{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{n \alpha - 1}}} \Biggr]} \,d\eta. \end{aligned}$$

Lemma 3.13

Let \(\alpha >0 \) and assume that \(g(\mu ) \), \(u_{1}(\mu ) \), and \(v_{1}(\mu ) \) are defined in the same way as in Lemma 3.10. Furthermore, if

$$\begin{aligned} g(\mu ) \le u_{2}(\mu ) + {\rho ^{1 - \alpha }}v_{2}(\mu ) \int _{\mu }^{T} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}}g( \eta ) \,d\eta ,\quad 0 \le \mu \le \eta , \end{aligned}$$

on \(\mu \in [0,T]\), then

$$\begin{aligned} g(\mu ) \le {{u_{2}}(\mu )} {E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{2}}( \mu )\Gamma (\alpha ){{\bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha }}} \bigr). \end{aligned}$$

Proof

From Lemma 3.12,

$$\begin{aligned} g(\mu ) \le u_{2}(\mu ) + \int _{\mu }^{T} { \Biggl[ {\sum _{j = 1}^{\infty }{ \frac{{{\rho ^{1 - n\alpha }}{{(v_{2}(T )\Gamma (\alpha ))}^{n}}}}{{\Gamma (n\alpha )}}} {\eta ^{\rho - 1}}u_{2}(\eta ){{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{n \alpha - 1}}} \Biggr]} \,d\eta. \end{aligned}$$

Since \(u_{2} \) is a nondecreasing function, therefore \({u_{2}}(\mu ) \leq {u_{2}}(\eta ) \) for all \(\eta \in [ {0,T } ] \), and hence

$$\begin{aligned} g(\mu ) &\leq {u_{2}}(\mu ) \Biggl\{ {1 + \int _{\mu }^{T} {\sum_{j = 1}^{\infty }{ \frac{{{\rho ^{1 - n\alpha }}{{({v_{2}}(T )\Gamma (\alpha ))}^{n}}}}{{\Gamma (n\alpha )}}} {\eta ^{\rho - 1}} {u_{2}}(\eta ){{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{n \alpha - 1}}} \,d\eta } \Biggr\} \\ &= {u_{2}}(\mu ) \Biggl\{ {1 + \sum_{j = 1}^{\infty }{ \frac{{{\rho ^{ - n\alpha }}{{({v_{2}}(T )\Gamma (\alpha ))}^{n}}}}{{\Gamma (n\alpha + 1)}}} {{\bigl({T ^{\rho }} - {\mu ^{\rho }} \bigr)}^{n\alpha }}} \Biggr\} \\ &={{u_{2}}(\mu )} \Biggl\{ {\sum_{j = 0}^{\infty }{ \frac{{{{ ( {{\rho ^{ - \alpha }}{v_{2}}(T )\Gamma (\alpha ){{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}} )}^{n}}}}{{\Gamma (n\alpha + 1)}}} } \Biggr\} \\ &= {{u_{2}}(\mu )} {E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{2}}( \mu )\Gamma (\alpha ){{\bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha }}} \bigr), \end{aligned}$$

and the proof is ended. □

Lemma 3.14

Let \(\alpha >0 \), \(0<\mu <T \), and assume that \(g(\mu ) \), \(u_{1}(\mu ) \), \(u_{2}(\mu ) \), \(v_{1}(\mu ) \), and \(v_{2}(\mu ) \) are defined in the same way as in Lemma 3.10and Lemma 3.12. Furthermore, if \(g(\mu ) \) satisfies the inequality

$$\begin{aligned} g(\mu ) \le{}& {u_{1}}(\mu ) + {\rho ^{1 - \alpha }} {v_{1}}(\mu ) \int _{0}^{\mu }{{\eta ^{\rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}}g( \eta ) \,d \eta + {u_{2}}(\mu ) + {\rho ^{1 - \alpha }} {v_{2}}(\mu ) \\ &{}\times \int _{\mu }^{T} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}}g(\eta ) \,d\eta , \end{aligned}$$

then the following inequality holds true:

$$\begin{aligned} g(\mu ) \le \bigl({u_{1}}(\mu ) + {u_{2}}(\mu ) \bigr){E_{\alpha ,1}} \bigl( {{ \rho ^{ - \alpha }} {v_{2}}(\mu ) \Gamma (\alpha ){{\bigl({T ^{\rho }} - {\mu ^{\rho }} \bigr)}^{\alpha }}} \bigr){E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{1}}( \mu )\Gamma (\alpha ){\mu ^{\alpha \rho }}} \bigr), \end{aligned}$$

where \({E_{\alpha ,1}} ( \cdot ) \) is a Mittag-Leffler function.

Proof

Conflating Lemma 3.10 and Lemma 3.13 gives

$$\begin{aligned} g(\mu ) \leq{}& \biggl( {{u_{1}}(\mu ) + {u_{2}}(\mu ) + { \rho ^{1 - \alpha }} {v_{2}}(\mu ) \int _{\mu }^{T} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} \,d\eta } \biggr) \\ &{}\times {E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{1}}( \mu ) \Gamma (\alpha ){\mu ^{\alpha \rho }}} \bigr) \\ \leq{}& \bigl({u_{1}}(\mu ) + {u_{2}}(\mu ) \bigr){E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{2}}(\mu ) \Gamma (\alpha ){{\bigl({T ^{\rho }} - {\mu ^{\rho }} \bigr)}^{\alpha }}} \bigr)\\ &{}\times{E_{\alpha ,1}} \bigl( {{\rho ^{ - \alpha }} {v_{1}}( \mu )\Gamma (\alpha ){\mu ^{\alpha \rho }}} \bigr). \end{aligned}$$

 □

Existence and stability

For the upcoming existence results and discussion for boundary value (1), we use the following conditions. Let \(J = [0,T] \) and \(C(J) \) be the space of all continuous functions defined on J. We define the space

$$\begin{aligned} X = \bigl\{ {\phi (\mu ) | \phi (\mu ) \in C(J) \text{ and } {}_{*}^{\rho }{D^{{\alpha ^{*}}}} \phi (\mu ) \in C(J)} \bigr\} \end{aligned}$$

characterized by the norm \({ \Vert {\phi (\mu )} \Vert _{X}} = \mathop{\max }_{ \mu \in J} \vert {\phi (\mu )} \vert + \mathop{\max }_{ \mu \in J} \vert {{}_{*}^{\rho }{D^{{\alpha ^{*}}}}\phi (\mu )} \vert \).

Lemma 4.1

\(( {X,{{ \Vert \cdot \Vert }_{X}}} )\) is a Banach space.

Proof

Let \(\{ {{\phi _{j}}} \}_{j = 0}^{\infty }\) be a Cauchy sequence in \(( {X,{{ \Vert \cdot \Vert }_{X}}} ) \). Then clearly \(\{ {{}_{*}^{\rho }{D^{{\alpha ^{*}}}}{\phi _{j}}} \} _{j = 0}^{\infty }\) is also a Cauchy sequence in the space \({C(J)} \). Therefore both \(\{{{\phi _{j}(\mu )}} \}_{j = 0}^{\infty }\) and \(\{ {{}_{*}^{\rho }{D^{{\alpha ^{*}}}}{\phi _{j}}(\mu )} \} _{j = 0}^{\infty }\) converge uniformly, say \(u(\mu ) \) and \(v(\mu ) \), respectively, in the space \(C(J) \). We just have to show that \(v = {{}_{*}^{\rho }{D^{{\alpha ^{*}}}}}u \). For this, consider

$$\begin{aligned} & \bigl\vert {{}^{\rho }I_{{0^{+} }}^{{\beta ^{*}}}{}_{*}^{\rho }D_{{0^{+} }}^{{ \alpha ^{*}}}{ \phi _{j}}(\mu ) - {}^{\rho }I_{{0^{+} }}^{{\alpha ^{*}}}v( \mu )} \bigr\vert \\ &\quad= \biggl\vert { \frac{{{\rho ^{1 - {\alpha ^{*}}}}}}{{\Gamma ({\alpha ^{*}})}} \int _{0}^{\mu }{ \frac{{{}_{*}^{\rho }D_{{0^{+} }}^{{\alpha ^{*}}}{\phi _{j}}(\eta ){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - {\alpha ^{*}}}}}}\,d \eta - } \frac{{{\rho ^{1 - {\alpha ^{*}}}}}}{{\Gamma ({\alpha ^{*}})}} \int _{0}^{\mu }{ \frac{{v(\eta ){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - {\alpha ^{*}}}}}}} \,d\eta } \biggr\vert \\ &\quad\leq \frac{{{\rho ^{1 - {\alpha ^{*}}}}}}{{\Gamma ({\alpha ^{*}})}} \int _{0}^{\mu }{ \biggl\vert { \frac{{({}_{*}^{\rho }D_{{0^{+} }}^{{\alpha ^{*}}}{\phi _{j}}(\eta ) - v(\eta )){\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - {\alpha ^{*}}}}}}} \biggr\vert } \,d\eta \\ &\quad\leq \frac{{{\mu ^{\rho }}^{{\alpha ^{*}}}}}{{\rho \Gamma ({\alpha ^{*}} + 1)}} \mathop{\max } _{\mu \in J} \bigl\vert {{}_{*}^{\rho }D_{{0^{+} }}^{{ \alpha ^{*}}}{\phi _{j}}(\mu ) - v(\mu )} \bigr\vert . \end{aligned}$$

Since \(\{ {{}_{*}^{\rho }{D^{{\alpha ^{*}}}}{\phi _{j}}(\mu )} \} _{j = 0}^{\infty }\) converges uniformly to \(v(\mu ) \) for \(\mu \in J \), hence

$$\begin{aligned} \bigl\vert {{}^{\rho }I_{{0^{+} }}^{{\alpha ^{*}}}{}_{*}^{\rho }D_{{0^{+} }}^{{ \alpha ^{*}}}{ \phi _{j}}(\mu ) - {}^{\rho }I_{{0^{+} }}^{{\beta ^{*}}}v( \mu )} \bigr\vert \to 0 \end{aligned}$$

as \(j \to \infty \), i.e., \(\mathop{\lim }_{j \to \infty } {}^{\rho }I_{{0^{+} }}^{{ \alpha ^{*}}}{}_{*}^{\rho }D_{{0^{+} }}^{{\alpha ^{*}}}{\phi _{j}}( \mu ) \cong {}^{\rho }I_{{0^{+} }}^{{\alpha ^{*}}}v(\mu ) \). Now considering

$$\begin{aligned} _{*}^{\rho }D_{{0^{+} }}^{{\alpha ^{*}}}\bigl(\mathop{\lim } _{j \to \infty } {}^{\rho }I_{{0^{+} }}^{{\alpha ^{*}}}{}_{*}^{\rho }D_{{0^{+} }}^{{\beta ^{*}}}{ \phi _{j}}(\mu )\bigr) = {}_{*}^{\rho }D_{{0^{+} }}^{{ \alpha ^{*}}}{}^{\rho }I_{{0^{+} }}^{{\alpha ^{*}}}v( \mu ) \end{aligned}$$

and taking into account Theorem 3.6 and Theorem 2.3, we get \(v(\mu ) = {}_{*}^{\rho }{D^{{\alpha ^{*}}}}u(\mu ) \). This completes the proof. □

Lemma 4.2

Let \(\alpha \in (1,2)\), \({\alpha ^{*}} \in (0,1)\), and \(g \in C(J)\). Then problem (1) is equivalent to the following integral equation:

$$\begin{aligned} \phi (\mu ) ={}& \frac{1}{2} ( {{\phi _{0}} + {\phi _{T} }} ) + \biggl( { \frac{{{\phi _{T} } - {\phi _{0}}}}{{2{T ^{\rho }}}}} \biggr) \bigl( {2{ \mu ^{\rho }} - {T ^{\rho }}} \bigr) \\ &{}+ \frac{{{\rho ^{1 - \alpha }} ( {{\mu ^{\rho }} - {T ^{\rho }}} )}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}g\bigl(\eta , \phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi ( \eta )\bigr)} \,d\eta \\ &{}- \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl( {{T ^{\rho }} - {\eta ^{\rho }}}\bigr)}^{ \alpha - 1}}g\bigl(\eta , \phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }\phi ( \eta )\bigr)} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{T} {{ \eta ^{\rho - 1}} {{ \bigl\vert {{\eta ^{\rho }} - {\mu ^{\rho }}} \bigr\vert }^{ \alpha - 1}}g\bigl(\eta ,\phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }\phi (\eta )\bigr)} \,d\eta \\ ={}& \frac{1}{2}({\phi _{0}} + {\phi _{T}}) + \psi ( \mu ), \end{aligned}$$
(3)

where, for \(\mu > \eta \),

$$\begin{aligned} \psi (\mu ) ={}& \frac{{({\phi _{T}} - {\phi _{0}}){\mu ^{\rho }}}}{{2{T ^{\rho }}}} - \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta , \end{aligned}$$

and for \(\eta > \mu \),

$$\begin{aligned} \psi (\mu ) ={}& \frac{{({\phi _{T}} - {\phi _{0}})({\mu ^{\rho }} - {T ^{\rho }})}}{{2{T ^{\rho }}}} + \frac{{({\mu ^{\rho }} - {T ^{\rho }}){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta. \end{aligned}$$

Proof

Let \(\phi (\mu )\in X \) be a solution of boundary value problem (1). Then, by applying the generalized Riesz-type integral operator on both sides of equation (1) and using Definition 3.2, Lemma 2.5, and Lemma 3.4, we obtain

$$\begin{aligned} &\frac{1}{2}\phi (\mu )- \frac{1}{2}\phi (0) - {c_{0}} \frac{{{\mu ^{\rho }}}}{{2\rho }} + \frac{1}{2}\phi (\mu ) - \frac{1}{2}\phi (T ) - \frac{1}{2}{c_{1}}\biggl( \frac{{{\mu ^{\rho }} - {T ^{\rho }}}}{\rho }\biggr) \\ &\quad= \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{T} {{ \eta ^{\rho - 1}}} { \bigl\vert {\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)} \bigr\vert ^{ \alpha - 1}}g\bigl(\eta ,\phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}} \phi (\eta )\bigr)\,d\eta \end{aligned}$$

or

$$\begin{aligned} \phi (\mu ) ={}& \frac{1}{2}({\phi _{0}} + {\phi _{T} }) + \frac{{{c_{0}}{\mu ^{\rho }}}}{{2\rho }} \\ &{}+ \frac{1}{2}{c_{1}}\biggl(\frac{{{\mu ^{\rho }} - {T ^{\rho }}}}{\rho }\biggr) + \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{{ \eta ^{\rho - 1}}} {\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)^{\alpha - 1}}g\bigl( \eta ,\phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}}\phi (\eta )\bigr)\,d \eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} {{ \eta ^{\rho - 1}}} {\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)^{\alpha - 1}}g\bigl( \eta ,\phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}}\phi (\eta )\bigr)\,d \eta. \end{aligned}$$

Using the boundary conditions \(\phi (0) = {\phi _{0}}\) and \(\phi (T ) = {\phi _{T} }\) into the above equation, we get

$$\begin{aligned} {c_{0}} = \frac{{\rho ({\phi _{T}} - {\phi _{0}})}}{{{T ^{\rho }}}} - \frac{{2{\rho ^{2 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \end{aligned}$$

and

$$\begin{aligned} {c_{1}} = \frac{{\rho ({\phi _{T}} - {\phi _{0}})}}{{{T ^{\rho }}}} + \frac{2{{\rho ^{2 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta. \end{aligned}$$

Now again substituting these values of constants into the above equation, we get

$$\begin{aligned} \phi (\mu ) ={}& \frac{1}{2}({\phi _{0}} + {\phi _{T}}) \frac{{({\phi _{T}} - {\phi _{0}}){\mu ^{\rho }}}}{{2{T ^{\rho }}}} \\ &{}- \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta + \frac{{({\phi _{T}} - {\phi _{0}})({\mu ^{\rho }} - {T ^{\rho }})}}{{2{T ^{\rho }}}} \\ &{}+ \frac{{({\mu ^{\rho }} - {T ^{\rho }}){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ ={}& \frac{1}{2} ( {{\phi _{0}} + {\phi _{T} }} ) + \biggl( { \frac{{{\phi _{T} } - {\phi _{0}}}}{{2{T ^{\rho }}}}} \biggr) \bigl( {2{ \mu ^{\rho }} - {T ^{\rho }}} \bigr) \\ &{}+ \frac{{{\rho ^{1 - \alpha }} ( {{\mu ^{\rho }} - {T ^{\rho }}} )}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}g\bigl(\eta , \phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi ( \eta )\bigr)} \,d\eta \\ &{}- \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl( {{T ^{\rho }} - {\eta ^{\rho }}}\bigr)}^{ \alpha - 1}}g\bigl(\eta , \phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }\phi ( \eta )\bigr)} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{T} {{ \eta ^{\rho - 1}} {{ \bigl\vert {{\eta ^{\rho }} - {\mu ^{\rho }}} \bigr\vert }^{ \alpha - 1}}g\bigl(\eta ,\phi (\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }\phi (\eta )\bigr)} \,d\eta \\ ={}&\frac{1}{2}({\phi _{0}} + {\phi _{T}}) + \psi (\mu ), \end{aligned}$$

where, for \(\mu > \eta \),

$$\begin{aligned} \psi (\mu ) ={}& \frac{{({\phi _{T}} - {\phi _{0}}){\mu ^{\rho }}}}{{2{T ^{\rho }}}} - \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta , \end{aligned}$$

and for \(\eta > \mu \),

$$\begin{aligned} \psi (\mu ) ={}& \frac{{({\phi _{T}} - {\phi _{0}})({\mu ^{\rho }} - {T ^{\rho }})}}{{2{T ^{\rho }}}} + \frac{{({\mu ^{\rho }} - {T ^{\rho }}){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta. \end{aligned}$$

Conversely, let \(\phi (\mu )\in X \) be a solution of the fractional integral operator (3), and we denote the right-hand side of equation (3) by \(\Phi (\mu )\), i.e.,

$$\begin{aligned} \Phi (\mu )= \frac{1}{2}({\phi _{0}} + {\phi _{T}}) + \psi (\mu ). \end{aligned}$$

Now taking the left and the right generalized Caputo derivative on both sides of the above equation, we get

$$\begin{aligned} {}_{*}^{\rho }D_{0,\mu }^{\alpha } \Phi (\mu ) ={}& {}_{*}^{\rho }D_{0,\mu }^{\alpha } \biggl(\frac{1}{2}({\phi _{0}} + {\phi _{T}})\biggr) + \frac{{({\phi _{T} } - {\phi _{0}})}}{{2{T ^{\rho }}}}{}_{*}^{\rho }D_{0, \mu }^{\alpha } \bigl({\mu ^{\rho }}\bigr) \\ &{}- \frac{{{}_{0}^{\rho }I_{T} ^{\alpha }g(T ,\phi (T ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (T ))}}{{{T ^{\rho }}}}{}_{*}^{\rho }D_{0,\mu }^{\alpha } \bigl({\mu ^{\rho }}\bigr) \\ &{}+ {}_{*}^{\rho }D_{0,\mu }^{\alpha }{}_{0}^{\rho }I_{\mu }^{\alpha }g \bigl(\mu , \phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}}\phi (\mu )\bigr) \\ ={}& g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}}\phi (\mu )\bigr) \end{aligned}$$
(4)

and

$$\begin{aligned} {}_{*}^{\rho }D_{\mu ,T }^{\alpha } \Phi (\mu ) ={}& {}_{*}^{\rho }D_{\mu ,T }^{\alpha } \biggl(\frac{1}{2}({\phi _{0}} + {\phi _{T}})\biggr) + \frac{{({\phi _{T} } - {\phi _{0}})}}{{2{T ^{\rho }}}}{}_{*}^{\rho }D_{ \mu ,T }^{\alpha } \bigl({\mu ^{\rho }} - {T ^{\rho }}\bigr) \\ &{}+ \frac{{{}_{0}^{\rho }I_{T} ^{\alpha }g(T ,\phi (T ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (T ))}}{{{T ^{\rho }}}}{}_{*}^{\rho }D_{\mu ,T }^{\alpha } \bigl({\mu ^{\rho }} - {T ^{\rho }}\bigr) \\ &{}+ {}_{*}^{\rho }D_{\mu ,T }^{\alpha }{}_{\mu }^{\rho }I_{T} ^{\alpha }g\bigl(\mu , \phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\mu )\bigr) \\ ={}& g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi ( \mu )\bigr). \end{aligned}$$
(5)

Here, we have used Theorem 2.3, and some simple calculation leads to the facts that \({}_{*}^{\rho }D_{0,\mu }^{\alpha }({\mu ^{\rho }}) = 0\) and \({}_{*}^{\rho }D_{\mu ,T }^{\alpha }({\mu ^{\rho }} - {T ^{\rho }}) = 0\). Consequently, from equations (4), (5) and Definition 3.3, the required result follows, i.e.,

$$\begin{aligned} \frac{1}{2} \bigl( {{}_{*}^{\rho }D_{0,\mu }^{\alpha } \Phi (\mu ) + {}_{*}^{\rho }D_{\mu ,T }^{\alpha }\Phi ( \mu )} \bigr) = {}_{0}^{RC}D_{T} ^{ \alpha ,\rho }\Phi (\mu ) = g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }\phi (\mu )\bigr), \end{aligned}$$

and the proof is completed. □

Now we present the existence and uniqueness results for the nonlinear boundary value problem (1). We define an operator \(\tilde{T}:X \to X\) by

$$\begin{aligned} \tilde{T}\bigl(\phi (\mu )\bigr) ={}& \frac{1}{2} ( {{\phi _{0}} + {\phi _{T} }} ) + \biggl( { \frac{{{\phi _{T} } - {\phi _{0}}}}{{2{T ^{\rho }}}}} \biggr) \bigl( {2{ \mu ^{\rho }} - {T ^{\rho }}} \bigr) \\ &{}+ \frac{{{\rho ^{1 - \alpha }} ( {{\mu ^{\rho }} - {T ^{\rho }}} )}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}g\bigl(\eta , \phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi ( \eta )\bigr)} \,d\eta \\ &{}- \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}g\bigl(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\eta ) \bigr)} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{T} {{ \eta ^{\rho - 1}} {{ \bigl\vert {{\eta ^{\rho }} - {\mu ^{\rho }}} \bigr\vert }^{ \alpha - 1}}g\bigl(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\eta )\bigr)} \,d\eta. \end{aligned}$$
(6)

Lemma 4.2 signifies that solutions of problem (1) coincide with the fixed points of the operator \(T(\phi (\mu )) \). Ahead of the detailed existence results, let us have the following considerations first:

\((H{_{1}^{*})}\):

Let \(1<\alpha <2\), \(0<\alpha ^{*}<1 \), and \({g}: [ {0,T} ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be a continuous function and \(U(\mu ) \in {L^{1}}[J,{\mathbb{R}_{+} }]\) be a nonnegative function such that \(U(\mu )\leq \phi (\mu ) \). Furthermore, g satisfies

$$\begin{aligned} \bigl\vert {g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\mu )\bigr)} \bigr\vert \leq {\rho ^{\alpha }} \bigl( {{a_{1}} \bigl\vert { \phi (\mu )} \bigr\vert + {a_{2}} \bigl\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }\phi ( \mu )} \bigr\vert } \bigr) + \frac{{b{\rho ^{\alpha }}}}{{{T ^{\rho }}}}U(\mu ), \end{aligned}$$

where \(a_{1}, a_{2}, b \in \mathbb{R}_{+}\).

\((H{_{2}^{*})}\):

Let \(1<\alpha <2\), \(0<\alpha ^{*}<1 \), and \({g}: [ {0,T} ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be a continuous function, and g satisfies the Lipschitz condition, i.e.,

$$\begin{aligned} &\bigl\vert g\bigl(\eta ,\phi _{1}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr) \bigr\vert \\ &\quad\leq {\lambda _{1}} \bigl( { \bigl\vert {{\phi _{1}}( \mu ) - {\phi _{2}}( \mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{ \phi _{1}}(\eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}( \eta ))} \bigr\vert } ), \end{aligned}$$

where \(0 < {\lambda _{1}} < \frac{1}{2}\max \{ {K_{1}},{K_{2}}\} \).

Let \({M_{1}} = \mathop{\max }_{\mu \in J} \{ {{h_{1}}( \mu ): \vert {{h_{1}}(\mu )} \vert \le {d_{2}}} \} \) and \({M^{*}} = \mathop{\max }_{\mu \in J} \{ {f(\mu ): \vert {f(\mu )} \vert \le {d_{1}}} \} \), where

$$\begin{aligned} {h_{1}}(\mu ) ={}& \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert {\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}K{\mu ^{\rho (\alpha - {\alpha ^{*}} - 1)}}}}{{{T ^{\rho }}{\rho ^{1 - {\alpha ^{*}}}}\Gamma (2 - {\alpha ^{*}})}} \\ &{}+ \frac{{2{a_{3}}{\rho ^{\alpha - 1}}K{\mu ^{\rho (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}} + 1)}} + \frac{{b{\rho ^{1 + {\alpha ^{*}}}}{K^{*}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} \end{aligned}$$

and

$$\begin{aligned} f(\mu ) = {T ^{\rho (\alpha - 1)}} + {\mu ^{\alpha \rho }} + {\bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr)^{\alpha }}. \end{aligned}$$

Furthermore, let

$$\begin{aligned} &{K^{*}}: = \frac{\rho }{{\Gamma (\alpha )}}\max \biggl( \int _{0}^{T} {{ \eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}U(\eta )} \,d\eta , \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}} U(\eta )\,d \eta \biggr), \\ &{L_{1}} := \sup \Bigl( \max_{\mu \in J} \bigl( {{T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{\bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha }}} \bigr) \Bigr) \end{aligned}$$

and

$$\begin{aligned} L_{2}:= {}&\sup \biggl[\mathop{\max } _{\mu \in J} \biggl\{ \frac{{{T ^{\rho (\alpha - 1)}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\mu ^{(\alpha - {\alpha ^{*}})}}}}{{\Gamma (2 - {\alpha ^{*}} + 1)}} \\ &{}+ \frac{{{T ^{\rho (\alpha - 1)}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}}{{\Gamma (\alpha - {\alpha ^{*}} + 1)}} \biggr\} \biggr]. \end{aligned}$$

By means of local integrability of \(U(\mu ) \), \({K^{*}} \) exists certainly. Define a set

$$\begin{aligned} {A_{r}} = \bigl\{ {\phi \in C(J): \Vert \phi \Vert < r} \bigr\} , \end{aligned}$$

where \(r = \{ {4\max ( { \vert {{\phi _{T}}} \vert , \vert {{ \phi _{0}}} \vert ,\frac{{2b{K^{*}}}}{{{T ^{\rho }}}}, \frac{{2K{a_{3}}{M^{*}}}}{{\Gamma (\alpha + 1)}}} )} \} E_{ \alpha ,1}^{2}(b) \). Then manifestly the set \({A_{r}}\) is a closed, bounded, and convex subset of the above defined Banach space \(( {X,{{ \Vert \cdot \Vert }_{X}}} )\).

Theorem 4.3

Assume that condition \((H{_{1}^{*})}\) holds. Then problem (1) has a solution in \({A_{r}} \).

Proof

We prove this result using the Schauder fixed point theorem. First we show that the operator \(\tilde{T}:A_{r} \to A_{r}\) is a self-map. Suppose \(\phi \in {A_{r}} \), and for \(L\in (0,1) \), the operator (6) satisfies \(\phi (\mu )=L\tilde{T}(\phi (\mu ))\). Then from (6) and using condition \((H{_{1}^{*})}\), we have that

$$\begin{aligned} \bigl\vert {\phi (\mu )} \bigr\vert \leq{}& \bigl\vert {\tilde{T}\phi (\mu )} \bigr\vert \\ \leq{}& \frac{1}{2} \bigl\vert {({\phi _{0}} + { \phi _{T} })} \bigr\vert + \frac{{ \vert {{\phi _{T}} - {\phi _{0}}} \vert {\mu ^{\rho }}}}{{2{T ^{\rho }}}} + \frac{{ \vert {{\phi _{0}} - {\phi _{T} }} \vert ({T ^{\rho }} - {\mu ^{\rho }})}}{{2{T ^{\rho }}}} \\ &{}+ \frac{{{a_{1}}\rho {\mu ^{\rho }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}} \bigl\vert { \phi (\eta )} \bigr\vert } \,d\eta \\ &{} + \frac{{{a_{2}}\rho {\mu ^{\rho }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}} \bigl\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta )} \bigr\vert } \,d\eta \\ &{}+ \frac{{b\rho {\mu ^{\rho }}}}{{{T ^{2\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}U( \eta )} \,d \eta \\ &{} + \frac{{{a_{1}}\rho }}{{\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}} \bigl\vert { \phi (\eta )} \bigr\vert } \,d\eta \\ &{}+ \frac{{{a_{2}}\rho }}{{\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}} \bigl\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta )} \bigr\vert } \,d\eta \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} U(\eta )\,d \eta \\ &{}+ \frac{{{a_{1}}\rho ({T ^{\rho }} - {\mu ^{\rho }})}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}} \bigl\vert { \phi (\eta )} \bigr\vert } \,d\eta \\ &{}+ \frac{{{a_{2}}\rho ({T ^{\rho }} - {\mu ^{\rho }})}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}} \bigl\vert {_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }\phi (\eta )} \bigr\vert } \,d\eta \\ &{}+ \frac{{b\rho ({T ^{\rho }} - {\mu ^{\rho }})}}{{\Gamma (\alpha ){T ^{2\rho }}}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}} U(\eta )\,d \eta + \frac{{{a_{1}}\rho }}{{\Gamma (\alpha )}} \int _{\mu }^{T} {{\eta ^{ \rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} \bigl\vert { \phi (\eta )} \bigr\vert \,d\eta \\ &{}+ \frac{{{a_{2}}\rho }}{{\Gamma (\alpha )}} \int _{\mu }^{T} {{\eta ^{ \rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} \bigl\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta )} \bigr\vert \,d\eta \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{\mu }^{T} {{ \eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} U( \eta )\,d\eta \\ \leq{}& \vert {{\phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{2{a_{3}}K\rho {\mu ^{\rho }}}}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \,d\eta \\ &{}+ \frac{{b\rho {\mu ^{\rho }}}}{{{T ^{2\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}U( \eta )} \,d \eta \\ &{}+ \frac{{2{a_{3}}K\rho }}{{\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} \,d\eta + \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} U(\eta )\,d \eta \\ &{}+ \frac{{2{a_{3}}K\rho ({T ^{\rho }} - {\mu ^{\rho }})}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}} \,d\eta + \frac{{b\rho ({T ^{\rho }} - {\mu ^{\rho }})}}{{\Gamma (\alpha ){T ^{2\rho }}}} \int _{0}^{T} {{\eta ^{\alpha \rho - 1}}} U(\eta )\,d \eta \\ &{}+ \frac{{2{a_{3}}K\rho }}{{\Gamma (\alpha )}} \int _{\mu }^{T} {{\eta ^{ \rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} \,d\eta + \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{\mu }^{T} {{\eta ^{ \rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} U(\eta )\,d \eta , \end{aligned}$$

where \(K = \mathop{\max }_{\mu \in [J} ( { \vert {\phi ( \mu )} \vert , \vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi ( \mu )} \vert } )\), and so

$$\begin{aligned} \bigl\vert {\tilde{T}\phi (\mu )} \bigr\vert \leq{}& \vert {{\phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{2{a_{3}}K{\mu ^{\rho }}}}{{{T ^{\rho (2 - \alpha )}}\Gamma (\alpha + 1)}} + \frac{{b{K^{*}}}}{{{T ^{\rho }}}} \\ &{}+ \frac{{2{a_{3}}K{\mu ^{\alpha \rho }}}}{{\Gamma (\alpha + 1)}} + \frac{{2{a_{3}}K({T ^{\rho }} - {\mu ^{\rho }})}}{{{T ^{\rho (2 - \alpha )}}\Gamma (\alpha + 1)}} + \frac{{2{a_{3}}K{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}}}{{\Gamma (\alpha + 1)}} \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} U(\eta )\,d \eta \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{\mu }^{T} {{ \eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} U( \eta )\,d\eta \\ ={}& \vert {{\phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{b{K^{*}}}}{{{T ^{\rho }}}} + \frac{{2K{a_{3}}}}{{\Gamma (\alpha + 1)}} \bigl\{ {{T ^{\rho (\alpha - 1)}} + {\mu ^{\alpha \rho }} + {{\bigl({T ^{\rho }} - {\mu ^{\rho }} \bigr)}^{\alpha }}} \bigr\} \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{\mu }{{\eta ^{ \rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{\alpha - 1}}} U(\eta )\,d \eta \\ &{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{\mu }^{T} {{ \eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha - 1}}} U( \eta )\,d \eta. \end{aligned}$$

Since the functions \({\mu ^{\alpha \rho }} \) and \({({T ^{\rho }} - {\mu ^{\rho }})^{\alpha }} \) are integrable, uniformly continuous, and nonnegative for \(\mu \in [0,T] \) and also \(U(\mu )\leq \phi (\mu ) \), hence applying Lemma 3.14 gives

$$\begin{aligned} \bigl\vert {\tilde{T}\phi (\mu )} \bigr\vert \leq {}&\biggl\{ { \vert {{ \phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{2b{K^{*}}}}{{{T ^{\rho }}}} + \frac{{2K{a_{3}}}}{{\Gamma (\alpha + 1)}} \bigl\{ {{T ^{\rho (\alpha - 1)}}+\mu ^{\alpha \rho }+ {{\bigl({T ^{\rho }} - {\mu ^{\rho }} \bigr)}^{\alpha }}} \bigr\} } \biggr\} \\ &{}\times {E_{\alpha ,1}} \biggl( { \frac{{b{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}}}{{{T ^{\rho }}}}} \biggr){E_{\alpha ,1}} \biggl( { \frac{{b{\mu ^{\alpha \rho }}}}{{{T ^{\rho }}}}} \biggr) \\ \leq{}& \biggl\{ { \vert {{\phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{b{K^{*}}}}{{{T ^{\rho }}}} + \frac{{2K{a_{3}}{M^{*}}}}{{\Gamma (\alpha + 1)}}} \biggr\} {E_{ \alpha ,1}} \biggl( { \frac{{b{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}}}{{{T ^{\rho }}}}} \biggr){E_{\alpha ,1}} \biggl( { \frac{{b{\mu ^{\alpha \rho }}}}{{{T ^{\rho }}}}} \biggr). \end{aligned}$$

Thus

$$\begin{aligned} \bigl\vert {\tilde{T}\phi (\mu )} \bigr\vert \leq \biggl\{ { \vert {{\phi _{T}}} \vert + \vert {{\phi _{0}}} \vert + \frac{{2b{K^{*}}}}{{{T ^{\rho }}}} + \frac{{2K{a_{3}}{M^{*}}}}{{\Gamma (\alpha + 1)}}} \biggr\} E_{\alpha ,1}^{2}(b)< r. \end{aligned}$$

Also,

$$\begin{aligned} & \bigl\vert {{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \\ &\quad \leq {}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \biggl(\frac{1}{2}({ \phi _{0}} + {\phi _{T}})\biggr) \\ &\qquad{}+ \frac{{({\phi _{T} } - {\phi _{0}})}}{{2{T ^{\rho }}}} \bigl( {{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \bigl({\mu ^{\rho }}\bigr)} \bigr) + \biggl( { \frac{{2{a_{3}}{\rho ^{\alpha }}{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}}({\mu ^{\rho }})}}{{{T ^{\rho }}}}} \biggr){}_{0}^{\rho }I_{T} ^{\alpha }K \\ &\qquad{}+ \biggl( {\frac{{b{\rho ^{\alpha }}}}{{{T ^{2\rho }}}}{}_{*}^{\rho }D_{0, \mu }^{{\alpha ^{*}}} \bigl({\mu ^{\rho }}\bigr)} \biggr){}_{0}^{\rho }I_{T} ^{\alpha }U(\mu ) + 2{a_{3}} {\rho ^{\alpha }} {}_{*}^{\rho }D_{0,\mu }^{{ \alpha ^{*}}}{}_{0}^{\rho }I_{\mu }^{\alpha }K + \frac{{b{\rho ^{\alpha }}}}{{{T ^{\rho }}}}{}_{*}^{\rho }D_{0,\mu }^{\alpha }{}_{0}^{\rho }I_{\mu }^{\alpha }U( \eta ). \end{aligned}$$

Using Theorem 2.3 and Lemma 2.5, we get

$$\begin{aligned} & \bigl\vert {{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \\ &\quad\leq \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert {\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}K{\mu ^{\rho (\alpha - {\alpha ^{*}} - 1)}}}}{{{T ^{\rho }}{\rho ^{1 - {\alpha ^{*}}}}\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{b{\rho ^{\alpha + {\alpha ^{*}}}}{\mu ^{\rho (1 - {\alpha ^{*}})}}{}_{0}^{\rho }I_{T} ^{\alpha }U(\eta )}}{{\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} + 2{a_{3}} {\rho ^{\alpha }} {}_{0}^{\rho }I_{\mu }^{\alpha - {\alpha ^{*}}}K + \frac{{b{\rho ^{\alpha }}}}{{{T ^{\rho }}}}{}_{0}^{\rho }I_{\mu }^{\alpha - { \alpha ^{*}}}U( \eta ) \\ &\quad= \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert {\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}K{\mu ^{\rho (\alpha - {\alpha ^{*}} - 1)}}}}{{{T ^{\rho }}{\rho ^{1 - {\alpha ^{*}}}}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}{\rho ^{\alpha - 1}}K{\mu ^{\rho (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}} + 1)}} \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}{K^{*}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} + \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{{T ^{\rho }}\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{\mu }{{\eta ^{\rho - 1}} {{\bigl({\mu ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - {\alpha ^{*}} - 1}}} U(\eta )\,d \eta. \end{aligned}$$

Since \(U(\mu )\leq \phi (\mu ) \) and \({{\mu ^{\rho (1 - {\alpha ^{*}})}}} \), \({{\mu ^{\rho (\alpha - {\alpha ^{*}} - 1)}}} \) are measurable and continuous functions for \(\mu \in [0,T] \), therefore by using the assumption \(\phi (\mu )=L\tilde{T}(\phi (\mu ))\) and Corollary 3.11, we get

$$\begin{aligned} \bigl\vert {{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \le {M_{1}} {E_{(\alpha - {\alpha ^{*}}),1}} \biggl( { \frac{{{\rho ^{{\alpha ^{*}}}}b}}{{{T ^{\rho }}}}{\mu ^{(\alpha - { \alpha ^{*}})\rho }}} \biggr)< r_{1}. \end{aligned}$$
(7)

Moreover,

$$\begin{aligned} & \bigl\vert {{}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \\ &\quad\leq {}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \biggl(\frac{1}{2}({ \phi _{0}} + {\phi _{T}})\biggr) \\ &\qquad{}+ \frac{{ \vert {{\phi _{T} } - {\phi _{0}}} \vert }}{{2{T ^{\rho }}}}{}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr) + \frac{{2{a_{3}}K}}{{\Gamma (\alpha ){T ^{\rho (1 - \alpha )}}}}{}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr) \\ &\qquad{}+ \frac{{b{K^{*}}}}{{{T ^{2\rho }}}}{}_{*}^{\rho }D_{\mu ,T }^{{ \alpha ^{*}}} \bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr) + 2{a_{3}} { \rho ^{\alpha }} {}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}}{}_{\mu }^{\rho }I_{T} ^{\alpha }K + \frac{{b{\rho ^{\alpha }}}}{{{T ^{\rho }}}}{}_{*}^{\rho }D_{\mu ,T }^{{ \alpha ^{*}}}I_{T} ^{\alpha }U(\eta ). \end{aligned}$$

Using Theorem 2.3 and Lemma 2.5, we get

$$\begin{aligned} & \bigl\vert {{}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \\ &\quad\leq \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert {{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}K{\rho ^{{\alpha ^{*}}}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{b{K^{*}}{\rho ^{{\alpha ^{*}}}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} + 2{a_{3}} {\rho ^{\alpha }} {}_{\mu }^{\rho }I_{T} ^{\alpha - {\alpha ^{*}}}K + \frac{{b{\rho ^{\alpha }}}}{{{T ^{\rho }}}}{}_{\mu }^{\rho }I_{T} ^{\alpha - {\alpha ^{*}}}U( \eta ) \\ &\quad= \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert {{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} + \frac{{2{a_{3}}K{\rho ^{{\alpha ^{*}}}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{b{K^{*}}{\rho ^{{\alpha ^{*}}}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} + \frac{{2{a_{3}}K{\rho ^{{\alpha ^{*}}}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}}{{\Gamma (\alpha - {\alpha ^{*}} + 1)}} \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{{T ^{\rho }}\Gamma (\alpha - {\alpha ^{*}})}} \int _{\mu }^{T} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - {\mu ^{\rho }}\bigr)}^{ \alpha - {\alpha ^{*}} - 1}}} U(\eta )\,d\eta. \end{aligned}$$

Since \(U(\mu )\leq \phi (\mu ) \) and \({{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}\in {L^{1}}[J,{ \mathbb{R}_{+} }] \) for \(\mu \in [0,T] \), therefore taking into account the assumption that \(\phi (\mu )=L\tilde{T}(\phi (\mu ))\) with \(L\in (0,1) \) and Corollary 3.11 yields

$$\begin{aligned} \bigl\vert {{}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert \le {M_{1}} {E_{(\alpha - {\alpha ^{*}}),1}} \biggl( { \frac{{{\rho ^{{\alpha ^{*}}}}b}}{{{T ^{\rho }}}}{{\bigl({T ^{\rho }} - { \mu ^{\rho }}\bigr)}^{\alpha - {\alpha ^{*}}}}} \biggr)< r_{2}. \end{aligned}$$
(8)

From Definition 3.3, inequalities (7) and (8) yield

$$\begin{aligned} \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \bigl(\tilde{T}\phi (\mu )\bigr)} \bigr\vert = \frac{1}{2} \bigl\vert {{}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}}\bigl( \tilde{T} \phi (\mu )\bigr) - {}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl( \tilde{T}\phi (\mu )\bigr)} \bigr\vert < r, \end{aligned}$$

which implies \(\tilde{T}\phi \in A_{r}\); that is, the operator \(\tilde{T}:A_{r} \to A_{r}\) is a self-map. Next we show that operator (6) is continuous. For this, let \({\phi _{1}}(\mu ),{\phi _{2}}(\mu ) \in {A_{r}} \). Then we have

$$\begin{aligned} & \bigl\vert \tilde{T}\phi _{1}(\mu ) - \tilde{T}\phi _{2}(\mu ) \bigr\vert \\ &\quad\leq \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}+ \frac{{({T ^{\rho }} - {\mu ^{\rho }}){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{ \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\quad= \frac{{ \{ {{T ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}} \} }}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \bigl\vert {g\bigl(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr)} \bigr\vert . \end{aligned}$$

Since g is continuous on \(A_{r} \), hence for all \(\mu \in [0,T] \) there exists \(\delta >0 \) such that \(\Vert {{\phi _{1}}(\eta ) - {\phi _{2}}(\eta )} \Vert < \delta \), and for any \(\epsilon >0 \),

$$\begin{aligned} \bigl\vert {g\bigl(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr)} \bigr\vert < \frac{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}}{{{T ^{\alpha \rho }}}} \varepsilon. \end{aligned}$$

Therefore,

$$\begin{aligned} & \bigl\vert {\tilde{T} {\phi _{1}}(\mu ) - \tilde{T} {\phi _{2}}(\mu )} \bigr\vert \\ &\quad \leq \frac{{ \{ {{T ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}} \} }}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \\ &\qquad{}\times \bigl\vert {g\bigl(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr)} \bigr\vert \\ &\quad\leq \frac{\varepsilon }{2} + \frac{{{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }}}}{{2{T ^{\alpha \rho }}}} \varepsilon < \varepsilon. \end{aligned}$$

Likewise one can prove \({{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }(\tilde{T}\phi (\mu ))} \) is continuous on \(A_{r}\). Moreover, we show that operator (6) is completely continuous. For this, let \(\eta _{1}, \eta _{2} \in J \) with \(\eta _{1}<\eta _{2} \) and \(\phi \in A_{r} \). Then we have

$$\begin{aligned} &\bigl\vert \tilde{T} \phi (\mu _{1}) - \tilde{T}\phi (\mu _{2}) \bigr\vert \\ &\quad\leq \frac{{ \vert {{\phi _{T} } - {\phi _{0}}} \vert (\mu _{1}^{\rho }- \mu _{2}^{\rho })}}{{{T ^{\rho }}}} \\ &\qquad{}+ \biggl\vert { \frac{{(\mu _{1}^{\rho }- \mu _{2}^{\rho }){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{{ \mu _{1}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{(\mu _{1}^{\rho }- {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta\\ &\qquad{} - \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{{ \mu _{2}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{(\mu _{2}^{\rho }- {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{(\mu _{1}^{\rho }- \mu _{2}^{\rho }){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{{ \mu _{1}}}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}- \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{{ \mu _{2}}}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - \mu _{2}^{\rho })}^{1 - \alpha }}}}} \,d\eta \biggr\vert \\ &\quad\leq \frac{{ \vert {{\phi _{T} } - {\phi _{0}}} \vert (\mu _{1}^{\rho }- \mu _{2}^{\rho })}}{{{T ^{\rho }}}} + \frac{{2{a_{3}}K{T ^{\alpha \rho }}(\mu _{1}^{\rho }- \mu _{2}^{\rho })}}{{\Gamma (\alpha + 1)}} \\ &\qquad{}+ \frac{{b\rho {K^{*}}(\mu _{1}^{\rho }- \mu _{2}^{\rho })}}{{{T ^{2\rho }}\Gamma (\alpha )}} + \frac{{2K{a_{3}}\rho }}{{\Gamma (\alpha )}} \int _{0}^{{\mu _{1}}} { \bigl\{ {{\eta ^{\rho - 1}} {{ \bigl(\mu _{1}^{\rho }- {\eta ^{\rho }} \bigr)}^{ \alpha - 1}} - {\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \bigr\} } \,d\eta \\ &\qquad{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{0}^{{\mu _{1}}} { \bigl\{ {{\eta ^{\rho - 1}} {{ \bigl(\mu _{1}^{\rho }- {\eta ^{\rho }} \bigr)}^{ \alpha - 1}} - {\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \bigr\} } U(\eta )\,d\eta \\ &\qquad{}+ \frac{{2K{a_{3}}\rho }}{{\Gamma (\alpha )}} \int _{{\mu _{1}}}^{{ \mu _{2}}} {{\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \,d \eta + \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{{\mu _{1}}}^{{\mu _{2}}} {{\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- { \eta ^{\rho }}\bigr)}^{\alpha - 1}}U( \eta )} \,d\eta \\ &\qquad{}+ \frac{{2K{a_{3}}(\mu _{1}^{\rho }- \mu _{2}^{\rho }){T ^{\rho (\alpha - 1)}}}}{{\Gamma (\alpha + 1)}} + \frac{{(\mu _{1}^{\rho }- \mu _{2}^{\rho }){K^{*}}}}{{{T ^{\rho }}}} \\ &\qquad{}+ \frac{{2K{a_{3}}\rho }}{{\Gamma (\alpha )}} \int _{{\mu _{2}}}^{T} { \bigl\{ {{\eta ^{\rho - 1}} {{ \bigl({\eta ^{\rho }} - \mu _{1}^{\rho } \bigr)}^{1 - \alpha }} - {\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - \mu _{2}^{\rho }\bigr)}^{1 - \alpha }}} \bigr\} } \,d\eta \\ &\qquad{}+ \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{{\mu _{2}}}^{T} { \bigl\{ {{\eta ^{\rho - 1}} {{ \bigl({\eta ^{\rho }} - \mu _{1}^{\rho } \bigr)}^{1 - \alpha }} - {\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - \mu _{2}^{\rho }\bigr)}^{1 - \alpha }}} \bigr\} } U(\eta )\,d \eta \\ &\qquad{}+ \frac{{2K{a_{3}}\rho }}{{\Gamma (\alpha )}} \int _{{\mu _{1}}}^{{ \mu _{2}}} { \frac{{{\eta ^{\rho - 1}}}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - \alpha }}}}} \,d\eta + \frac{{b\rho }}{{{T ^{\rho }}\Gamma (\alpha )}} \int _{{\mu _{1}}}^{{ \mu _{2}}} { \frac{{{\eta ^{\rho - 1}}U(\eta )}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - \alpha }}}}} \,d\eta. \end{aligned}$$

Since \(U(\mu ) \in {L^{1}}[J,{\mathbb{R}_{+} }]\), therefore the functions

$$\begin{aligned} \bigl( {{\eta ^{\rho - 1}} {{\bigl(\mu _{1}^{\rho }- {\eta ^{\rho }}\bigr)}^{ \alpha - 1}} - {\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \bigr)U(\eta ), \end{aligned}$$

\(( {{\eta ^{\rho - 1}}{{(\mu _{2}^{\rho }- {\eta ^{\rho }})}^{ \alpha - 1}}U(\eta )} ) \), and \(( {{\eta ^{\rho - 1}}{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - \alpha }} - {\eta ^{\rho - 1}}{{({\eta ^{\rho }} - \mu _{2}^{\rho })}^{1 - \alpha }}} )U(\eta ) \) are Lebesgue integrable in η. Also \({(\mu _{1}^{\rho }- \mu _{2}^{\rho }){T ^{\rho (\alpha - 1)}}}\) and \({T ^{\alpha \rho }}(\mu _{1}^{\rho }- \mu _{2}^{\rho }) \) are uniformly continuous for \(\mu _{1}, \mu _{2} \in J \). So we see through that the right-hand side of the above inequality tends to zero as \({\mu _{1}} \to {\mu _{2}} \). Furthermore, we prove that \(\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }(\tilde{T}\phi ({\mu _{1}})) - _{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }(\tilde{T}\phi ({\mu _{2}}))} \vert \to 0 \) as \({\mu _{1}} \to {\mu _{2}} \) for all \(\mu _{1},\mu _{2} \in [0,T] \) with \(\mu _{1}<\mu _{2} \). For this, let us compute first the left and the right generalized Caputo derivatives of operator (6).

$$\begin{aligned} {}_{*}^{\rho }D_{0,\mu }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr) = {}&\frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{T} } - {\phi _{0}})\mu _{1}^{\rho (1 - {\alpha ^{*}})}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} \\ &{}- \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}\mu _{1}^{\rho (1 - {\alpha ^{*}})}}}{{{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{{\mu _{1}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{(\mu _{1}^{\rho }- {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta \end{aligned}$$
(9)

and

$$\begin{aligned} {}_{*}^{\rho }D_{\mu ,T }^{{\alpha ^{*}}} \bigl(\tilde{T}\phi (\mu )\bigr) ={}& \frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{0}} - {\phi _{T} }){{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} \\ &{}- \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}}}}{{T^{\rho }\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta. \end{aligned}$$
(10)

From Definition 3.3 and equations (9) and (10), we have

$$\begin{aligned} & {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\bigl(\tilde{T} \phi ({\mu _{1}})\bigr) \\ &\quad= \frac{1}{2} \bigl( {{}_{*}^{\rho }D_{0,\mu _{1} }^{{\alpha ^{*}}}\bigl( \tilde{T} \phi (\mu _{1} )\bigr) - {}_{*}^{\rho }D_{\mu _{1} ,T }^{{\alpha ^{*}}} \bigl( \tilde{T}\phi (\mu _{1} )\bigr)} \bigr) \\ &\quad= \frac{1}{2} \biggl[ \frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{T} } - {\phi _{0}})\mu _{1}^{\rho (1 - {\alpha ^{*}})}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} - \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}\mu _{1}^{\rho (1 - {\alpha ^{*}})}}}{{{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{{\mu _{1}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{(\mu _{1}^{\rho }- {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta + \frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{T} } - {\phi _{0}}){{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}}}}{{{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &\qquad{}- \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta \biggr]. \end{aligned}$$

Therefore, by using the above equation, we establish that

$$\begin{aligned} & \bigl\vert {}_{0}^{RC} D_{T}^{\alpha ^{*},\rho } \bigl(\tilde{T}\phi ( \mu _{1})\bigr) - {}_{0}^{RC} D_{T}^{\alpha ^{*},\rho }\bigl(\tilde{T}\phi ( \mu _{2})\bigr) \bigr\vert \\ &\quad \leq \biggl\vert { \frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{T} } - {\phi _{0}})(\mu _{1}^{\rho (1 - {\alpha ^{*}})} - \mu _{1}^{\rho (1 - {\alpha ^{*}})})}}{{4{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}}} \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}(\mu _{2}^{\rho (1 - {\alpha ^{*}})} - \mu _{1}^{\rho (1 - {\alpha ^{*}})})}}{{2{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{{\mu _{1}}} { \biggl\{ { \frac{{{\eta ^{\rho - 1}}}}{{{{(\mu _{1}^{\rho }- {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}} - \frac{{{\eta ^{\rho - 1}}}}{{{{(\mu _{2}^{\rho }- {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \biggr\} } \\ &\qquad{}\times g\bigl(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\eta )\bigr)\,d\eta \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{{\mu _{2}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{(\mu _{2}^{\rho }- {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{{\rho ^{{\alpha ^{*}}}}({\phi _{T} } - {\phi _{0}}) \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{1 - {\alpha ^{*}}}}} \} }}{{4{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}}} \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}} \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{1 - {\alpha ^{*}}}}} \} }}{{2T^{\rho }\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} { \frac{{g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert { \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{{\mu _{2}}} { \frac{{{\eta ^{\rho - 1}}g(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\phi (\eta ))}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta } \biggr\vert \\ &\qquad{}+ \biggl\vert \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \biggl\{ { \int _{{\mu _{2}}}^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{({\eta ^{\rho }} - \mu _{2}^{\rho })}^{1 - (\alpha - {\alpha ^{*}})}}}}} - \int _{{\mu _{2}}}^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{({\eta ^{\rho }} - \mu _{1}^{\rho })}^{1 - (\alpha - {\alpha ^{*}})}}}}} } \biggr\} \\ &\qquad{}\times g \bigl(\eta ,\phi (\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\eta )\bigr)\,d\eta \biggr\vert . \end{aligned}$$

Using condition \((H{_{1}^{*})}\),

$$\begin{aligned} & \bigl\vert {}_{0}^{RC} D_{T}^{\alpha ^{*},\rho } \bigl(\tilde{T}\phi ( \mu _{1})\bigr) - {}_{0}^{RC} D_{T}^{\alpha ^{*},\rho }\bigl(\tilde{T}\phi ( \mu _{2})\bigr) \bigr\vert \\ &\quad \leq \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert (\mu _{1}^{\rho (1 - {\alpha ^{*}})} - \mu _{2}^{\rho (1 - {\alpha ^{*}})})}}{{4{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{{\rho ^{1 + {\alpha ^{*}}}}{a_{3}}K(\mu _{1}^{\rho (1 - {\alpha ^{*}})} - \mu _{2}^{\rho (1 - {\alpha ^{*}})})}}{{{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}} \,d\eta \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}(\mu _{1}^{\rho (1 - {\alpha ^{*}})} - \mu _{2}^{\rho (1 - {\alpha ^{*}})}}}{{2{T ^{2\rho }}\Gamma (\alpha )}} \int _{0}^{T} {{\eta ^{\rho - 1}} {{\bigl({T ^{\rho }} - {\eta ^{\rho }}\bigr)}^{ \alpha - 1}}U(\eta )} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 + {\alpha ^{*}}}}{a_{3}}K}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{{\mu _{1}}} { \bigl\{ {{{\bigl(\mu _{1}^{\rho }- {\eta ^{\rho }}\bigr)}^{( \alpha - {\alpha ^{*}}) - 1}} { \eta ^{\rho - 1}} - {\eta ^{\rho - 1}} {{\bigl( \mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{(\alpha - {\alpha ^{*}}) - 1}}} \bigr\} } \,d\eta \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (\alpha - {\alpha ^{*}})}} \int _{0}^{{\mu _{1}}} { \bigl\{ {{{\bigl(\mu _{1}^{\rho }- {\eta ^{\rho }}\bigr)}^{( \alpha - {\alpha ^{*}}) - 1}} { \eta ^{\rho - 1}} - {\eta ^{\rho - 1}} {{\bigl( \mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{(\alpha - {\alpha ^{*}}) - 1}}} \bigr\} } U(\eta )\,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 + {\alpha ^{*}}}}{a_{3}}K}}{{\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{{\mu _{2}}} {{\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- { \eta ^{\rho }}\bigr)}^{(\alpha - {\alpha ^{*}}) - 1}}} \,d\eta \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (\alpha - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{{\mu _{2}}} {\eta ^{\rho - 1}}\bigl(\mu _{2}^{\rho }- \eta ^{\rho }\bigr)^{(\alpha - \alpha ^{*})-1} U(\eta ) \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{{\alpha ^{*}}}} \vert {{\phi _{T} } - {\phi _{0}}} \vert \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{1 - {\alpha ^{*}}}}} \} }}{{4{T ^{\rho }}\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{{a_{3}}K{\rho ^{{\alpha ^{*}}}} \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{1 - {\alpha ^{*}}}}} \} }}{{\Gamma (\alpha )(\alpha - {\alpha ^{*}})\Gamma (2 - {\alpha ^{*}}){T ^{\rho (1 - \alpha + {\alpha ^{*}})}}}} \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}} \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{1 - {\alpha ^{*}}}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{1 - {\alpha ^{*}}}}} \} }}{{2\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}}){T ^{2\rho }}}} \int _{0}^{T} {{\eta ^{\rho (\alpha - {\alpha ^{*}}) - 1}}} U(\eta )\,d \eta \\ &\qquad{}+ \frac{{{a_{3}}K{\rho ^{{\alpha ^{*}}}} \{ {{{({T ^{\rho }} - \mu _{1}^{\rho })}^{(\alpha - {\alpha ^{*}})}} - {{({T ^{\rho }} - \mu _{2}^{\rho })}^{(\alpha - {\alpha ^{*}})}} } \} }}{{\Gamma (\alpha )(\alpha - {\alpha ^{*}})\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{{\mu _{2}}}^{T} \bigl\{ {\eta ^{\rho - 1}} {{ \bigl({\eta ^{\rho }} - \mu _{1}^{\rho } \bigr)}^{(\alpha - {\alpha ^{*}}) - 1}} \\ &\qquad{}- {\eta ^{\rho - 1}} {{\bigl({ \eta ^{\rho }} - \mu _{2}^{\rho }\bigr)}^{(\alpha - {\alpha ^{*}}) - 1}} \bigr\} U(\eta )\,d \eta \\ &\qquad{}+ \frac{{b{\rho ^{1 + {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \int _{{\mu _{1}}}^{{\mu _{2}}} {{\eta ^{\rho - 1}} {{\bigl({\eta ^{\rho }} - \mu _{1}^{\rho }\bigr)}^{(\alpha - {\alpha ^{*}}) - 1}}} U( \eta )\,d\eta. \end{aligned}$$

Since \(U(\mu ) \in {L^{1}}[J,{\mathbb{R}_{+} }] \) and the functions

$$\begin{aligned} {{\eta ^{\rho - 1}} {{\bigl(\mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{(\alpha - { \alpha ^{*}}) - 1}}}, \end{aligned}$$

\(({{\eta ^{\rho (\alpha - {\alpha ^{*}}) - 1}}}U({\eta }) \), and

$$\begin{aligned} \bigl({{{\bigl(\mu _{1}^{\rho }- {\eta ^{\rho }} \bigr)}^{(\alpha - {\alpha ^{*}}) - 1}} { \eta ^{\rho - 1}} - {\eta ^{\rho - 1}} {{\bigl( \mu _{2}^{\rho }- {\eta ^{\rho }}\bigr)}^{( \alpha - {\alpha ^{*}}) - 1}}}U( \eta )\bigr) \end{aligned}$$

are Lebesgue integrable on \([0,T] \), so the right-hand side of the above inequality tends to zero as \({\mu _{1}} \to {\mu _{2}} \). Hence the set of operators \(\tilde{T}A_{r} \) is equicontinuous. Also \(\tilde{T}{A_{r}} \subseteq {A_{r}} \) implies that \(\tilde{T}{A_{r}} \) is uniformly bounded. Henceforth, is completely continuous and thus Schauder’s fixed point theorem assures the existence of at least one fixed point of operator (6). Hence, by taking into account Lemma 4.2, the proof is finalized. □

Theorem 4.4

Assume that conditions \((H{_{1}^{*})}\) and \((H{_{2}^{*})}\) hold. Then equation (3) comports as a unique solution of Problem (1).

Proof

To prove this theorem, we use the Banach fixed point theorem. For this, we first necessitate to confirm that (6) is a self-mapped operator, and afterwards we show that satisfies the contraction mapping principle. Since we have shown in Theorem 4.3 that \(\tilde{T}\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }( \tilde{T}\phi (\mu )) \in {A_{r}} \), so the operator satisfies the self-mappedness property under these conditions. Hence, the only stipulation that we need to verify here is contraction. For this, consider

$$\begin{aligned} & \bigl\vert \tilde{T}\phi _{1}(\mu ) - \tilde{T}\phi _{2}(\mu ) \bigr\vert \\ &\quad\leq \bigl\vert g\bigl(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr) \bigr\vert \\ &\qquad{}\times \biggl( \frac{{{\mu ^{\rho }}{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta + \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}}}}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}+ \frac{{({T ^{\rho }} - {\mu ^{\rho }}){\rho ^{1 - \alpha }}}}{{\Gamma (\alpha ){T ^{\rho }}}} \int _{0}^{T} {\frac{1}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta + \frac{{{\rho ^{1 - \alpha }}}}{{\Gamma (\alpha )}} \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}}}}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \biggr) \\ &\quad\leq \frac{{{\lambda _{1}} ( { \vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \vert + \vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert } )}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \\ &\qquad{}\times \bigl({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{\bigl({T ^{\rho }} - {\mu ^{\rho }}\bigr)}^{\alpha }}\bigr) \\ &\quad\leq \frac{{{\lambda _{1}}}}{{{K_{1}}}} \bigl\Vert {{\phi _{1}}(\mu ) - { \phi _{2}}(\mu )} \bigr\Vert , \end{aligned}$$

where \({K_{1}} = \frac{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}}{{{L_{1}}}} \). Moreover,

$$\begin{aligned} & \bigl\vert {}_{0}^{RC} D_{T}^{\alpha ^{*},\rho } \bigl(\tilde{T}\phi _{1}( \mu )\bigr) - {}_{0}^{RC}D_{T}^{\alpha ^{*},\rho } \bigl(\tilde{T}\phi _{2}(\mu )\bigr) \bigr\vert \\ &\quad\leq \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}\times \int _{0}^{T} { \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({T ^{\rho }} - {\eta ^{\rho }})}^{1 - \alpha }}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \\ &\qquad{}\times \int _{0}^{\mu }{ \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({\mu ^{\rho }} - {\eta ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2{T ^{\rho }}\Gamma (\alpha )\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}\times \int _{0}^{T} { \frac{{ \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{\eta ^{1 - \alpha \rho }}}}} \,d\eta \\ &\qquad{}+ \frac{{{\rho ^{1 - (\alpha - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - {\alpha ^{*}})}} \\ &\qquad{}\times \int _{\mu }^{T} { \frac{{{\eta ^{\rho - 1}} \vert {g(\eta ,{\phi _{1}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert }}{{{{({\eta ^{\rho }} - {\mu ^{\rho }})}^{1 - (\alpha - {\alpha ^{*}})}}}}} \,d\eta \\ &\quad\leq {\lambda _{1}} \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}\times \bigl( { \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}( \eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \bigr\vert } ) \\ &\qquad{}+ {\lambda _{1}} \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{\mu ^{(\alpha - {\alpha ^{*}})}}}}{{2\Gamma (2 - {\alpha ^{*}} + 1)}} \\ &\qquad{}\times \bigl( { \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}( \eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \bigr\vert } ) \\ &\qquad{}+ {\lambda _{1}} \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} \\ &\qquad{}\times \bigl( { \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}( \eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \bigr\vert } ) \\ &\qquad{}+ {\lambda _{1}} \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}}{{2\Gamma (\alpha - {\alpha ^{*}} + 1)}} \\ &\qquad{}\times \bigl( { \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}( \eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \bigr\vert } ) \\ &\quad= \frac{{{\lambda _{1}}}}{{{K_{2}}}} \bigl\Vert {{\phi _{1}}(\mu ) - { \phi _{2}}(\mu )} \bigr\Vert , \end{aligned}$$

where \({K_{2}} = \frac{{2{\rho ^{(\alpha - {\alpha ^{*}})}}}}{{{L_{2}}}} \). Therefore

$$\begin{aligned} \bigl\vert {\tilde{T} {\phi _{1}}(\mu ) - \tilde{T} {\phi _{2}}(\mu )} \bigr\vert \le \frac{{2{\lambda _{1}}}}{M} \bigl\Vert {{\phi _{1}}(\mu ) - { \phi _{2}}(\mu )} \bigr\Vert \end{aligned}$$

where \(M = \max ({K_{1}},{K_{2}}) \). Thus the Banach fixed point theorem assures the existence of a unique fixed point of operator (6). So, in consequence of Lemma 4.2, we concluded that (3) is the unique solution of boundary value problem (1). □

Lemma 4.5

Assume that \(1<\alpha <2\), \(0<\beta ^{*}<1 \) and \({g}: [ {0,1} ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) is a continuous function. Furthermore, g satisfies

$$\begin{aligned} \bigl\vert {g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\mu )\bigr)} \bigr\vert \le {a_{3}} + {a_{4}}\max \bigl\vert {\phi (\mu )} \bigr\vert + {b_{2}}\max \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{2}}(\mu )} \bigr\vert \end{aligned}$$

for all \(a_{3}, a_{4}, b_{2} \in \mathbb{R}_{+}\). Then the solution \(\phi (\mu ) \) of (1) exists in \(A_{r} \).

Proof

The result follows from Theorem 4.3. □

Lemma 4.6

Assume that \(1<\alpha <2\), \(0<\beta ^{*}<1 \) and \({g}: [ {0,1} ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) is a continuous function. Furthermore, g satisfies the following condition:

$$\begin{aligned} \bigl\vert {g\bigl(\mu ,\phi (\mu ),{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \phi (\mu )\bigr)} \bigr\vert \le \frac{{{\rho ^{\alpha }}}}{{{T ^{\rho }}}} \bigl\vert {\phi (\mu )} \bigr\vert . \end{aligned}$$

Then problem (1) has at least one solution in \(A_{r} \).

Proof

Let \(a_{1}=a_{2}=0 \) and \(U(\mu ) = \vert \phi \vert \). Then, taking into account Theorem 4.3, the result holds. □

Example 4.7

Consider the following fractional differential equation:

$$\begin{aligned} \textstyle\begin{cases} _{0}^{RC}D_{\pi }^{\frac{7}{4},2}u(\mu ) = \frac{{ \vert u \vert }}{{{{(\mu + 4)}^{2}} ( {1 + \vert u \vert } )}},\quad \mu \in [0,\pi ], \\ u(0) = 0, \qquad u(\pi ) = 1, \end{cases}\displaystyle \end{aligned}$$

where \(g(\mu ,u) = \frac{{ \vert u \vert }}{{{{(\mu + 4)}^{2}} ( {1 + \vert u \vert } )}} \), \(\alpha = \frac{7}{4} \), and \(T=\pi \). Also, since \(\Vert {g(\mu ,u) - g(\mu ,v)} \Vert \le {\lambda _{1}} \Vert {u - v} \Vert \) with \({\lambda _{1}} = \frac{1}{{16}} \), therefore Theorem 4.4 assures that the boundary value problem has a unique solution on \([0, \pi ] \).

Dependence of solutions on the parameters

The stability analysis of fractional differential equations has been carried out by many mathematicians. For details, one can see [36, 3942] and the references therein. The solutions satisfy various types of stability, and continuous dependence on the initial data is one of them. This section demonstrates that the solution of problem (1) depends on the parameters α, \({\phi _{0}}\), \({\phi _{T} } \), and g provided that the function g satisfies conditions \((H{_{2}^{*})}\) and \((H{_{2}^{*})}\). Continuous dependence of solutions on the parameters indicates the stability of solutions.

Theorem 4.8

Assume that \({{\phi _{1}}(\eta )} \) is the solution of BVP (1) and \({{\phi _{2}}(\eta )} \) is the solution of the following problem:

$$\begin{aligned} \textstyle\begin{cases} {}_{0}^{RC}D_{T} ^{{\alpha -\epsilon },\rho } \phi (\mu ) = g(\mu , \phi (\mu ), _{0}^{RC}D_{T} ^{{\alpha {^{*},\rho }}}\phi (\mu )),\quad \mu \in [0,T ], \\ \phi (0) = {\phi _{0}}, \qquad \phi (T ) = {\phi _{T} }, \end{cases}\displaystyle \end{aligned}$$

where \(1<{\alpha -\epsilon } < \alpha \le 2 \), \(0 < \alpha ^{*} \le 1 \), and g is continuous. Then \(\Vert {{\phi _{1}} - {\phi _{2}}} \Vert = \mathrm{O}( \varepsilon ) \).

Proof

Using equation (3), we have

$$\begin{aligned} & \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert \\ &\quad\leq \bigl\vert {g\bigl(\eta ,{\phi _{1}}(\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr)} \bigr\vert \\ &\qquad{}\times \biggl\{ \frac{{({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} + \frac{{({T ^{\rho (\alpha - \varepsilon )}} + {\mu ^{\rho (\alpha - \varepsilon )}} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - \varepsilon }})}}{{{\rho ^{\alpha - \varepsilon }}\Gamma (\alpha - \varepsilon + 1)}} \biggr\} \\ &\quad\leq {\lambda _{1}} \biggl\{ { \frac{{({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} + \frac{{({T ^{\rho (\alpha - \varepsilon )}} + {\mu ^{\rho (\alpha - \varepsilon )}} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - \varepsilon }})}}{{{\rho ^{\alpha - \varepsilon }}\Gamma (\alpha - \varepsilon + 1)}}} \biggr\} \\ &\qquad{}\times \Vert {{\phi _{1}} - {\phi _{2}}} \Vert = \mathrm{O}( \varepsilon ). \end{aligned}$$

Also

$$\begin{aligned} \bigl\vert {_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \bigl({\phi _{1}}(\mu )\bigr) - _{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\bigl({\phi _{2}}(\mu )\bigr)} \bigr\vert &\le {\lambda _{1}} \Vert {{\phi _{1}} - {\phi _{2}}} \Vert \bigl( {H(\mu ) + H( \mu ,\varepsilon )} \bigr) \\ &= \mathrm{O}(\varepsilon ), \end{aligned}$$

where

$$\begin{aligned} H(\mu )={}& \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{\mu ^{(\alpha - {\alpha ^{*}})}}}}{{2\Gamma (2 - {\alpha ^{*}} + 1)}} \\ &{}+ \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}}{{2\Gamma (\alpha - {\alpha ^{*}} + 1)}} \end{aligned}$$

and

$$\begin{aligned} H(\mu ,\varepsilon ) ={}& \frac{{{\rho ^{({\alpha ^{*}} - \alpha - \varepsilon )}}{T ^{\rho (\alpha - \varepsilon - 1)}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2\Gamma (\alpha - \varepsilon + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha - \varepsilon )}}{\mu ^{(\alpha - \varepsilon - {\alpha ^{*}})}}}}{{2\Gamma (2 - {\alpha ^{*}} + 1)}} \\ &{}+ \frac{{{\rho ^{({\alpha ^{*}} - \alpha - \varepsilon )}}{T ^{\rho (\alpha - \varepsilon - 1)}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2\Gamma (\alpha - \varepsilon + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha - \varepsilon )}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - \varepsilon - {\alpha ^{*}}}}}}{{2\Gamma (\alpha - {\alpha ^{*}} + 1)}}. \end{aligned}$$

This completes the proof. □

Theorem 4.9

Assume that the conditions of Theorem 4.4hold and if \({{\phi _{1}}(\eta )} \) is the solution of BVP (1) and \({{\phi _{2}}(\eta )} \) is the solution of the following problem:

$$\begin{aligned} \textstyle\begin{cases} {}_{0}^{RC}D_{T} ^{{\alpha },\rho } \phi (\mu ) = g(\mu ,\phi (\mu ), _{0}^{RC}D_{T} ^{{\alpha {^{*},\rho }}}\phi (\mu )), \quad \mu \in [0,T ] \\ \phi (0) = {\phi _{0}}+\varepsilon _{1}, \qquad \phi (T ) = {\phi _{T} }+ \varepsilon _{2}, \end{cases}\displaystyle \end{aligned}$$

then \(\Vert {{\phi _{1}} - {\phi _{2}}} \Vert = \mathrm{O}(\max \{ { \varepsilon _{1}},{\varepsilon _{2}}\})\).

Proof

We have

$$\begin{aligned} & \bigl\vert \phi _{1}(\mu ) - \phi _{2}(\mu ) \bigr\vert \\ &\quad \leq \frac{(\varepsilon _{1} + \varepsilon _{2})\mu ^{\rho }}{2} + \frac{(\varepsilon _{1} + \varepsilon _{2})}{2T^{\rho }} \\ &\qquad{}+ \frac{ \vert {g(\eta ,{\phi _{1}}(\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{1}}(\eta )) - g(\eta ,{\phi _{2}}(\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{\phi _{2}}(\eta ))} \vert ({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \\ &\quad\leq \frac{{({\varepsilon _{1}} + {\varepsilon _{2}}){\mu ^{\rho }}}}{2} + \frac{{({\varepsilon _{1}} + {\varepsilon _{2}})}}{{2{T ^{\rho }}}} + \frac{{{\lambda _{1}}({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \bigl\Vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\Vert \\ &\quad= \mathrm{O}\bigl( \{ \varepsilon _{1},\varepsilon _{2} \} \bigr). \end{aligned}$$

This gives the desired result. □

Theorem 4.10

Assume that \({{\phi _{1}}(\eta )} \) is the solution of BVP (1) and \({{\phi _{2}}(\eta )} \) is the solution of the following problem:

$$\begin{aligned} \textstyle\begin{cases} {}_{0}^{RC}D_{T} ^{{\alpha },\rho } \phi (\mu ) = g(\mu ,\phi (\mu ), _{0}^{RC}D_{T} ^{{\alpha {^{*},\rho }}}\phi (\mu ))+\epsilon , \quad \mu \in [0,T ] \\ \phi (0) = {\phi _{0}}, \qquad \phi (T ) = {\phi _{T} }, \end{cases}\displaystyle \end{aligned}$$

where \(1<{\alpha -\epsilon } < \alpha \le 2 \) and \(0 < \alpha ^{*} \le 1 \) and g is continuous. Then \(\Vert {{\phi _{1}} - {\phi _{2}}} \Vert = \mathrm{O}( \varepsilon ) \).

Proof

From Lemma 4.2, we have

$$\begin{aligned} & \bigl\vert {{\phi _{1}}(\mu ) - {\phi _{2}}(\mu )} \bigr\vert \\ &\quad\leq \bigl\vert {g\bigl(\eta ,{\phi _{1}}(\eta ),_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - g\bigl(\eta ,{\phi _{2}}(\eta ),_{0}^{RC}D_{T} ^{{ \alpha ^{*}},\rho }{\phi _{2}}(\eta )\bigr)} \bigr\vert \\ &\qquad{}\times \biggl( \frac{{({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \biggr) + \frac{{\varepsilon ({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \\ &\quad\leq \bigl\{ {{\lambda _{1}} \bigl( { \bigl\vert {{\phi _{1}}(\mu ) - { \phi _{2}}(\mu )} \bigr\vert + \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}}, \rho }{\phi _{1}}(\eta )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }{ \phi _{2}}(\eta ))} \bigr\vert + \varepsilon } )} \bigr\} \\ &\qquad{}\times \frac{{({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \\ &\quad\leq \frac{{({T ^{\alpha \rho }} + {\mu ^{\alpha \rho }} + {{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha }})}}{{{\rho ^{\alpha }}\Gamma (\alpha + 1)}} \bigl\{ {{\lambda _{1}} \Vert {{\phi _{1}} - {\phi _{2}}} \Vert + \varepsilon } \bigr\} = \mathrm{O}(\varepsilon ). \end{aligned}$$

Moreover,

$$\begin{aligned} \bigl\vert {{}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho } \bigl(\tilde{T} {\phi _{1}}( \mu )\bigr) - {}_{0}^{RC}D_{T} ^{{\alpha ^{*}},\rho }\bigl(\tilde{T} {\phi _{2}}( \mu )\bigr)} \bigr\vert \le H(\mu ) \bigl\{ {{\lambda _{1}} \Vert {{\phi _{1}} - { \phi _{2}}} \Vert + \varepsilon } \bigr\} = \mathrm{O}( \varepsilon ), \end{aligned}$$

where

$$\begin{aligned} H(\mu ) ={}& \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{\mu ^{\rho (1 - {\alpha ^{*}})}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{\mu ^{(\alpha - {\alpha ^{*}})}}}}{{2\Gamma (2 - {\alpha ^{*}} + 1)}} \\ &{}+ \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{T ^{\rho (\alpha - 1)}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{1 - {\alpha ^{*}}}}}}{{2\Gamma (\alpha + 1)\Gamma (2 - {\alpha ^{*}})}} + \frac{{{\rho ^{({\alpha ^{*}} - \alpha )}}{{({T ^{\rho }} - {\mu ^{\rho }})}^{\alpha - {\alpha ^{*}}}}}}{{2\Gamma (\alpha - {\alpha ^{*}} + 1)}}. \end{aligned}$$

This completes the proof. □

Concluding remarks

We presented a generalization of the Riesz fractional operator in this work. We provided some results and inequalities for the new generalized Riesz fractional operators. Furthermore, we proved some equivalence results for the nonlinear fractional differential equation involving the generalized Riesz derivative operator. By using suitable fixed point theorems, we provided the uniqueness of solution of the problem and some several mathematical techniques. Also, we discussed the stability of solutions and showed continuous dependence onto given parameters. An instructive comparison with literature shows that these results present the generalization of various old theorems in the related areas.

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Acknowledgements

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17. The fourth and fifth authors were supported by Azarbaijan Shahid Madani University. The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.

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Aleem, M., Ur Rehman, M., Alzabut, J. et al. On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator. Adv Differ Equ 2021, 303 (2021). https://doi.org/10.1186/s13662-021-03459-w

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  • DOI: https://doi.org/10.1186/s13662-021-03459-w

MSC

  • 26A33
  • 34A08
  • 34A12

Keywords

  • Fixed point theory
  • Gronwall inequality
  • Stability of solutions
  • The generalized Riesz fractional operator
  • The generalized Caputo fractional operator