Skip to main content

Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via ψ-Hilfer fractional derivative

Abstract

In this paper, we investigate the existence and uniqueness of a solution for a class of ψ-Hilfer implicit fractional integro-differential equations with mixed nonlocal conditions. The arguments are based on Banach’s, Schaefer’s, and Krasnosellskii’s fixed point theorems. Further, applying the techniques of nonlinear functional analysis, we establish various kinds of the Ulam stability results for the analyzed problem, that is, the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we provide some examples to illustrate the applicability of our results.

Introduction

Fractional calculus is a generalization of ordinary differentiation and integration of arbitrary order, which can be noninteger. Differential equations of fractional order have attracted the attention of several researchers; see the monographs [18] and references therein. In the literature, there exist several definitions of fractional integrals and derivatives, from the most popular Riemann–Liouville and Caputo-type fractional derivatives to the other ones such as Hadamard fractional derivative, the Erdélyi–Kober fractional derivative, and so forth. A generalization of both Riemann–Liouville and Caputo derivatives was given by Hilfer [9], which is known as the Hilfer fractional derivative \(D^{\alpha , \beta }x(t)\) of order α and type \(\beta \in [0, 1]\). The Hilfer fractional derivative interpolates between the Riemann–Liouville and Caputo derivatives as it reduces to the Riemann–Liouville and Caputo fractional derivatives for \(\beta =0\) and \(\beta =1\), respectively. The Hilfer fractional derivative is used in theoretical simulation of dielectric relaxation in glass-forming materials and in fractional diffusion equations; see [10, 11]. Some properties and applications of the Hilfer derivative can be found in [1216] and references therein.

The fractional derivative with another function, in the Hilfer sense, called the ψ-Hilfer fractional derivative and introduced in [17], generalizes the Hilfer fractional derivative [9]. The ψ-Hilfer fractional derivative is defined with respect to another function and unifies several definitions of fractional derivatives available in the literature. Thus the ψ-Hilfer fractional derivative covers a wide class of fractional derivatives and provides a platform to obtain a particular one by fixing the function ψ; see Remark 2.4. For some recent results on the existence and uniqueness of solutions of initial value problems and on the Ulam–Hyers–Rassias stability, see [10, 11, 1827] and references therein.

Nonlocal boundary value problems have become a rapidly growing area of research. The study of this type of problems is driven not only by theoretical interest, but also by the fact that several phenomena in engineering, physics, and life sciences can be modeled in this way. The idea of nonlocal conditions dates back to the work of Hilb [28]. However, the systematic investigation of a certain class of spatial nonlocal problems was carried out by Bitsadze and Samarskii [29]. We refer the reader to [30, 31] and references therein for a motivation regarding nonlocal conditions.

In [32] the authors considered fractional differential equations with mixed nonlocal fractional derivatives, integrals, and multipoint conditions of the form

$$ \textstyle\begin{cases} {}^{c}\mathfrak{D}^{\alpha }x(t) = f(t,x(t)), \quad t \in (0,T], \\ \sum_{i=1}^{m}{{{\gamma }_{i}}x(\eta _{i})} + \sum_{j=1}^{n}\lambda _{j}{}^{c}\mathfrak{D}^{\beta _{j}}x( \xi _{i}) + \sum_{r=1}^{k}\sigma _{r}\mathcal{I}^{\delta _{r}}x( \phi _{r}) = A, \end{cases} $$
(1.1)

where \(x \in C^{1}([0,T],\mathbb{R})\), \({}^{c}\mathfrak{D}^{\alpha }\) and \({}^{c}\mathfrak{D}^{\beta _{j}}\) denote the Caputo fractional derivatives of orders α and \(\beta _{j}\), respectively, \(0< \beta _{j} \leq \alpha \leq 1\) for \(j = 1,2,\ldots ,n\), \(\mathcal{I}^{\delta _{r}}\) is the Riemann–Liouville fractional integral operator of order \(\delta _{r} > 0\) for \(r = 1,2,\ldots ,k\), \(\gamma _{i}\), \(\lambda _{j}\), \(\sigma _{r}\), \(A \in \mathbb{R}\), \(\eta _{i}, \xi _{j}, \phi _{r} \in [0,T]\), \(i = 1,2,\ldots ,m\), and \(f\in C([0,T]\times \mathbb{R}, \mathbb{R})\). The existence and uniqueness results were obtained by applying Schaefer’s fixed point theorem and Banach’s contraction mapping principle. In addition, the authors established different kinds of Ulam stability for the problem.

In [33] the authors studied the existence, uniqueness, and Ulam–Hyers–Rassias stability for a class of ψ-Hilfer fractional differential equations described by

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{a^{+}}^{\alpha ,\rho ;\psi }x(t) = f(t,x(t),{}^{H} \mathfrak{D}_{{{a}^{+}}}^{\alpha ,\rho ;\psi }x(t)), \quad t\in J=(a,T], \\ \mathcal{I}_{a^{+}}^{1-\gamma ;\psi }x(a) = x_{a},\quad \alpha \leq \gamma = \alpha + \rho - \alpha \rho , T>a, \end{cases} $$
(1.2)

where \({}^{H}\mathfrak{D}_{a^{+}}^{\alpha ,\rho ;\psi }\) is the ψ-Hilfer fractional derivative of order \(\alpha \in (0,1]\) and type \(\rho \in [0,1]\), \(\mathcal{I}_{a^{+}}^{1-\gamma ;\psi }\) is the Riemann–Liouville fractional integral of order \(1 - \gamma \) with respect to the function ψ, \(f\in C(J\times \mathbb{R}^{2}, \mathbb{R})\), and \(x_{a} \in \mathbb{R}\).

Harikrishman et al. [34] discussed the existence and uniqueness of nonlocal initial value problems for Pantograph equations with ψ-Hilfer fractional derivative of the form

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{a^{+}}^{\alpha ,\rho ;\psi }x(t) = f(t,x(t),x( \lambda t)), \quad t\in J=(a,b], 0 < \lambda < 1, \\ \mathcal{I}_{a^{+}}^{1-\gamma ;\psi }x(a) = \sum_{i=1}^{k}c_{i}x({ \tau _{i}}), \quad \tau _{i}\in (a,b], \alpha \leq \gamma = \alpha + \rho - \alpha \rho , \end{cases} $$
(1.3)

where \({}^{H}\mathfrak{D}_{a^{+}}^{\alpha ,\rho ;\psi }\) is the ψ-Hilfer fractional derivative of order \(\alpha \in (0,1)\) and type \(\rho \in [0,1]\), \(\mathcal{I}_{a^{+}}^{1-\gamma ;\psi }\) is the Riemann–Liouville fractional integral of order \(1 - \gamma \) with respect to the continuous function ψ such that \(\psi ^{\prime } > 0\), and \(f\in C(J\times \mathbb{R}^{2}, \mathbb{R})\).

In [35] the authors established existence, uniqueness and Ulam–Hyers stability of implicit Pantograph fractional differential equations involving ψ-Hilfer fractional derivatives of the form

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }x(t) = f(t,x(t),x( \lambda t), {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }x( \lambda t)), \quad t\in J=(0,T], T > 0, 0 < \lambda < 1, \\ \mathcal{I}_{0^{+}}^{1-\gamma ;\psi }x(0^{+}) = \sum_{i=1}^{m}b_{i} \mathcal{I}_{0^{+}}^{\beta ;\psi }x({\xi _{i}}),\quad \xi _{i}\in J, \alpha \leq \gamma = \alpha + \rho - \alpha \rho , \end{cases} $$
(1.4)

where \({}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }\) is the ψ-Hilfer fractional derivative of order \(\alpha \in (0,1)\) and type \(\rho \in [0,1]\), \(\mathcal{I}_{0^{+}}^{1-\gamma ;\psi }\) and \(\mathcal{I}_{0^{+}}^{\beta ;\psi }\) are the ψ-Riemann–Liouville fractional integrals of orders \(1 - \gamma \) and \(\beta >0\), respectively, with respect to the continuous function ψ such that \(\psi ^{\prime } \neq 0\), \(f\in C(J\times \mathbb{R}^{3},\mathbb{R})\), \(b_{i}\in \mathbb{R}\), and \(0 < \xi _{1} \leq \xi _{2} \leq \cdots \leq \xi _{m} < T\).

Motivated by papers [3335] and some familiar results on fractional integro-differential equations, we establish the existence and uniqueness results and different types of Ulam stability, such as Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability for a class of ψ-Hilfer implicit fractional integro-differential equations with mixed nonlocal boundary conditions of the form

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }x(t) = f(t,x(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }x(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }x(t)), \quad t\in (0,T], \\ \sum_{i=1}^{m}{{{\omega }_{i}}x({{\eta }_{i}})} + \sum_{j=1}^{n}{{{\kappa }_{j}}{}^{H}\mathfrak{D}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }x({{\zeta }_{j}})} + \sum_{r=1}^{k} \sigma _{r}\mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }x({{\theta }_{r}}) = A, \end{cases} $$
(1.5)

where \({}^{H}\mathfrak{D}_{0^{+}}^{u ,\rho ;\psi }\) is the ψ-Hilfer fractional derivative of order \(u = \{\alpha ,\beta _{j}\}\) with \(0< \alpha , \beta _{j}\leq 1\), \(\alpha \geq \beta _{j} +\rho (1-\beta _{j} )\), \(j=1,\ldots ,n\), and \(0\leq \rho \leq 1\), \(\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }\) and \(\mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }\) are the ψ-Riemann–Liouville fractional integrals of orders α and \(\delta _{r} > 0\), respectively, \(\omega _{i}, \kappa _{j}, \sigma _{r}, A \in \mathbb{R}\), \(\eta _{i}, \zeta _{j}, \theta _{r} \in J\), \(i=1, 2,\ldots , m\), \(j=1, 2,\ldots , n\), \(r=1,2,\ldots ,k\), \(f : J\times \mathbb{R}^{3} \to \mathbb{R}\) is a given continuous function, and \(J := [0,T]\), \(T> 0\). We emphasize that the mixed nonlocal boundary conditions include multipoint, fractional derivative multiorder, and fractional integral multiorder boundary conditions.

The paper is organized as follows: In Sect. 2, we recall some basic and essential definitions and lemmas. In Sect. 3, we obtain the existence and uniqueness results for problem (1.5) via Banach’s, Schaefer’s, and Krasnosel’skiĭ’s fixed point theorems. In Sect. 4, we discuss the Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias stability results. Finally, in Sect. 5, we give some examples to illustrate the benefit of our main results.

Background material and auxiliary results

In this section, we introduce some notation, spaces, definitions, and some useful fundamental lemmas.

We denote \(\mathcal{C}[J ,\mathbb{R}]\) the Banach space of all continuous functions from an interval J into \(\mathbb{R}\) with the norm defined by

$$ \Vert f \Vert =\sup_{t\in J}\bigl\{ \bigl\vert f(t) \bigr\vert \bigr\} . $$

The weighted space \(\mathcal{C}_{\gamma ,\psi }[J ,\mathbb{R}]\) of continuous functions f on J is defined by

$$ \mathcal{C}_{\gamma ,\psi }[J ,\mathbb{R}] = \bigl\{ f(t):(0,T]:\bigl(\psi (t)- \psi (0)\bigr)^{\gamma }f(t) \in \mathcal{C}[J ,\mathbb{R}]\bigr\} $$

with the norm

$$ \Vert f \Vert _{\mathcal{C}_{\gamma ,\psi }[J ,\mathbb{R}]} = \bigl\Vert \bigl( \psi (t)-\psi (0) \bigr)^{\gamma }f(t) \bigr\Vert =\sup_{t\in J}\bigl\{ \bigl\vert \bigl(\psi (t)- \psi (0)\bigr)^{\gamma }f(t) \bigr\vert \bigr\} . $$

Definition 2.1

([2])

Let \((0,b]\) be a finite or infinite interval on the half-axis \(\mathbb{R}^{+}\), and let \(\alpha \in \mathbb{R}^{+}\). Also, let \(\psi (x)\) be an increasing positive function on \((0,b]\), having a continuous derivative \(\psi '(x)\) on \((0,b)\). The ψ-Riemann–Liouville fractional integral of a function f with respect to another function ψ on \([0,b]\) is defined by

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }f(t) = \frac{1}{\Gamma (\alpha )} \int _{0}^{t}{{\psi }'(\tau ){{ \bigl( \psi (t)-\psi (\tau ) \bigr)}^{\alpha -1}}f(\tau )\,d\tau },\quad t>0, $$
(2.1)

where Γ is the gamma function.

Definition 2.2

([2])

Let \(\psi '(x)\neq 0\), \(\alpha > 0\), and \(n \in \mathbb{N}\). The Riemann–Liouville fractional derivatives of a function f with respect to another function ψ of order α is defined by

$$ \mathfrak{D}_{{{0}^{+}}}^{\alpha ;\psi }f(t) = {{ \biggl( \frac{1}{{\psi }'(t)}\frac{d}{dt} \biggr)}^{n}} \mathcal{I}_{{{0}^{+}}}^{n- \alpha ;\psi }f(t), $$
(2.2)

where \(n = [\alpha ]+1\), and \([\alpha ]\) represents the integer part of the real number α.

Definition 2.3

([17])

Let \(f, \psi \in \mathcal{C}^{n} (J,\mathbb{R})\) be two functions such that \(\psi \geq 0\) and \(\psi '(t)\neq 0\) for all \(t \in J\) and \(n-1<\alpha <n\) with \(n \in \mathbb{N}\). The ψ-Hilfer fractional derivative of a function f of order α and type \(0\leq \rho \leq 1\) is defined by

$$ {}^{H}\mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\rho (n-\alpha );\psi }{{ \biggl( \frac{1}{{\psi }'(t)} \frac{d}{dt} \biggr)}^{n}}\mathcal{I}_{{{0}^{+}}}^{(1- \rho )(n-\alpha );\psi }f(t), $$
(2.3)

where \(n = [\alpha ]+1\), and \([\alpha ]\) represents the integer part of the real number α.

Remark 2.4

The operator \({}^{H}\mathfrak{D}_{a^{+}}^{\alpha ,\rho ;\psi }\) is reduced to the fractional derivative of Hilfer when \(\psi (t)\to t\) [9], of Hilfer–Hadamard when \(\psi (t)\to \log t\) [36], of Hilfer–Katugampola when \(\psi (t)\to t^{\rho }\), \(\rho > 0\) [37], of Riemann–Liouville when \(\psi (t)\to t\), \(\beta \to 0\) [2], of Caputo type when \(\psi (t)\to t\), \(\beta \to 1\) [2], of generalized Riemann–Liouville when \(\beta \to 0\) [2], and of generalized Caputo when \(\beta \to 1\) [38].

The following lemma presents the semigroup properties of the ψ-Hilfer fractional integral and derivative.

Lemma 2.5

([2])

Let \(\alpha \geq 0\), \(0 \leq \rho < 1\), and \(f\in L^{1}[J,\mathbb{R}]\). Then

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }\mathcal{I}_{{{0}^{+}}}^{ \beta ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha +\beta ;\psi }f(t)\quad \textit{for a.e. } t\in J. $$
(2.4)

In particular, if \(f\in \mathcal{C}_{\gamma ,\psi }[J,\mathbb{R}]\) and \(f \in \mathcal{C}[J,\mathbb{R}]\), then

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }\mathcal{I}_{{{0}^{+}}}^{ \beta ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha +\beta ;\psi }f(t)\quad \textit{for all } t\in (0,T], $$

and

$$ ^{H}\mathfrak{D}_{{{0}^{+}}}^{\alpha ;\psi } \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }f(t) = f(t)\quad \textit{for all } t\in J. $$

The composition of the ψ-Hilfer fractional integral and derivative operators is given by the following lemma.

Lemma 2.6

([2])

Let \(0 < \alpha \leq 1 \), \(0 \leq \rho < 1\), and \(\gamma = \alpha + \rho - \alpha \rho \). If \(f(t) \in \mathcal{C}_{1-\gamma }^{\gamma }[J,\mathbb{R}]\), then

$$ \mathcal{I}_{{{0}^{+}}}^{\gamma ;\psi } {^{H} \mathfrak{D}}_{{{0}^{+}}}^{ \gamma ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } {^{H} \mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }f(t), \quad \textit{and}\quad {^{H}\mathfrak{D}}_{{{0}^{+}}}^{\gamma ;\psi } \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } f(t) = {^{H} \mathfrak{D}}_{{{0}^{+}}}^{\rho (1-\alpha ) ;\psi }f(t). $$

Next, we take into account some important properties of the ψ-fractional derivative and integral operators.

Proposition 2.7

([2, 17])

Let \(\alpha \geq 0\), \(\upsilon > 0\), and \(t > 0\). Then the ψ-fractional integral and derivative of a power function are given by

  1. (i)

    \(\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{ ( \psi (s)- \psi (0) )}^{\upsilon -1}}(t) = \frac{\Gamma (\upsilon )}{\Gamma (\upsilon +\alpha )}{{ ( \psi (t)- \psi (0) )}^{\upsilon +\alpha -1}}\);

  2. (ii)

    \(\mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }{{ ( \psi (s)-\psi (0) )}^{\upsilon -1}}(t) = \frac{\Gamma (\upsilon )}{\Gamma (\upsilon -\alpha )}{{ ( \psi (t)- \psi (0) )}^{\upsilon -\alpha -1}}\);

  3. (iii)

    \({{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }{{ ( \psi (s)-\psi (0) )}^{\upsilon -1}}(t) = \frac{\Gamma (\upsilon )}{\Gamma (\upsilon -\alpha )}{{ ( \psi (t)- \psi (0) )}^{\upsilon -\alpha -1}}\).

    In particular, for \(n \leq k \in \mathbb{N}\) and \(\upsilon > n\),

    $$ {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }{{ \bigl( \psi (s)- \psi (0) \bigr)}^{\upsilon -1}}(t) = \frac{k!}{\Gamma (k+1 -\alpha )}{{ \bigl( \psi (t)-\psi (0) \bigr)}^{k -\alpha }}. $$

    On the other hand, for \(n \geq k\),

    $$ {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }{{ \bigl( \psi (s)- \psi (0) \bigr)}^{k}}(t) = 0. $$

Lemma 2.8

Let \(0<\alpha \), \(\beta \leq 1\), \(0\leq \rho \leq 1\), and \(\alpha \geq \beta +\rho (1-\beta )\). If \(f\in {\mathcal{C}_{1-\gamma ,\psi }}[0,T]\), then

$$ {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\beta ,\rho ;\psi } \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha -\beta ;\psi }f(t). $$
(2.5)

Proof

Letting \(\lambda =\beta +\rho (1-\beta ) \), we get

$$\begin{aligned} {}^{H}\mathfrak{D}_{{{0}^{+}}}^{\beta ,\rho ;\psi } \bigl( \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } f(t) \bigr) =&\mathcal{I}_{{{0}^{+}}}^{ \lambda -\beta ;\psi } \mathfrak{D}_{{{0}^{+}}}^{\lambda ;\psi } \bigl(\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }f(t) \bigr) \\ =&\mathcal{I}_{{{0}^{+}}}^{\lambda -\beta ;\psi } \biggl( \frac{1}{{\psi }'(t)} \frac{d}{dt} \biggr) \mathcal{I}_{{{0}^{+}}}^{1- \lambda ;\psi } \bigl( \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } f(t) \bigr) \\ =&\mathcal{I}_{{{0}^{+}}}^{\lambda -\beta ;\psi } \biggl( \frac{1}{{\psi }'(t)} \frac{d}{dt} \biggr) \mathcal{I}_{{{0}^{+}}}^{1- \lambda +\alpha ;\psi }f(t). \end{aligned}$$

By Definition 2.1 we obtain

$$\begin{aligned}& \biggl( \frac{1}{{\psi }'(t)}\frac{d}{dt} \biggr)\mathcal{I}_{{{0}^{+}}}^{1- \lambda +\alpha ;\psi } f(t) \\& \quad = \frac{1}{{\psi }'(t)\Gamma (1-\lambda +\alpha )} \frac{d}{dt} \int _{0}^{t}{{\psi }'(\tau ){{ \bigl( \psi (t)-\psi ( \tau ) \bigr)}^{\alpha -\lambda }} f(\tau )\,d\tau } \\& \quad = \frac{1}{{\psi }'(t)\Gamma (1-\lambda +\alpha )} \int _{0}^{t} {( \alpha -\lambda ){\psi }'(t){\psi }'(\tau ){{ \bigl( \psi (t)-\psi ( \tau ) \bigr)} ^{\alpha -\lambda -1}}f(\tau )\,d\tau } \\& \quad = \frac{1}{\Gamma (\alpha -\lambda )} \int _{0}^{t}{{\psi }'(\tau ){{ \bigl( \psi (t)-\psi (\tau ) \bigr)}^{\alpha -\lambda -1}}f(\tau )\,d\tau } \\& \quad = I_{{{0}^{+}}}^{\alpha -\lambda ;\psi }f(t). \end{aligned}$$

Then

$$ {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\beta ,\rho ;\psi } \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }f(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha -\beta ;\psi }f(t). $$

 □

Lemma 2.9

([17])

If \(f \in \mathcal{C}^{n} [J,\mathbb{R}]\), \(n-1 < \alpha < n\), \(0 \leq \beta \leq 1\), and \(\gamma =\alpha +\rho (n-\alpha )\), then

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{}^{H} \mathfrak{D}_{{{0}^{+}}}^{ \alpha ,\rho ;\psi }f(t) = f(t) - \sum _{k=1}^{n}{ \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -k}}}{\Gamma (\gamma -k+1)}}f_{ \psi }^{[n-k]} \mathcal{I}_{{{0}^{+}}}^{(1-\rho )(n-\alpha );\psi }f(0) $$
(2.6)

for all \(t\in J\), where \(f_{\psi }^{[n]}f(t):={{ ( \frac{1}{{\psi }'(t)}\frac{d}{dt} )}^{n}}f(t)\). Moreover, if \(0< \alpha <1\), then

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{} ^{H} \mathfrak{D}_{{{0}^{+}}}^{ \alpha ,\rho ;\psi }f(t) = f(t) - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Gamma (\gamma )} \mathcal{I}_{{{0}^{+}}}^{(1-\rho )(1-\alpha );\psi }f(0) $$
(2.7)

for all \(0 < \gamma < 1\) and \(t\in J\).

In addition, if \(f \in C_{1-\gamma ;\psi }[J,\mathbb{R}]\) and \(\mathcal{I}_{0^{+}}^{1-\gamma ;\psi } f\in C_{1-\gamma ;\psi }^{1}[J, \mathbb{R}]\), then

$$ \mathcal{I}_{{{0}^{+}}}^{\gamma ;\psi }{} ^{H} \mathfrak{D}_{{{0}^{+}}}^{ \gamma ;\psi }f(t) = f(t) - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Gamma (\gamma )} \mathcal{I}_{{{0}^{+}}}^{(1-\gamma );\psi }f(0) $$

for all \(0 < \gamma < 1\) and \(t\in J\).

To transform problem (1.5) into a fixed point problem, problem (1.5) must be converted to an equivalent Volterra integral equation. We provide the following lemma, which is important in our main results and concerns a linear variant of problem (1.5).

Lemma 2.10

Let \(h \in \mathcal{C}(J, \mathbb{R})\), \(\alpha \in (0,1]\), \(\rho \in [0,1)\), \(\gamma =\alpha +\rho (1-\alpha )\), \(\beta _{j}\in (0,1]\), \(\alpha \geq \beta _{j} +\rho (1-\beta _{j} )\), \(j=1,2,\ldots ,n\), \(\delta _{r}>0\), \(r=1,2,\ldots ,k\), and \(\Omega \ne 0\). Then the function \(x\in {\mathcal{C}_{1-\gamma ,\psi }}(J,\mathbb{R})\) is a solution of the linear ψ-Hilfer fractional differential equation equipped with mixed nonlocal conditions

$$ \textstyle\begin{cases} {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }x(t) = h(t), \quad t\in (0,T], \\ \sum_{i=1}^{m}{{{\omega }_{i}}x({{\eta }_{i}})} + \sum_{j=1}^{n}{{{\kappa }_{j}}{{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }x({{\zeta }_{j}})} + \sum_{r=1}^{k}{ \sigma _{r}}\mathcal{I}_{0^{+}}^{{{\delta }_{r}};\psi }x({{\theta }_{r}}) = A, \end{cases} $$
(2.8)

if and only if x satisfies the integral equation

$$\begin{aligned} x(t) =& \mathcal{I}_{0^{+}}^{\alpha ;\psi }h(t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl[A- \sum_{i=1}^{m}{{{ \omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }h({{\eta }_{i}})}- \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }h({{ \theta }_{r}})} \\ &{}- \sum_{j=1}^{n}{\kappa }_{j}\mathcal{I}_{{0}^{+}}^{ \alpha -{\beta }_{j};\psi }h({\zeta }_{j}) \Biggr], \end{aligned}$$
(2.9)

where

$$\begin{aligned} \Omega =& \sum_{i=1}^{m}{{{\omega }_{i}} {{ \bigl( \psi ({{ \eta }_{i}})-\psi (0) \bigr)}^{\gamma -1}}}+\sum_{r=1}^{k}{ \frac{{{\sigma }_{r}}\Gamma (\gamma ){{ ( \psi ({{\theta }_{r}})-\psi (0) )}^{\gamma +{{\delta }_{r}}-1}}}{\Gamma (\gamma +{{\delta }_{r}})}} \\ &{}+\sum_{j=1}^{n}{ \frac{{{\kappa }_{j}}\Gamma (\gamma ){{ ( \psi ({{\zeta }_{j}})-\psi (0) )} ^{\gamma -{{\beta }_{j}}-1}}}{\Gamma (\gamma -{{\beta }_{j}})}}. \end{aligned}$$
(2.10)

Proof

Let x be a solution of problem (2.8). By Lemma 2.9 we have

$$ x(t) = \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }h(t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Gamma (\gamma )}{{c}_{1}}, $$
(2.11)

where \(c_{1} \in \mathbb{R}\) is an arbitrary constant.

Taking the operators \({{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{\beta }_{j}},{{\rho }_{j}};\psi }\) and \(\mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }\) on (2.11), we obtain

$$\begin{aligned} {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{\beta }_{j}},{{\rho }};\psi }x(t) =& \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }h(t) + {{c}_{1}} \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -{{\beta }_{j}}-1}}}{\Gamma (\gamma -{{\beta }_{j}})}, \\ \mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }x(t) =& \mathcal{I}_{{{0}^{+}}}^{{{ \delta }_{r}}+\alpha ;\psi }h(t) + {{c}_{1}} \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma +{{\delta }_{r}}-1}}}{\Gamma (\gamma +{{\delta }_{r}})}. \end{aligned}$$

Applying the given boundary condition in (2.8), we get

$$\begin{aligned} A =& \sum_{i=1}^{m}{{{\omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }h({{\eta }_{i}})} + \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }h({{ \theta }_{r}})} + \sum_{j=1}^{n}{{{ \kappa }_{j}}\mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }h({{\zeta }_{j}})} \\ &{}+{{c}_{1}} \Biggl[ \sum_{i=1}^{m}{ \frac{{{\omega }_{i}}{{ ( \psi ({{\eta }_{i}})-\psi (0) )}^{\gamma -1}}}{\Gamma (\gamma )}}+ \sum_{r=1}^{k}{ \frac{{{\sigma }_{r}}{{ ( \psi ({{\theta }_{r}})-\psi (0) )}^{\gamma +{{\delta }_{r}}-1}}}{\Gamma (\gamma +{{\delta }_{r}})}} \\ &{}+ \sum_{j=1}^{n}{ \frac{{{\kappa }_{j}}{{ ( \psi ({{\zeta }_{j}})-\psi (0) )}^{\gamma -{{\beta }_{j}}-1}}}{\Gamma (\gamma -{{\beta }_{j}})}} \Biggr], \end{aligned}$$

from which we get

$$ c_{1} = \frac{\Gamma (\gamma )}{\Omega } \Biggl[ A - \sum _{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }h({{\eta }_{i}})} - \sum_{r=1}^{k}{{{\sigma }_{r}}\mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }h({{\theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}}\mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }h({{ \zeta }_{j}})} \Biggr], $$

where Ω is defined by (2.10). Inserting this value of \(c_{1}\) into (2.11), we get (2.9).

Conversely, suppose that x is a solution of problem (2.8). Taking the ψ-Hilfer fractional derivative \({{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }\) into both sides of the Volterra integral equation (2.9) and using Proposition 2.7 with Lemma 2.8, it follows that

$$\begin{aligned} {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi } x(t) =& {{}^{H} \mathfrak{D}}_{{{0}^{+}}}^{\alpha ,\rho ;\psi } \mathcal{I}_{0^{+}}^{ \alpha ;\psi }h(t) + \frac{1}{\Omega } \Biggl[A - \sum_{i=1}^{m}{{{ \omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }h({{\eta }_{i}})} - \sum_{r=1}^{k}{{{ \sigma }_{r}}\mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }h({{ \theta }_{r}})} \\ &{} - \sum_{j=1}^{n}{\kappa }_{j}\mathcal{I}_{{0}^{+}}^{ \alpha -{\beta }_{j};\psi }h({\zeta }_{i}) \Biggr]{{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{ \alpha ,\rho ;\psi }{{ \bigl( \psi (t)-\psi (0) \bigr)}^{\gamma -1}} \\ =& h(t), \end{aligned}$$

for \(t\in J\), where \(0 < \gamma = \alpha + \rho (1-\alpha ) \leq 1\). Next, we show that x satisfies the boundary conditions. Applying the operator \({{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{\beta }_{j}},{{\rho }};\psi }\) and \(\mathcal{I}_{0^{+}}^{{{\delta }_{r}};\psi }\) to both sides of (2.9) with Lemma 2.8 and Proposition 2.7, for \(i = 1,2,\ldots , m\), \(j = 1,2,\ldots , n\), and \(r = 1, 2,\ldots , k\), we obtain

$$\begin{aligned}& \begin{aligned} \sum_{i=1}^{m}{{{\omega }_{i}}x({{\eta }_{i}})} ={}& \sum _{i=1}^{m}{{\omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{\alpha ; \psi }h({\eta }_{i}) + \sum _{i=1}^{m} \frac{{{\omega }_{i}}{{ ( \psi ({\eta }_{i})-\psi (0) )}^{\gamma -1}}}{\Omega \Gamma (\gamma )} \\ &{} \times \Biggl[ A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }h({{\eta }_{i}})} - \sum_{r=1}^{k}{{{ \sigma }_{r}}\mathcal{I}_{{{0}^{+}}}^{\alpha +{{ \delta }_{r}};\psi }h({{ \theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}}\mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }h({{ \zeta }_{j}})} \Biggr], \end{aligned} \\& \begin{aligned} \sum_{j=1}^{n}{{{\kappa }_{j}} {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }x({{ \zeta }_{j}})} ={}& \sum_{j=1}^{n}{{ \kappa }_{j}}\mathcal{I}_{0^{+}}^{\alpha -{\beta }_{j};\psi }h({\zeta }_{j}) + \sum_{j=1}^{n} \frac{{{\kappa }_{j}}\Gamma (\gamma ) ( \psi ({\zeta }_{j})-\psi (0) )^{\gamma - \beta _{j} -1}}{\Omega \Gamma (\gamma -\beta _{j})} \\ &{} \times \Biggl[A- \sum_{i=1}^{m}{{{ \omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }h({{\eta }_{i}})}- \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }h({{ \theta }_{r}})} \\ &{}- \sum_{j=1}^{n}{ \kappa }_{j}\mathcal{I}_{{0}^{+}}^{\alpha -{ \beta }_{j};\psi }h({\zeta }_{i}) \Biggr], \end{aligned} \\& \begin{aligned} \sum_{r=1}^{k}{\sigma _{r}}\mathcal{I}_{0^{+}}^{{{\delta }_{r}}; \psi }x({{\theta }_{r}}) ={}& \sum_{r=1}^{k}{ \sigma _{r}} \mathcal{I}_{0^{+}}^{\alpha + {\delta }_{r};\psi }h({\theta }_{r}) + \sum_{r=1}^{k} \frac{{\sigma _{r}}\Gamma (\gamma ) ( \psi ({\theta }_{r})-\psi (0) )^{\gamma + \delta _{r}-1}}{\Omega \Gamma (\gamma +\delta _{r})} \\ &{} \times \Biggl[A- \sum_{i=1}^{m}{{{ \omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }h({{\eta }_{i}})}- \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }h({{ \theta }_{r}})} - \sum_{j=1}^{n}{ \kappa }_{j}\mathcal{I}_{{0}^{+}}^{\alpha -{ \beta }_{j};\psi }h({\zeta }_{i}) \Biggr], \end{aligned} \end{aligned}$$

where Ω is given by (2.10). Therefore

$$ \sum_{i=1}^{m}{{{\omega }_{i}}x({{\eta }_{i}})} + \sum _{j=1}^{n}{{{\kappa }_{j}} {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }x({{\zeta }_{j}})} + \sum_{r=1}^{k}{ \sigma _{r}}\mathcal{I}_{0^{+}}^{{{\delta }_{r}};\psi }x({{ \theta }_{r}}) = A. $$

The lemma is proved. □

Fixed point theorems play a major role in establishing the existence theory for problem (1.5). We collect here some well-known fixed point theorems used in this paper.

Lemma 2.11

(Banach contraction principle [39])

Let D be a nonempty closed subset of a Banach space E. Then any contraction mapping T from D into itself has a unique fixed point.

Lemma 2.12

(Schaefer’s fixed point theorem [39])

Let \(\mathbb{M}\) be a Banach space, let \(T:\mathbb{M}\rightarrow \mathbb{M}\) be a completely continuous operator, and let the set \(D=\{x\in \mathbb{M}:x=\kappa Tx, 0<\kappa \leq 1\}\) be bounded. Then T has a fixed point in \(\mathbb{M}\).

Lemma 2.13

(Krasnosel’skiĭ’s fixed point theorem [40])

Let M be a closed, bounded, convex, and nonempty subset of a Banach space. Let A, B be the operators such that (i) \(Ax+By \in \mathcal{M}\) whenever \(x,y \in \mathcal{M}\); (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists \(z \in \mathcal{M}\) such that \(z=Az+Bz\).

Existence results

In this section, we present results on the existence of a solution of problem (1.5).

For simplicity, we set

$$ {{F}_{x}}(t) = f\bigl(t,x(t),{{}^{H} \mathfrak{D}}_{{{0}^{+}}}^{\alpha , \rho ;\psi }x(t), \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }x(t) \bigr),\quad t\in J. $$

In this paper, the expression \(\mathcal{I}_{{{0}^{+}}}^{\alpha ,\rho }F_{x}(s)(c)\) means that

$$ \mathcal{I}_{{{0}^{+}}}^{q ;\psi } {{F}_{x}}(s ) (c) = \frac{1}{\Gamma (q)} \int _{0}^{c}{{\psi }'(s ){{ \bigl( \psi (c)- \psi (s ) \bigr)}^{q -1}} {{F}_{x}}(s ) \,ds }, $$

where \(q = \{\alpha , \alpha -\beta _{j}, \alpha +\delta _{r}\}\) and \(c = \{t, \eta _{i}, \zeta _{j}, \theta _{r}\}\), \(i = 1,2,\ldots ,m\), \(j = 1,2,\ldots ,n\), \(r = 1,2,\ldots ,k\).

In view of Lemma 2.10, the operator \(\mathcal{Q}:{{C}_{1-\gamma ,\psi }}[J,\mathbb{R}] \to {{C}_{1- \gamma ,\psi }}[J,\mathbb{R}]\) is defined by

$$\begin{aligned} (\mathcal{Q}x) (t) =& \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}(s) (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl[A - \sum_{i=1}^{m}{{{ \omega }_{i}} I_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}(s) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{\sigma }_{r}} I_{{{0}^{+}}}^{\alpha +{{ \delta }_{r}};\psi }{{F}_{x}}(s) ({{\theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}} I_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }{{F}_{x}}(s) ({{\zeta }_{j}})} \Biggr]. \end{aligned}$$
(3.1)

Note that problem (1.5) has solutions if and only if the operator \(\mathcal{Q}\) has fixed points. In the following subsection, we establish the existence of solutions for the problem (1.5) by applying Banach’s, Schaefer’s, and Krasnosel’skiĭ’s fixed point theorems.

We list here the necessary assumptions to prove our main results.

\((H_{1})\):

There exist constants \(L_{1}, L_{3}>0\), and \(0< L_{2}<1\) such that

$$ \bigl\vert f(t,u_{1},v_{1},w_{1}) - f(t,u_{2},v_{2},w_{2}) \bigr\vert \leq L_{1} \vert u_{1} - u_{2} \vert + L_{2} \vert v_{1} - v_{2} \vert + L_{3} \vert w_{1} - w_{2} \vert $$

for any \(u_{i}\), \(v_{i}\), \(w_{i} \in \mathbb{R}\), \(i=1,2\), and \(t\in J\).

\((H_{2})\):

There exist nonnegative continuous functions \(h_{1}\), \(h_{2}\), \(h_{3}\), \(h_{4} \in \mathbb{R}\) such that

$$ \bigl\vert f(t,u,v,w) \bigr\vert \leq h_{1}(t)+h_{2}(t) \vert u \vert +h_{3}(t) \vert v \vert +h_{4}(t) \vert w \vert , \quad u,v,w\in \mathbb{R}, t\in J, $$

with \(h_{1}^{\ast }=\sup_{t\in J}h_{1}(t)\), \(h_{2}^{\ast }=\sup_{t\in J}h_{2}(t)\), \(h_{3}^{\ast }=\sup_{t\in J}h_{3}(t)\), \(h_{4}^{\ast }=\sup_{t\in J}h_{4}(t)\).

\((H_{3})\):

\(f(t,u,v,w) \leq q(t)\), \((t,u,v,w) \in J \times \mathbb{R}^{3}\), where \(q\in \mathcal{C}(J,\mathbb{R}^{+})\).

For computational convenience, we use of the following notations:

$$\begin{aligned}& {\Psi }(u,B) = \frac{{{ ( \psi (B)-\psi (0) )}^{u+\gamma -1}}}{\Gamma (u+\gamma )}, \end{aligned}$$
(3.2)
$$\begin{aligned}& {{\Lambda }_{1}}(u,B) = \frac{\Gamma (\gamma )}{1-{L_{2}}} \bigl[ {L}_{1}{{ \Psi }(u,B)}+{L}_{3}{\Psi }(\alpha +u,B) \bigr], \end{aligned}$$
(3.3)
$$\begin{aligned}& {{\Lambda }_{2}}(u,B) = \frac{\Gamma (\gamma )}{1-{h_{3}^{*}}} \bigl[ h_{2}^{*}{{\Psi }(u,B)}+h_{4}^{*}{ \Psi }(\alpha +u,B) \bigr], \end{aligned}$$
(3.4)
$$\begin{aligned}& \Theta _{\varepsilon } = \frac{1}{ \vert \Omega \vert } \Biggl[ \sum _{i=1}^{m} { \vert {\omega }_{i} \vert {{\Lambda }_{\varepsilon }}( \alpha ,{{\eta }_{i}})}+\sum _{r=1}^{k}{ \vert {\sigma }_{r} \vert {{\Lambda }_{\varepsilon }}(\alpha +{{\delta }_{r}},{{\theta }_{r}})}+ \sum _{j=1}^{n}{ \vert {\kappa }_{j} \vert {{\Lambda }_{ \varepsilon }}(\alpha -{{\beta }_{j}},{{ \zeta }_{j}})} \Biggr], \end{aligned}$$
(3.5)
$$\begin{aligned}& {{\Phi }_{\varrho }} = {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1- \gamma }} {{\Lambda }_{\varrho }}(\alpha ,T) +\Theta _{\varrho } , \end{aligned}$$
(3.6)
$$\begin{aligned}& \Xi = {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }}\Psi ( \alpha - \gamma +1 ,T) +\frac{1}{ \vert \Omega \vert } \Biggl[ \sum_{i=1}^{m} { \vert {\omega }_{i} \vert \Psi (\alpha -\gamma +1 ,{{ \eta }_{i}})} \\& \hphantom{\Xi ={}}{} +\sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \Psi (\alpha +{{ \delta }_{r}-\gamma +1}, {{\theta }_{r}})}+\sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \Psi (\alpha -{{\beta }_{j}-\gamma +1},{{ \zeta }_{j}})} \Biggr], \end{aligned}$$
(3.7)

where \(\varepsilon =\{1,2\}\) and \(\varrho =\{1,2\}\).

Existence and uniqueness via Banach contraction mapping principle

We will first prove the existence and uniqueness of a solution for problem (1.5) by using the Banach contraction mapping principle (Banach’s fixed point theorem).

Theorem 3.1

Let \(f: J\times \mathbb{R}^{3} \to \mathbb{R}\) be a continuous function satisfying \((H_{1})\). If

$$ \Phi _{1} < 1, $$
(3.8)

where \(\Phi _{1}\) is given by (3.6), then problem (1.5) has a unique solution \(x \in \mathcal{C}_{1-\gamma ,\psi }\) on J.

Proof

Firstly, we transform problem (1.5) into a fixed point problem, \(x = \mathcal{Q}x\), where the operator \(\mathcal{Q}\) is defined as in (3.1). It is clear that the fixed points of the operator \(\mathcal{Q}\) are solutions of problem (1.5). Applying the Banach contraction mapping principle, we will show that the operator \(\mathcal{Q}\) has a fixed point, which is a unique solution of problem (1.5).

Let \(\sup_{t\in [0,T]}|f(t,0,0,0)| := M_{1} < \infty \). Next, we set \(B_{\Upsilon _{1}} := \{x \in \mathcal{C}_{1-\gamma ,\psi }: {{\Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}} \leq \Upsilon _{1}\}\) with

$$ {{\Upsilon }_{1}} \geq \frac{1}{1 - {\Phi }_{1}} \biggl( \frac{{{M}_{1}}\Xi }{1-{L}_{2}} \biggr), $$
(3.9)

where Ω, \(\Phi _{1}\), and Ξ are given by (2.10), (3.6), and (3.7), respectively. Observe that \(B_{\Upsilon _{1}}\) is a bounded, closed, and convex subset of \(\mathcal{C}_{1-\gamma ,\psi }\). The proof is divided into two steps.

Step I. We show that \(\mathcal{Q}B_{\Upsilon _{1}} \subset B_{\Upsilon _{1}}\).

For any \(x \in B_{\Upsilon _{1}}\), we have

$$\begin{aligned}& \bigl\vert { \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma } ( Qx ) (t) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert (T) + \frac{1}{ \vert \Omega \vert } \Biggl( \vert A \vert + \sum _{i=1}^{m}{{{ \vert \omega }_{i}\vert }\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{{{\vert \sigma }_{r}\vert }\mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\theta }_{r}})} + \sum_{j=1}^{n}{{{\vert \kappa }_{j}\vert }\mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr). \end{aligned}$$

Consider

$$\begin{aligned} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert x(\tau ) \bigr\vert (s) =& \frac{1}{\Gamma (\alpha )} \int _{0}^{s}{{\psi }'(\tau ){{ \bigl( \psi (s) - \psi (\tau ) \bigr)}^{\alpha -1}} \bigl\vert x(\tau ) \bigr\vert \,d\tau } \\ \leq & \frac{1}{\Gamma (\alpha )} \int _{0}^{s}{{\psi }'(\tau ){{ \bigl( \psi (s)-\psi (\tau ) \bigr)}^{\alpha -1}} {{ \bigl( \psi ( \tau )- \psi (0) \bigr)}^{\gamma -1}} {{ \Vert x \Vert }_{ \mathcal{C}_{1-\gamma ,\psi }}}\,d\tau } \\ =& \frac{\Gamma (\gamma ){{ ( \psi (s)-\psi (0) )}^{\alpha +\gamma -1}}}{\Gamma (\alpha +\gamma )}{{ \Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}}. \end{aligned}$$

It follows from condition \((H_{1})\) that

$$\begin{aligned} \bigl\vert {{F}_{x}}(t) \bigr\vert \leq & \bigl\vert f \bigl(t,x(t),{{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{ \alpha ,\rho ;\psi }x(t), \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }x(t)\bigr) - f(t,0,0,0) \bigr\vert + \bigl\vert f(t,0,0,0) \bigr\vert \\ \leq & {{L}_{1}} \bigl\vert x(t) \bigr\vert + {{L}_{2}} \bigl\vert {{}^{H}\mathfrak{D}}_{{{0}^{+}}}^{ \alpha ,\rho ;\psi }x(t) \bigr\vert + {{L}_{3}} \bigl\vert \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }x(t) \bigr\vert +{{M}_{1}}. \end{aligned}$$

Then

$$ \bigl\vert {{F}_{x}}(t) \bigr\vert \leq \frac{1}{1-{{L}_{2}}} \biggl( {{L}_{1}} {{ \bigl( \psi (t)-\psi (0) \bigr)}^{\gamma -1}} + \frac{{{L}_{3}}\Gamma (\gamma ){{ ( \psi (t)-\psi (0) )}^{\alpha +\gamma -1}}}{\Gamma (\alpha +\gamma )} \biggr){{ \Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}} + \frac{{{M}_{1}}}{1-{{L}_{2}}}. $$

Thus we get

$$\begin{aligned}& \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert (T) \leq {{\Lambda }_{1}}(\alpha ,T){{ \Vert x \Vert }_{\mathcal{C}_{1- \gamma ,\psi }}} + \frac{{{M}_{1}}{{ ( \psi (T)-\psi (0) )}^{\alpha }}}{(1-{{L}_{2}})\Gamma (\alpha +1)}, \end{aligned}$$
(3.10)
$$\begin{aligned}& \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{ \eta }_{i}}) \leq {{\Lambda }_{1}}(\alpha ,{{\eta }_{i}}){{ \Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}}+ \frac{{{M}_{1}}{{ ( \psi ({{\eta }_{i}})-\psi (0) )}^{\alpha }}}{(1-{{L}_{2}})\Gamma (\alpha +1)}, \end{aligned}$$
(3.11)
$$\begin{aligned}& \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\theta }_{r}}) \leq {{\Lambda }_{1}}(\alpha +{{\delta }_{r}},{{ \theta }_{r}}){{ \Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}}+ \frac{{{M}_{1}}{{ ( \psi ({{\theta }_{r}})-\psi (0) )}^{\alpha +{{\delta }_{r}}}}}{(1-{{L}_{2}})\Gamma (\alpha +{{\delta }_{r}}+1)}, \end{aligned}$$
(3.12)
$$\begin{aligned}& \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\zeta }_{j}}) \leq {{\Lambda }_{1}}(\alpha -{{\beta }_{j}},{{ \zeta }_{j}}){{ \Vert x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}} + \frac{{{M}_{1}}{{ ( \psi ({{\zeta }_{j}})-\psi (0) )}^{\alpha -{{\beta }_{j}}}}}{(1-{{L}_{2}})\Gamma (\alpha -{{\beta }_{j}}+1)}. \end{aligned}$$
(3.13)

From (3.10)–(3.13) we obtain

$$ \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} ( \mathcal{Q}x ) (t) \bigr\vert \leq {{\Phi }_{1}} {{\Upsilon }_{1}} + \frac{{{M}_{1}}\Xi }{1-{{L}_{2}}}, $$

which implies that \({{\Vert \mathcal{Q}x \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}}\le {{ \Upsilon }_{1}}\). Therefore \(\mathcal{Q}B_{\Upsilon _{1}} \subset B_{\Upsilon _{1}}\).

Step II. We show that the operator \(\mathcal{Q}:{\mathcal{C}_{1-\gamma ,\psi }}\to {\mathcal{C}_{1- \gamma ,\psi }}\) is a contraction.

For all x, \(y \in {\mathcal{C}_{1-\gamma ,\psi }}\) and \(t \in J\), we have

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} \bigl( ( \mathcal{Q}x ) (t)- ( \mathcal{Q}y ) (t) \bigr) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {{F}_{x}}(s)-{{F}_{y}}(s) \bigr\vert (T) + \frac{1}{ \vert \Omega \vert } \Biggl(\sum _{i=1}^{m}{ \vert { \omega }_{i} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{x}}(s)-{{F}_{y}}(s) \bigr\vert ({{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{x}}(s)-{{F}_{y}}(s) \bigr\vert ({{ \theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{x}}(s)-{{F}_{y}}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr) . \end{aligned}$$
(3.14)

From \((H_{1})\) we obtain

$$\begin{aligned} \bigl\vert {{F}_{x}}(t)-{{F}_{y}}(t) \bigr\vert =& \bigl\vert f\bigl(t,x(t),{{F}_{x}}(t), \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }x(t) \bigr) - f\bigl(t,y(t),{{F}_{y}}(t), \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }y(t) \bigr) \bigr\vert \\ \leq & {{L}_{1}} \bigl\vert x(t)-y(t) \bigr\vert + {{L}_{2}} \bigl\vert {{F}_{x}}(t)-{{F}_{y}}(t) \bigr\vert + {{L}_{3}} \bigl\vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }x(t) - \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }y(t) \bigr\vert , \end{aligned}$$

and thus

$$\begin{aligned} &\bigl\vert {{F}_{x}}(t)-{{F}_{y}}(t) \bigr\vert \\ &\quad \leq \frac{{{ \Vert x-y \Vert }_{\mathcal{C}_{1-\gamma ,\psi }}}}{1-L_{2}} \biggl( {{L}_{1}} {{ \bigl( \psi (t) - \psi (0) \bigr)}^{\gamma -1}} + \frac{{{L}_{3}}\Gamma (\gamma ){{ ( \psi (t)-\psi (0) )}^{\alpha +\gamma -1}}}{\Gamma (\alpha +\gamma )} \biggr). \end{aligned}$$
(3.15)

Then by substituting (3.15) into (3.14) we get

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} \bigl( ( \mathcal{Q}x ) (t)- ( \mathcal{Q}y ) (t) \bigr) \bigr\vert \\& \quad \leq \Biggl\{ {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} {{ \Lambda }_{1}}(\alpha ,T) + \frac{1}{ \vert \Omega \vert } \Biggl[ \sum _{i=1}^{m}{ \vert {\omega }_{i} \vert \Lambda _{1} (\alpha ,{{ \eta }_{i}})} + \sum _{r=1}^{k} \vert {\sigma }_{r} \vert {{ \Lambda }_{1}}(\alpha +{{\delta }_{r}},{{\theta }_{r}}) \\& \qquad {} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert {{\Lambda }_{1}}( \alpha -{{\beta }_{j}},{{\zeta }_{j}}) \Biggr] \Biggr\} { \Vert x-y \Vert }_{ \mathcal{C}_{1-\gamma ,\psi }}} , \end{aligned}$$

which implies that \({\Vert \mathcal{Q}x-\mathcal{Q}y \Vert }_{\mathcal{C}_{1-\gamma , \psi }}\leq {{\Phi }_{1}}{\Vert x-y \Vert }_{\mathcal{C}_{1-\gamma , \psi }}\). Since \({\Phi }_{1}< 1\), the operator \(\mathcal{Q}\) is a contraction map. Therefore by the Banach contraction mapping principle (Lemma 2.11) problem (1.5) has a unique solution in \(\mathcal{C}_{1-\gamma ,\psi }\). The proof is completed. □

Existence result via Schaefer’s fixed point theorem

The next existence result is based on Schaefer’s fixed point theorem.

Theorem 3.2

Let \(f : J\times \mathbb{R}^{3} \to \mathbb{R}\) be a continuous function satisfying \((H_{2})\). Then problem (1.5) has at least one solution on J.

Proof

We show that the operator \(\mathcal{Q}\) defined in (3.1) has at least one fixed point in \({\mathcal{C}_{1-\gamma ,\psi }}\). The proof is divided into four steps.

Step I. The operator \(\mathcal{Q}\) is continuous.

Let \(x_{n}\) be a sequence such that \(x_{n} \to x\) in \({\mathcal{C}_{1-\gamma ,\psi }}\). Then for each \(t\in J\), we obtain

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} \bigl( ( \mathcal{Q} {{x}_{n}} ) (t) - ( \mathcal{Q}x ) (t) \bigr) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {{F}_{{{x}_{n}}}}(s) - {{F}_{x}}(s) \bigr\vert (T) + \frac{1}{ \vert \Omega \vert } \Biggl[ \sum_{i=1}^{m}{ \vert { \omega }_{i} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{{{x}_{n}}}}(s)-{{F}_{x}}(s) \bigr\vert ({{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{{{x}_{n}}}}(s) - {{F}_{x}}(s) \bigr\vert ({{ \theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{{{x}_{n}}}}(s)-{{F}_{x}}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr] \\& \quad \leq \Biggl[ {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{ \bigl( \psi (s) - \psi (0) \bigr)}^{\gamma -1}}(T) \\& \qquad {}+ \frac{1}{ \vert \Omega \vert } \Biggl( \sum _{i=1}^{m}{ \vert {\omega }_{i} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{ \bigl( \psi (s)-\psi (0) \bigr)}^{\gamma -1}}({{ \eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k} \vert {\sigma }_{r} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }{{ \bigl( \psi (s) - \psi (0) \bigr)}^{ \gamma -1}}({{\theta }_{r}}) + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }{{ \bigl( \psi (s)-\psi (0) \bigr)}^{\gamma -1}}({{\zeta }_{j}})} \Biggr) \Biggr] \\& \qquad {} \times {{ \Vert {{F} {{{x}_{n}}}}-{{F} {x}} \Vert }_{{\mathcal{C}_{1- \gamma ,\psi }}}}. \end{aligned}$$

Since f is a continuous, this implies that \({{F}_{x}}\) is also continuous. Hence we obtain

$$ {{ \Vert {\mathcal{Q} {{{x}_{n}}}} - {\mathcal{Q} {x}} \Vert }_{{ \mathcal{C}_{1-\gamma ,\psi }}}}\to 0\quad \text{as } n \to \infty . $$

Step II. The operator \(\mathcal{Q}\) maps bounded sets into bounded sets in \({{\mathcal{C}_{1-\gamma ,\psi }}}\).

For \(\Upsilon _{2}>0\), there exists a constant \(\mu > 0\) such that, for each \(x\in \bar{B}_{\Upsilon _{2}}=\{x\in {{\mathcal{C}_{1-\gamma ,\psi }}}: \Vert x\Vert _{{\mathcal{C}_{1-\gamma ,\psi }}} \leq \Upsilon _{2}\}\), we have \(\Vert \mathcal{Q}x\Vert _{{\mathcal{C}_{1-\gamma ,\psi }}}\leq \mu \).

Indeed, for any \(t\in J\) and \(x\in \bar{B}_{\Upsilon _{2}}\), we have

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} ( \mathcal{Q}x ) (t) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert (T) + \frac{1}{ \vert \Omega \vert } \Biggl[ \vert A \vert + \sum _{i=1}^{m}{{{ \vert \omega }_{i}\vert }\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr]. \end{aligned}$$
(3.16)

It follows from condition \((H_{2})\) that

$$ \bigl\vert {{F}_{x}}(t) \bigr\vert \leq h_{1}^{*}+h_{2}^{*} \bigl\vert x(t) \bigr\vert +h_{3}^{*} \bigl\vert {{F}_{x}}(t) \bigr\vert +h_{4}^{*} \bigl\vert \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }x(t) \bigr\vert , $$

and thus

$$ \bigl\vert {{F}_{x}}(t) \bigr\vert \leq \frac{1}{1-h_{3}^{*}} \biggl[ h_{1}^{*}+ \biggl( h_{2}^{*}{{ \bigl( \psi (t)-\psi (0) \bigr)}^{\gamma -1}}+ \frac{h_{4}^{*}\Gamma (\gamma ){{ ( \psi (t)-\psi (0) )}^{\alpha +\gamma -1}}}{\Gamma (\alpha +\gamma )} \biggr){{ \Vert x \Vert }_{{\mathcal{C}_{1-\gamma ,\psi }}}} \biggr]. $$
(3.17)

Then by substituting (3.17) into (3.16) we get

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} ( \mathcal{Q}x ) (t) \bigr\vert \\& \quad \leq {{ \Vert x \Vert }_{1-\gamma , \psi }} \Biggl[ {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} {{ \Lambda }_{2}}(\alpha ,T) + \frac{1}{ \vert \Omega \vert } \Biggl( \sum _{i=1}^{m}{ \vert {\omega }_{i} \vert {{\Lambda }_{2}}(\alpha ,{{ \eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert {{\Lambda }_{2}}( \alpha +{{\delta }_{r}},{{\theta }_{r}})} + \sum _{j=1}^{n}{ \vert {\kappa }_{j} \vert {{\Lambda }_{2}}(\alpha -{{\beta }_{j}},{{ \zeta }_{j}})} \Biggr) \Biggr] + \frac{h_{1}^{*}\Xi }{1-h_{3}^{*}}, \end{aligned}$$

from which we get

$$ { \Vert \mathcal{Q}x \Vert }_{{\mathcal{C}_{1-\gamma ,\psi }}} \leq {{ \Phi }_{2}}\Upsilon _{2} + \frac{h_{1}^{*}\Xi }{1-h_{3}^{*}} :=\mu . $$

Step III. The operator \(\mathcal{Q}\) maps bounded sets into equicontinuous sets of \({{\mathcal{C}_{1-\gamma ,\psi }}}\).

For \(0 \leq t_{1} < t_{2}\leq T\) and \(x\in \bar{B}_{\Upsilon _{2}}\) where \(\bar{B}_{\Upsilon _{2}}\) is as defined in Step II, since f is bounded on the compact set \(J\times \bar{B}_{\Upsilon _{2}}^{3}\), we have

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi ({{t}_{2}})-\psi (0) \bigr)}^{1-\gamma }} (\mathcal{Q}x ) ({{t}_{2}})-{{ \bigl( \psi ({{t}_{1}}) - \psi (0) \bigr)}^{1-\gamma }} (\mathcal{Q}x ) ({{t}_{1}}) \bigr\vert \\& \quad = \bigl\vert {{ \bigl( \psi ({{t}_{2}})-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}({{t}_{2}}) - {{ \bigl( \psi ({{t}_{1}})-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}({{t}_{1}}) \bigr\vert \\& \quad = \biggl\vert \bigl(\psi (t_{2})-\psi (0)\bigr)^{1-\gamma } \int _{0}^{t_{2}} \psi '(\tau ) \bigl(\psi (t_{2})-\psi (\tau )\bigr)^{\alpha -1} {{F}_{x}}(\tau ) \,d\tau \\& \qquad {} - {{ \bigl( \psi ({{t}_{1}})-\psi (0) \bigr)}^{1-\gamma }} \int _{0}^{t_{1}} \psi '(\tau ) \bigl(\psi (t_{1})-\psi (\tau )\bigr)^{\alpha -1}{{F}_{x}}( \tau ) \,d\tau \biggr\vert . \end{aligned}$$

Set \(\sup_{(t,u,v,w) \in J\times B_{\Upsilon _{2}}^{3}} \vert f(t,u,v,w) \vert =\widehat{f} < \infty \). Since \((\psi (t)-\psi (0))^{1-\gamma }(\psi (t)-\psi (\tau ))^{\alpha -1}\) is a decreasing function on \(t \in (0,T)\), it follows that

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi ({{t}_{2}})-\psi (0) \bigr)}^{1-\gamma }} (\mathcal{Q}x ) ({{t}_{2}})-{{ \bigl( \psi ({{t}_{1}}) - \psi (0) \bigr)}^{1-\gamma }} (\mathcal{Q}x ) ({{t}_{1}}) \bigr\vert \\& \quad \leq \frac{\widehat{f}}{\Gamma (\alpha )} \biggl\vert \bigl(\psi (t_{2})- \psi (0)\bigr)^{1-\gamma } \int _{t_{1}}^{t_{2}}\psi '(\tau ) \bigl(\psi (t_{2})- \psi (\tau )\bigr)^{\alpha -1}\,d\tau \\& \qquad {}+{{ \bigl( \psi ({{t}_{1}})-\psi (0) \bigr)}^{1-\gamma }} \int _{0}^{t_{1}} \psi '(\tau ) \bigl(\psi (t_{1})-\psi (\tau )\bigr)^{\alpha -1}\,d\tau \\& \qquad {}- \bigl(\psi (t_{2})-\psi (0)\bigr)^{1-\gamma } \int _{0}^{t_{1}}\psi '(\tau ) \bigl( \psi (t_{2})-\psi (\tau )\bigr)^{\alpha -1}\,d\tau \biggr\vert \\& \quad = \frac{\widehat{f}}{\Gamma (\alpha +1 )} \bigl[ \bigl(\psi (t_{2})-\psi (0) \bigr)^{1- \gamma }\bigl(\psi (t_{2})-\psi (t_{1}) \bigr)^{\alpha } +\bigl(\psi (t_{1})-\psi (0) \bigr)^{1- \gamma }\bigl(\psi (t_{1})-\psi (0) \bigr)^{\alpha } \\& \qquad {}+\bigl(\psi (t_{2})-\psi (0)\bigr)^{1-\gamma } \bigl\{ \bigl( \psi (t_{2})-\psi (t_{1})\bigr)^{ \alpha }- \bigl( \psi (t_{2})-\psi (0)\bigr)^{\alpha } \bigr\} \bigr]. \end{aligned}$$

This inequality is independent of x and tends to zero as \(t_{2}\rightarrow t_{1}\), which implies that \(\Vert (\mathcal{Q}x)(t _{2})-(\mathcal{Q}x)(t_{1})\Vert _{{ \mathcal{C}_{1-\gamma ,\psi }}} \to 0\) as \(t_{2}\to t_{1}\). Thus, Steps I to III, together with the Arzelá–Ascoli theorem, we conclude that the operator \(\mathcal{Q}\) is completely continuous.

Step IV. The set \(\mathbb{E} =\{x\in {{\mathcal{C}_{1-\gamma ,\psi }}}: x = \varrho \mathcal{Q}x, 0 < \varrho \leq 1\}\) is bounded (a priori bounds).

Let \(x\in \mathbb{E}\). Then \(x = \varrho \mathcal{Q}x\) for some \(0<\varrho \leq 1\). From \((H_{2})\), for each \(t\in J\), we can get the estimate

$$\begin{aligned} x(t) =& \varrho \Biggl( \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}(t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl[A - \sum_{i=1}^{m}{{{ \omega }_{i}}\mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl( {{F}_{x}}(s) \bigr) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{\sigma }_{r}}\mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl( {{F}_{x}}(s) \bigr) ({{\theta }_{r}})} - \sum _{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl( {{F}_{x}}(s) \bigr) ({{\zeta }_{j}})} \Biggr] \Biggr). \end{aligned}$$

It follows from Step II that for each \(t\in J\), \(\Vert \mathcal{Q}x\Vert _{{\mathcal{C}_{1-\gamma ,\psi }}} \leq \mu < \infty \). This implies that the set \(\mathcal{C}_{1-\gamma ,\psi }\) is bounded.

By all hypotheses of Theorem 3.2 we conclude that there exists a positive constant N such that \(\Vert x\Vert _{\mathcal{C}_{1-\gamma ,\psi }}\leq N<\infty \). By Schaefer’s fixed point theorem (Lemma 2.12) the operator \(\mathcal{Q}\) has at least one fixed point, which is a solution of problem (1.5). This completes the proof. □

Existence result via Krasnosel’skiĭ’s fixed point theorem

By using Krasnosel’skiĭ’s fixed point theorem, we obtain the final existence theorem.

Theorem 3.3

Let \(f: J\times \mathbb{R}^{3} \to \mathbb{R}\) be a continuous function satisfying \((H_{1})\) and \((H_{3})\). If

$$ \Theta _{1} < 1, $$
(3.18)

where \(\Theta _{1}\) is defined by (3.5), then problem (1.5) has at least one solution on J.

Proof

Let \(\sup_{t \in J} \vert q(t) \vert = q^{*}\). By choosing a suitable \(B_{\Upsilon _{3}}:= \{ x \in {{\mathcal{C}_{1-\gamma ,\psi }}}: \Vert x \Vert _{{\mathcal{C}_{1-\gamma ,\psi }}} \leq \Upsilon _{3} \}\), where \(\Upsilon _{3} \geq \frac{|A|}{\vert \Omega \vert }+{{q}^{*}}{{\Xi }}\), we define the operators \(\mathcal{Q}_{1}\) and \(\mathcal{Q}_{2}\) on \(B_{\Upsilon _{3}}\) by

$$\begin{aligned}& (\mathcal{Q}_{1}x) (t) = \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}(s) (t),\quad t\in J, \\& \begin{aligned} (\mathcal{Q}_{2}x) (t) ={}& \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl[A - \sum _{i=1}^{m}{{{\omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}(s) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{\sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }{{F}_{x}}(s) ({{\theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}}\mathcal{I}_{{{0}^{+}}}^{\alpha -{{ \beta }_{j}};\psi }{{F}_{x}}(s) ({{\zeta }_{j}})} \Biggr],\quad t\in J. \end{aligned} \end{aligned}$$

Note that \(\mathcal{Q}=\mathcal{Q}_{1}+\mathcal{Q}_{2}\). For any \(x,y \in B_{\Upsilon _{3}}\), we have

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} \bigl( ( { \mathcal{Q}_{1}}x ) (t) + ( {\mathcal{Q}_{2}}y ) (t) \bigr) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {{F}_{x}}(s) \bigr\vert (T)+ \frac{1}{ \vert \Omega \vert } \Biggl[ \vert A \vert + \sum _{i=1}^{m}{ \vert {\omega }_{i} \vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{y}}(s) \bigr\vert ({{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {{F}_{y}}(s) \bigr\vert ({{\theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl\vert {{F}_{y}}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr] \\& \quad \leq {{q}^{*}} {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \Psi (\alpha -\gamma +1,T) + \frac{1}{ \vert \Omega \vert } \Biggl[ \vert A \vert +{{q}^{*}} \Biggl( \sum_{i=1}^{m} { \vert {\omega }_{i} \vert \Psi (\alpha - \gamma +1 ,{{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \Psi (\alpha + {{ \delta }_{r}-\gamma +1}, {{\theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \Psi (\alpha -{{\beta }_{j}-\gamma +1}, {{ \zeta }_{j}})} \Biggr) \Biggr] \\& \quad \leq \frac{ \vert A \vert }{ \vert \Omega \vert }+{{q}^{*}} \Biggl[ {{ \bigl( \psi (T)- \psi (0) \bigr)}^{1-\gamma }}\Psi (\alpha -\gamma +1 ,T)+ \frac{1}{ \vert \Omega \vert } \Biggl( \sum_{i=1}^{m} { \vert { \omega }_{i} \vert \Psi (\alpha -\gamma +1 ,{{\eta }_{i}})} \\& \qquad {} + \sum_{r=1}^{k}{ \vert {\sigma }_{r} \vert \Psi (\alpha +{{ \delta }_{r}-\gamma +1}, {{\theta }_{r}})} + \sum_{j=1}^{n}{ \vert {\kappa }_{j} \vert \Psi (\alpha -{{\beta }_{j}-\gamma +1}, {{ \zeta }_{j}})} \Biggr) \Biggr] \\& \quad \leq \frac{ \vert A \vert }{ \vert \Omega \vert } + {{q}^{*}}\Xi \le {{\Upsilon }_{3}}. \end{aligned}$$

This implies that \(\mathcal{Q}_{1}x+\mathcal{Q}_{2}y \in B_{\Upsilon _{3}}\), which satisfies assumption (i) of Lemma 2.13.

We now show that assumption (ii) of Lemma 2.13 is satisfied.

Let \(x_{n}\) be a sequence such that \(x_{n} \to x\) in \({{\mathcal{C}_{1-\gamma ,\psi }}}\). Then for each \(t \in J\), we have

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi (t)-\psi (0) \bigr)}^{1-\gamma }} \bigl( ( \mathcal{Q}_{1}{{x}_{n}} ) (t) - ( \mathcal{Q}_{1}x ) (t) \bigr) \bigr\vert \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {{F}_{{{x}_{n}}}}(s)-{{F}_{x}}(s) \bigr\vert (T) \\& \quad \leq {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }} \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{ \bigl( \psi (s)-\psi (0) \bigr)}^{\gamma -1}}(T){{ \Vert {{F} {{{x}_{n}}}}-{{F} {x}} \Vert }_{{ \mathcal{C}_{1-\gamma ,\psi }}}}. \end{aligned}$$

Since f is continuous, this implies that the operator \(F_{x}\) is also continuous. Hence we obtain

$$ {{ \Vert {{F}_{{{x}_{n}}}} - {{F}_{x}} \Vert }_{{\mathcal{C}_{1-\gamma , \psi }}}}\to 0 \quad \text{as } n \to \infty . $$

This shows that the operator \(\mathcal{Q}_{1}x\) is continuous, since \(\|\mathcal{Q}_{1} x_{n}-\mathcal{Q}_{1} x\|_{1-\gamma ,\psi }\to 0\) as \(n \to \infty \). Also, the set \(\mathcal{Q}_{1}B_{\Upsilon _{3}}\) is uniformly bounded as

$$ \Vert {\mathcal{Q}_{1}}x \Vert _{\mathcal{C}_{1-\gamma ,\psi }} \leq {{q}^{*}} {{ \bigl( \psi (T)-\psi (0) \bigr)}^{1-\gamma }}\Psi (\alpha -\gamma +1,T). $$

Next, we prove the compactness of \(\mathcal{Q}_{1}\). For each \(t_{1}, t_{2} \in J\) with \(0 \leq t_{1} < t_{2} \leq T\), we have (see Step III of Theorem 3.2)

$$\begin{aligned}& \bigl\vert {{ \bigl( \psi ({{t}_{1}})-\psi (0) \bigr)}^{1-\gamma }} ( \mathcal{Q}_{1}x ) ({{t}_{1}}) - {{ \bigl( \psi ({{t}_{2}})- \psi (0) \bigr)}^{1-\gamma }} ( { \mathcal{Q}_{1}}x ) ({{t}_{2}}) \bigr\vert \\& \quad \leq \frac{q^{*}}{\Gamma (\alpha +1 )} \bigl[ \bigl(\psi (t_{2})-\psi (0) \bigr)^{1- \gamma }\bigl(\psi (t_{2})-\psi (t_{1}) \bigr)^{\alpha } +\bigl(\psi (t_{1})-\psi (0) \bigr)^{1- \gamma }\bigl(\psi (t_{1})-\psi (0) \bigr)^{\alpha } \\& \qquad {}+\bigl(\psi (t_{2})-\psi (0)\bigr)^{1-\gamma } \bigl\{ \bigl( \psi (t_{2})-\psi (t_{1})\bigr)^{ \alpha }- \bigl( \psi (t_{2})-\psi (0)\bigr)^{\alpha } \bigr\} \bigr]. \end{aligned}$$

Obviously, the right-hand side in this inequality is independent of x and tends to zero as \(t_{2} \to t_{1}\). Therefore the operator \(\mathcal{Q}_{1}\) is equicontinuous, and so by the Arzelà–Ascoli theorem, \(\mathcal{Q}_{1}\) is relatively compact.

Moreover, it is easy to prove using condition (3.18) that the operator \(\mathcal{Q}_{2}\) is a contraction mapping, and thus assumption (iii) of Lemma 2.13 holds. Thus all the assumptions of Lemma 2.13 are satisfied. So the conclusion of Lemma 2.13 implies that the boundary value problem (1.5) has at least one solution on J. The proof is completed. □

Stability analysis

In this section, we develop some sufficient conditions under which the concerned problem (1.5) satisfies the hypotheses of different types of Ulam stability such as the Ulam–Hyers stability (UH), generalized Ulam–Hyers stability (GUH), Ulam–Hyers–Rassias stability (UHR), and generalized Ulam–Hyers–Rassias stability (GUHR).

Before stating the main theorem, we need the following definitions. Let \(\epsilon > 0\), and let \(\mathcal{B} : [0,T] \to [0,\infty )\) be a continuous function. We consider the following inequalities:

$$\begin{aligned}& \bigl\vert {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) - f\bigl(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t)\bigr) \bigr\vert \leq \epsilon , \end{aligned}$$
(4.1)
$$\begin{aligned}& \bigl\vert {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) - f\bigl(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t)\bigr) \bigr\vert \leq \epsilon \mathcal{B}(t), \end{aligned}$$
(4.2)
$$\begin{aligned}& \bigl\vert {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) - f\bigl(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t)\bigr) \bigr\vert \leq \mathcal{B}(t). \end{aligned}$$
(4.3)

Definition 4.1

([41])

Problem (1.5) is said to be UH stable if there exists a constant \(\tau > 0\) such that for \(\epsilon > 0\) and each solution \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of inequality (4.1), there exists a solution \(x\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of problem (1.5) with

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq \tau \epsilon ,\quad t\in J. $$
(4.4)

Definition 4.2

([41])

Problem (1.5) is said to be GUH stable if there exists a function \(\mathcal{B}\in \mathcal{C}_{1-\gamma ,\psi }(\mathbb{R}^{+}, \mathbb{R}^{+})\) with \(\mathcal{B}(0)=0\) such that for each solution \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of inequality (4.2), there exists a solution \(x\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of problem (1.5) with

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq \mathcal{B}(\epsilon ),\quad t \in J. $$
(4.5)

Definition 4.3

([41])

Problem (1.5) is said to be UHR stable with respect to \(\mathcal{B}\in \mathcal{C}_{1-\gamma ,\psi }(J,\mathbb{R}^{+})\) if there exists a real number \(m_{f,\mathcal{B}}>0\) such that for each solution \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}[J,\mathbb{R}]\) of inequality (4.2), there exists a solution \(x\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of problem (1.5) with

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq m_{f,\mathcal{B}}\epsilon \mathcal{B}(t),\quad t\in J. $$
(4.6)

Definition 4.4

([41])

Problem (1.5) is said to be GUHR stable with respect to \(\mathcal{B}\in \mathcal{C}_{1-\gamma ,\psi }(J,\mathbb{R}^{+})\) if there exists a real number \(m_{f,\mathcal{B}}>0\) such that for each solution \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of inequality (4.3), there exists a solution \(x\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) of problem (1.5) with

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq m_{f,\mathcal{B}}\mathcal{B}(t),\quad t\in J. $$
(4.7)

Remark 4.5

It is clear that (i) Definition 4.1 Definition 4.2; (ii) Definition 4.3 Definition 4.4; (iii) Definition 4.3 for \(\mathcal{B}(\cdot ) = 1\) Definition 4.1.

Remark 4.6

A function \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) is a solution of inequality (4.1) if and only if there exists a function \(w\in \mathcal{C}_{1-\gamma ,\psi }(J,\mathbb{R})\) (dependent on z) such that:

  1. (i)

    \(|w(t)|\leq \epsilon \), \(t \in J\).

  2. (ii)

    \({}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) = f(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t)) + w(t)\), \(t\in J\).

Firstly, we present an important lemma that will be used in the proofs of UH and GUH stability.

Lemma 4.7

Let \(\alpha \in (0,1]\), \(\rho \in [0,1)\). If \(z \in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) is a solution of inequality (4.1), then z is a solution of the inequality

$$ \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert \leq { \bigl( \psi (T)- \psi (0) \bigr)}^{\gamma -1}\Xi \epsilon , $$
(4.8)

where

$$\begin{aligned} \mathcal{H}_{z}(t) =& \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{z}}(s) (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{z}}(s) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }{{F}_{z}}(s) ({{\theta }_{r}})}- \sum_{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }{{F}_{z}}(s) ({{\zeta }_{j}})} \Biggr), \end{aligned}$$

and Ξ is given by (3.7).

Proof

Let z be a solution of inequality (4.1). So, in view of Remark 4.6(ii) and Lemma 2.10, we have

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) = f(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t))+\omega (t),\quad t\in (0,T], \\ \sum_{i=1}^{m}{{{\omega }_{i}}z({{\eta }_{i}})} + \sum_{j=1}^{n}{{{\kappa }_{j}}{}^{H}\mathfrak{D}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }z({{\zeta }_{j}})} + \sum_{r=1}^{k} \sigma _{r}\mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }z({{\theta }_{r}}) = A. \end{cases} $$
(4.9)

Thus the solution of (4.9) is of the form

$$\begin{aligned} z(t) =& \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{z}}(s) (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{z}}(s) ({{\eta }_{i}})} - \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }{{F}_{z}}(s) ({{\theta }_{r}})} \\ &{} - \sum_{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }{{F}_{z}}(s) ({{\zeta }_{j}})} \Biggr) + \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }w(s) (t) - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(\sum_{i=1}^{m}{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }w(s) ( \eta _{i}) \\ &{} + \sum_{r=1}^{k}{{\sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }w(s) (\theta _{r}) + \sum_{j=1}^{n}{{ \kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{\alpha -{{\beta }_{j}};\psi }w(s) ({{ \zeta }_{j}}) \Biggr). \end{aligned}$$

Then by using (i) of Remark 4.6 it follows that

$$\begin{aligned} \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert =& \Biggl\vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }w(s) (t) - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(\sum_{i=1}^{m}{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }w(s) ( \eta _{i}) \\ &{} + \sum_{r=1}^{k}{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }w(s) ( \theta _{r}) + \sum_{j=1}^{n}{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }w(s) ({{\zeta }_{j}}) \Biggr) \Biggr\vert \\ \leq& \Biggl(\Psi (\alpha - \gamma + 1, T) + \frac{{{ ( \psi (T)-\psi (0) )}^{\gamma -1}}}{ \vert \Omega \vert } \Biggl(\sum _{i=1}^{m} \vert {{\omega }_{i}} \vert \Psi (\alpha - \gamma + 1,\eta _{i}) \\ &{} + \sum_{r=1}^{k} \vert {{\sigma }_{r}} \vert \Psi (\alpha - \gamma + \delta _{r} + 1, \theta _{r}) + \sum_{j=1}^{n} \vert {{ \kappa }_{j}} \vert \Psi (\alpha - \gamma - \beta _{j}+ 1, \zeta _{j}) \Biggr) \Biggr) \epsilon \\ =& { \bigl( \psi (T)-\psi (0) \bigr)}^{\gamma -1}\Xi \epsilon , \end{aligned}$$

from which we obtain inequality (4.8). The proof is completed. □

Now we present the UH and GUH stability results.

Theorem 4.8

Let \(f: [0,T]\times \mathbb{R}^{3} \to \mathbb{R}\) be a continuous function. If \((H_{1})\) is satisfied with \(\Phi _{1}<1\), where \(\Phi _{1}\) is given by (3.6), then problem (1.5) is UH stable and GUH stable on J.

Proof

Suppose that \(\epsilon > 0\) and \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) is any solution of inequality (4.1), that is,

$$ \bigl\vert {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }z(t) - f\bigl(t,z(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }z(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }z(t)\bigr) \bigr\vert \leq \epsilon ,\quad t\in J. $$

Let \(x \in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) be the unique solution of problem (1.5). Then we have

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\alpha ,\rho ;\psi }x(t) = f(t,x(t),{}^{H} \mathfrak{D}_{{{0}^{+}}}^{\alpha ,\rho ;\psi }x(t), \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }x(t)), \quad t\in J, \\ \sum_{i=1}^{m}{{{\omega }_{i}}x({{\eta }_{i}})} + \sum_{j=1}^{n}{{{\kappa }_{j}}{}^{H}\mathfrak{D}_{{{0}^{+}}}^{{{ \beta }_{j}},{{\rho }};\psi }x({{\zeta }_{j}})} + \sum_{r=1}^{k} \sigma _{r}\mathcal{I}_{{{0}^{+}}}^{{{\delta }_{r}};\psi }x({{\theta }_{r}}) = A. \end{cases} $$

By applying the triangle inequality we get

$$ \bigl\vert z(t) - x(t) \bigr\vert = \bigl\vert z(t) - \mathcal{H}_{z}(t) + \mathcal{H}_{z}(t) - x(t) \bigr\vert \leq \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert + \bigl\vert \mathcal{H}_{z}(t) - x(t) \bigr\vert . $$
(4.10)

By using Lemma 4.7 with (4.10) we obtain

$$\begin{aligned} \bigl\vert z(t) - x(t) \bigr\vert \leq & { \bigl( \psi (T)-\psi (0) \bigr)}^{\gamma -1} \Xi \epsilon \\ &{}+ \Biggl\vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{z}}(s) (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A- \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{z}}(s) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{\sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }{{F}_{z}}(s) ({{\theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{\alpha -{{ \beta }_{j}};\psi }{{F}_{z}}(s) ({{\zeta }_{j}})} \Biggr)- \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}(s) (t) \\ &{} - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}(s) ({{\eta }_{i}})}- \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }{{F}_{x}}(s) ({{\theta }_{r}})} \\ &{} - \sum_{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }{{F}_{x}}(s) ({{\zeta }_{j}})} \Biggr) \Biggr\vert \\ \leq & { \bigl( \psi (T)-\psi (0) \bigr)}^{\gamma -1}\Xi \epsilon + \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert (t) \\ &{} + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{ \vert \Omega \vert } \Biggl( \sum_{i=1}^{m}{{ \vert {\omega }_{i} \vert } \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{ \eta }_{i}})} \\ &{} + \sum_{r=1}^{k}{{ \vert {\sigma }_{r} \vert } \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{ \theta }_{r}})} + \sum_{j=1}^{n}{{ \vert {\kappa }_{j} \vert } \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{ \zeta }_{j}})} \Biggr) \\ \leq & { \bigl( \psi (T)-\psi (0) \bigr)}^{\gamma -1}\Xi \epsilon + \Phi _{1} \bigl\vert z(t) - x(t) \bigr\vert . \end{aligned}$$

This implies that

$$ \bigl\vert z(t) - x(t) \bigr\vert \leq \tau \epsilon , $$

where

$$ \tau = \frac{{ ( \psi (T)-\psi (0) )}^{\gamma -1}\Xi }{1-\Phi _{1}}. $$

Hence problem (1.5) is UH stable. Now setting \(\mathcal{B} = \tau \epsilon \) such that \(\mathcal{B}(0) = 0\) yields that problem (1.5) is GUH stable. The proof is completed. □

Remark 4.9

Let \(\mathcal{B}\in \mathcal{C}_{1-\gamma ,\psi }(J,\mathbb{R}^{+})\) be an increasing function. Then there exists \(\lambda _{\mathcal{B}} > 0\) such that for each \(t\in J\), we have the integral inequality

$$ \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }\mathcal{B}(t) \leq \lambda _{ \mathcal{B}} \mathcal{B}(t). $$
(4.11)

Lemma 4.10

Let \(\alpha \in (0,1]\) and \(\rho \in [0,1)\). If \(z \in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) is a solution of inequality (4.2), then z is a solution of the inequality

$$ \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert \leq \mathcal{K} \lambda _{\mathcal{B}}\mathcal{B}(t)\epsilon , $$
(4.12)

where

$$ \mathcal{K} = 1 - \frac{{{ ( \psi (T)-\psi (0) )}^{\gamma -1}}}{ \vert \Omega \vert } \Biggl(\sum _{i=1}^{m} \vert {{\omega }_{i}} \vert + \sum_{r=1}^{k} \vert {{ \sigma }_{r}} \vert + \sum_{j=1}^{n} \vert {{\kappa }_{j}} \vert \Biggr). $$
(4.13)

Proof

From Lemma 4.7, using Remarks 4.6(i) and 4.9, we obtain

$$\begin{aligned} \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert =& \Biggl\vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }w(s) (t) - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(\sum_{i=1}^{m}{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }w(s) ( \eta _{i}) \\ & {}+ \sum_{r=1}^{k}{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }w(s) ( \theta _{r}) + \sum_{j=1}^{n}{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }w(s) ({{\zeta }_{j}}) \Biggr) \Biggr\vert \\ \leq & \Biggl[1 - \frac{{{ ( \psi (T)-\psi (0) )}^{\gamma -1}}}{ \vert \Omega \vert } \Biggl(\sum _{i=1}^{m} \vert {{\omega }_{i}} \vert + \sum_{r=1}^{k} \vert {{ \sigma }_{r}} \vert + \sum_{j=1}^{n} \vert {{\kappa }_{j}} \vert \Biggr) \Biggr]\lambda _{\mathcal{B}} \epsilon \mathcal{B}(t) \\ =& \mathcal{K}\lambda _{\mathcal{B}}\mathcal{B}(t)\epsilon , \end{aligned}$$

from which we obtain inequality (4.12). This completes the proof. □

Next, we are ready to prove UlHR and GUHR stability results.

Theorem 4.11

Let \(f: J\times \mathbb{R}^{3} \to \mathbb{R}\) be a continuous function, and let \((H_{1})\) and (4.11) be satisfied. If \(\Phi _{1} < 1\), where \(\Phi _{1}\) is given by (3.6), then problem (1.5) is UHR stable and GUHR stable on J.

Proof

Let \(z\in \mathcal{C}_{1-\gamma ,\psi }^{1}(J,\mathbb{R})\) be a solution of inequality (4.2), and let x be the unique solution of problem (1.5). By applying the triangle inequality and Lemma 4.10 we get

$$\begin{aligned} \bigl\vert z(t) - x(t) \bigr\vert =& \bigl\vert z(t) - \mathcal{H}_{z}(t) + \mathcal{H}_{z}(t) - x(t) \bigr\vert \\ \leq & \bigl\vert z(t) - \mathcal{H}_{z}(t) \bigr\vert + \bigl\vert \mathcal{H}_{z}(t) - x(t) \bigr\vert \\ \leq & \mathcal{K}\lambda _{\mathcal{B}}\mathcal{B}(t)\epsilon + \Biggl\vert \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{z}}(s) (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{z}}(s) ({{\eta }_{i}})} \\ &{} - \sum_{r=1}^{k}{{{\sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{ \alpha +{{\delta }_{r}};\psi }{{F}_{z}}(s) ({{\theta }_{r}})} - \sum_{j=1}^{n}{{{ \kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{\alpha -{{ \beta }_{j}};\psi }{{F}_{z}}(s) ({{\zeta }_{j}})} \Biggr) - \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}(s) (t) \\ &{} - \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{\Omega } \Biggl(A - \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi }{{F}_{x}}(s) ({{\eta }_{i}})} - \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{\delta }_{r}};\psi }{{F}_{x}}(s) ({{\theta }_{r}})} \\ &{} - \sum_{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi }{{F}_{x}}(s) ({{\zeta }_{j}})} \Biggr) \Biggr\vert \\ \leq & \mathcal{K}\lambda _{\mathcal{B}}\mathcal{B}(t)\epsilon + \mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert (t) + \frac{{{ ( \psi (t)-\psi (0) )}^{\gamma -1}}}{ \vert \Omega \vert } \\ &{} \times \Biggl( \sum_{i=1}^{m}{{{ \omega }_{i}} \mathcal{I}_{{{0}^{+}}}^{ \alpha ;\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{\eta }_{i}})} + \sum_{r=1}^{k}{{{ \sigma }_{r}} \mathcal{I}_{{{0}^{+}}}^{\alpha +{{ \delta }_{r}};\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{\theta }_{r}})} \\ &{} + \sum_{j=1}^{n}{{{\kappa }_{j}} \mathcal{I}_{{{0}^{+}}}^{ \alpha -{{\beta }_{j}};\psi } \bigl\vert {F}_{z}(s) - {F}_{x}(s) \bigr\vert ({{\zeta }_{j}})} \Biggr) \\ \leq & \mathcal{K}\lambda _{\mathcal{B}}\mathcal{B}(t)\epsilon + \Phi _{1} \bigl\vert z(t) - x(t) \bigr\vert , \end{aligned}$$

where \(\Phi _{1}\) is defined by (3.6), which implies that

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq \frac{\mathcal{K}\lambda _{\mathcal{B}}}{1-\Phi _{1}}\mathcal{B}(t) \epsilon . $$

By setting

$$ m_{f,\mathcal{B}} = \frac{\mathcal{K}\lambda _{\mathcal{B}}}{1-\Phi _{1}} $$

we get the inequality

$$ \bigl\vert z(t)-x(t) \bigr\vert \leq m_{f,\mathcal{B}} \mathcal{B}(t)\epsilon . $$
(4.14)

Hence problem (1.5) is UHR stable. Moreover, if we set \(\epsilon = 1\) in (4.14) with \(\mathcal{B}(0) = 0\), then problem (1.5) is GUHR stable. The proof is completed. □

Examples

In this section, we present two examples, which illustrate the validity and applicability of main results.

Example 5.1

Consider the nonlocal boundary problem

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\frac{3}{10}, \frac{1}{4}; e^{ \frac{t}{2}}}x(t) = \frac{10t+2}{(5-\sin ^{2}\pi t)^{2}}\cdot \frac{ \vert x(t) \vert }{4+ \vert x(t) \vert } + \frac{5\cos \pi t-2}{5} \cdot \frac{ \vert {}^{H}\mathfrak{D}_{{{0}^{+}}}^{\frac{3}{10}, \frac{1}{4}; e^{\frac{t}{2}}}x(t) \vert }{6+ \vert {}^{H}\mathfrak{D}_{{{0}^{+}}}^{\frac{3}{10}, \frac{1}{4}; e^{\frac{t}{2}}}x(t) \vert } \\ \hphantom{ {}^{H}\mathfrak{D}_{0^{+}}^{\frac{3}{10}, \frac{1}{4}; e^{ \frac{t}{2}}}x(t) ={}}{} + (2t-1)\cdot \frac{ \vert \mathcal{I}_{{{0}^{+}}}^{\frac{3}{10}; e^{\frac{t}{2}}}x(t) \vert }{5+ \vert \mathcal{I}_{{{0}^{+}}}^{\frac{3}{10}; e^{\frac{t}{2}}}x(t) \vert }, \quad t\in [0, 4/5], \\ \sum_{i=1}^{m}{ ({\frac{i+1}{10}} )x ({\frac{i}{10}} )} + \sum_{j=1}^{n}{{ ( \frac{4-j}{10} )}{}^{H}\mathfrak{D}_{{{0}^{+}}}^{{ \frac{14+j}{100}},{{\frac{1}{4}}};e^{\frac{t}{2}} }x ({ \frac{3}{25}j} )} \\ \quad {} + \sum_{r=1}^{k} (\frac{r+1}{r+4} ) \mathcal{I}_{{{0}^{+}}}^{{\frac{r+2}{r+9}};e^{\frac{t}{2}}}x ({ \frac{r}{5}} ) = 0, \end{cases} $$
(5.1)

where \(\alpha = 3/10\), \(\rho = 1/4\), \(T = 4/5\), \(m = 2\), \(n = 3\), \(k = 1\), \(\omega _{i} = (i+1)/10\), \(\kappa _{j} = (4-j)/10\), \(\sigma _{r} = (r+1)/(r+4)\), \(\eta _{i} = i/10\), \(\zeta _{j} = 3j/25\), \(\theta _{r} = r/5\), \(\beta _{j} = (14+j)/100\), \(\delta _{r} = (r+2)/(r+9)\), and \(A=0\). From the given data we obtain that \(\Omega \approx 3.593684625 \neq 0\) and

$$ f(t,u,v,w) = \frac{10t+2}{(5-\sin ^{2}\pi t)^{2}}\cdot \frac{ \vert u \vert }{4+ \vert u \vert }+ \frac{5\cos \pi t-2}{5}\cdot \frac{ \vert v \vert }{6+ \vert v \vert } +(2t-1)\cdot \frac{ \vert w \vert }{5+ \vert w \vert }. $$

For \(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\), \(z_{1}\), \(z_{2}\in \mathbb{R}\) and \(t\in [0, 4/5]\), we have

$$ \bigl\vert f(t,x_{1},y_{1},z_{1})-f(t,x_{2},y_{2},z_{2}) \bigr\vert \leq \frac{1}{10} \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert \bigr)+ \frac{3}{25} \vert z_{1}-z_{2} \vert . $$

Assumption \((H_{1})\) is satisfied with \(L_{1} = \frac{1}{10}\), \(L_{2} = \frac{1}{10}\), \(L_{3} = \frac{3}{25}\). Hence

$$ \Phi _{1} \approx 0.5220929551 < 1. $$

Since all the assumptions of Theorem 3.1 are satisfied, problem (5.1) has a unique solution on \([0, 4/5]\). Furthermore, we can also compute that

$$ \tau := \frac{(\psi (T)-\psi (0))^{\gamma -1}\Xi }{1-\Phi _{1}} \approx 2.590447084 > 0. $$

Hence by Theorem 4.8 problem (5.1) is both UH and GUH stable.

Example 5.2

Consider the nonlocal boundary problem

$$ \textstyle\begin{cases} {}^{H}\mathfrak{D}_{0^{+}}^{\frac{3}{5}, \frac{1}{2}; \sqrt[3]{t} }x(t) = \frac{4t-3}{5^{3t+2} (1+ \vert x(t) \vert + \vert {}^{H}\mathfrak{D}_{{{0}^{+}}}^{\frac{3}{5}, \frac{1}{2}; \sqrt[3]{t}}x(t) \vert + \vert \mathcal{I}_{{{0}^{+}}}^{\frac{3}{5}; \sqrt[3]{t}}x(t) \vert )}, \quad t\in [0,3/2], \\ \sum_{i=1}^{m}{{\frac{i}{5}}x ( \frac{i}{i+4} )} + \sum_{j=1}^{n}{ ({ \frac{4-j}{10}} ){}^{H}\mathfrak{D}_{{{0}^{+}}}^{\frac{3j}{20}, \frac{1}{2}; \sqrt[3]{t} }x (\frac{j+1}{20} )} + \sum_{r=1}^{k} (\frac{6-r}{50} )\mathcal{I}_{{{0}^{+}}}^{{ \frac{r+1}{5}}; \sqrt[3]{t}}x (\frac{r}{5} ) = \frac{\sqrt{\pi }}{2}, \end{cases} $$
(5.2)

where \(\alpha = 3/5\), \(\rho = 1/2\), \(T = 3/5\), \(m=1\), \(n = 2\), \(k = 2\), \(\omega _{i} = i/5\), \(\kappa _{j} = (4-j)/10\), \(\sigma _{r} = (6-r)/50\), \(\eta _{i} = i/(i+4)\), \(\zeta _{j} = (j+1)/20\), \(\theta _{r} = r/5\), \(\beta _{j} = 3j/20\), \(\delta _{r} = (r+1)/5\), and \(A = \sqrt{\pi }/2\). From the given data we obtain that \(\Omega \approx 0.8384823809 \neq 0\) and

$$ f(t,u,v,w) = \frac{4t-3}{5^{3t+2} (1+ \vert u \vert + \vert v \vert + \vert w \vert )}. $$

For \(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\), \(z_{1}\), \(z_{2}\in \mathbb{R}\) and \(t\in [0,3/2]\), we have

$$ \bigl\vert f(t,x_{1},y_{1},z_{1})-f(t,x_{2},y_{2},z_{2}) \bigr\vert \leq \frac{3}{25} \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert \bigr). $$

Assumption \((H_{1})\) is satisfied with \(L_{1} = L_{2}= L_{3} = \frac{3}{25}\). Hence

$$ \Phi _{1} \approx 0.6225720515 < 1,\quad \text{and}\quad \mathcal{K} \approx 2.021525739. $$

Set \(\mathcal{B}(t) = \psi (t) - \psi (0)\). By using Proposition 2.7(i), we can simply calculate that

$$ \mathcal{I}_{{{0}^{+}}}^{\frac{3}{5}; \sqrt[3]{t} }\mathcal{B}(t) = \frac{1}{\Gamma (\frac{13}{5})} \bigl[\psi (t) - \psi (0)\bigr]^{\frac{3}{5}} \mathcal{B}(t) \leq \frac{\sqrt[5]{1.5}}{\Gamma (\frac{13}{5})} \mathcal{B}(t). $$

Thus inequality (4.11) is satisfied with \(\lambda _{\mathcal{B}} = \frac{\sqrt[5]{1.5}}{\Gamma (\frac{13}{5})} > 0\). It follows that

$$ m_{f,\mathcal{B}} = \frac{\mathcal{K}\lambda _{\mathcal{B}}}{1-\Phi _{1}} \approx 4.062949933 > 0. $$

Hence by Theorem 4.11 problem (5.2) is both UHR and GUHR stable.

Conclusion

We have proved the existence and uniqueness of a solution for a class of ψ-Hilfer fractional integro-differential equations with mixed nonlocal conditions. We emphasize that the nonlocal boundary condition is very general, including multipoint, fractional derivative multiorder, and fractional integral multiorder conditions. We used the fixed point theorems of Banach, Schaefer, and Krasnosel’ski to investigate the existence and uniqueness of solutions. Our results are not only new in the given setting but also provide some new special cases by fixing the parameters involved in the problem at hand. For instance, by fixing \(\omega _{j}=0\), \(\lambda _{k}=0\) for all \(j=1, 2,\ldots , n\), \(k=1,2,\ldots ,r\) our results correspond to boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations supplemented with multipoint boundary conditions. In case we take \(\delta _{i}=0\), \(\lambda _{k}=0\) for all \(i=1, 2,\ldots , m\), \(k=1,2,\ldots ,r\), we obtain results for boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations equipped with multiterm integral boundary conditions.

We also investigated different kinds of Ulam stability, such as the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. The obtained results are well illustrated by examples.

The work accomplished in this paper is new and enriches the literature on boundary value problems for nonlinear ψ-Hilfer fractional differential equations.

Availability of data and materials

The authors declare that all data and materials in this paper are available and veritable.

References

  1. 1.

    Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York (2010)

    Google Scholar 

  2. 2.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of the Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (2006)

    Google Scholar 

  3. 3.

    Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    Google Scholar 

  4. 4.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    Google Scholar 

  5. 5.

    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  6. 6.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon & Breach, Yverdon (1993)

    Google Scholar 

  7. 7.

    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Google Scholar 

  8. 8.

    Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham (2017)

    Google Scholar 

  9. 9.

    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics World Scientific, Singapore (2000)

    Google Scholar 

  10. 10.

    Almalahi, M.A., Abdo, M.S., Panchal, S.K.: Existence and Ulam–Hyers–Mittag-Leffler stability results of ψ-Hilfer nonlocal Cauchy problem. Rend. Circ. Mat. Palermo, II Ser. (2020). https://doi.org/10.1007/s12215-020-00484-8

    Article  Google Scholar 

  11. 11.

    Almalahi, M.A., Abdo, M.S., Panchal, S.K.: ψ-Hilfer fractional functional differential equation by Picard operator method. J. Appl. Nonlinear Dyn. 9(4), 685–702 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. J. Chem. Phys. 284, 399–408 (2002)

    Google Scholar 

  13. 13.

    Hilfer, R., Luchko, Y., Tomovski, Z.: Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, 299–318 (2009)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Wongcharoen, A., Ahmad, B., Ntouyas, S.K., Tariboon, J.: Three-point boundary value problems for Langevin equation with Hilfer fractional derivative. Adv. Math. Phys. 2020, Article ID 9606428 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Wongcharoen, A., Ntouyas, S.K., Tariboon, J.: Nonlocal boundary value problems for Hilfer type pantograph fractional differential equations and inclusions. Adv. Differ. Equ. 2020, 279 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wongcharoen, A., Ntouyas, S.K., Tariboon, J.: On coupled systems for Hilfer fractional differential equations with nonlocal integral boundary conditions. J. Math. 2020, Article ID 2875152 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Vanterler da C. Sousa, J., Capelas de Oliveira, E.: On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Vanterler da C. Sousa, J., Capelas de Oliveira, E.: Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81, 50–56 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Vanterler da C. Sousa, J., Kucche, K.D., Capelas de Oliveira, E.: On the Ulam–Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Methods Appl. Sci. 42(9), 3021–3032 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Vanterler da C. Sousa, J., Capelas de Oliveira, E., Rodrigues, F.G.: Ulam–Hyers stabilities of fractional functional differential equations. AIMS Math. 5(2), 1346–1358 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Abdo, M.S., Thabet, S.T.M., Ahmad, B.: The existence and Ulam–Hyers stability results for ψ-Hilfer fractional integrodifferential equations. J. Pseudo-Differ. Oper. Appl. 11, 1757–1780 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Abdo, M.S., Panchal, S.K., Wahash, H.A.: Ulam–Hyers–Mittag-Leffler stability for a ψ-Hilfer problem with fractional order and infinite delay. Res. Appl. Math. 7, 100–115 (2020)

    Google Scholar 

  23. 23.

    Abdo, M.S., Shah, K., Panchal, S.K., Wahash, H.A.: Existence and Ulam stability results of a coupled system for terminal value problems involving ψ-Hilfer fractional operator. Adv. Differ. Equ. 2020, 316 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Wahash, H.A., Abdo, M.S., Panchal, S.K.: Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative. Ufa Math. J. 11, 151–171 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Almalahi, M.A., Abdo, M.S., Panchal, S.K.: On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative. AIMS Math. 5(5), 4889–4908 (2020)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kavitha, K., Vijayakumar, V., Udhayakumar, R., Nisar, K.S.: Results on the existence of Hilfer fractional neutral evolution equations with infinite delay via measures of noncompactness. Math. Methods Appl. Sci. 44, 1438–1455 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Subashini, R., Jothimani, K., Nisar, K.S., Ravichandran, C.: New results on nonlocal functional integro-differential equations via Hilfer fractional derivative. Alex. Eng. J. 59, 2891–2899 (2020)

    Article  Google Scholar 

  28. 28.

    Hilb, E.: Zur Theorie der Entwicklungen willkürlicher Funktionen nach Eigenfunktionen. Math. Z. 58, 1–9 (1918)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Bitsadze, A.V., Samarskii, A.A.: Some elementary generalizations of linear elliptic boundary value problems. Sov. Math. Dokl. 10, 398–400 (1969)

    MATH  Google Scholar 

  30. 30.

    Balachandran, K., Uchiyama, K.: Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. Proc. Indian Acad. Sci. Math. Sci. 110, 225–232 (2000)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Asawasamrit, S., Nithiarayaphaks, W., Ntouyas, S.K., Tariboon, J.: Existence and stability analysis for fractional differential equations with mixed nonlocal conditions. Mathematics 7, 117 (2019)

    Article  Google Scholar 

  33. 33.

    Vanterler da C. Sousa, J., Capelas de Oliveira, E.: On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl. 20, 96 (2018)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Harikrishman, S., Elsayed, E., Kanagarajan, K.: Existence and uniqueness results for fractional Pantograph equations involving ψ-Hilfer fractional derivative. Dyn. Contin. Discrete Impuls. Syst. 25, 319–328 (2018)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Ahmed, I., Kuman, P., Shah, K., Borisut, P., Sitthithakerngkiet, K., Demba, M.A.: Stability results for implicit fractional pantograph differential equations via ϕ-Hilfer fractional derivative with a nonlocal Riemann–Liouville fractional integral condition. Mathematics 8, 94 (2020)

    Article  Google Scholar 

  36. 36.

    Kassim, M.D., Tatar, N.E.: Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2013, Article ID 605029 (2013)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Oliveira, D.S., de Oliveira, E.C.: Hilfer–Katugampola fractional derivative. Comput. Appl. Math. 37, 3672–3690 (2018)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Google Scholar 

  40. 40.

    Krasnosel’skiĭ, M.A.: Two remarks on the method of successive approximation. Usp. Mat. Nauk 10, 123–127 (1955)

    MathSciNet  Google Scholar 

  41. 41.

    Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math. 26, 103–107 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first and second authors would like to thank for funding this work through the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand. W. Sudsutad would like to thank Navamindradhiraj University and King Mongkut’s University of Technology North Bangkok for supporting this work. Moreover, the authors would like to thank the referees for their careful reading of the paper and insightful comments.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

We declare that all authors have contributed equally and significantly to the contents of the paper. All authors have read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to Weerawat Sudsutad.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Thaiprayoon, C., Sudsutad, W. & Ntouyas, S.K. Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via ψ-Hilfer fractional derivative. Adv Differ Equ 2021, 50 (2021). https://doi.org/10.1186/s13662-021-03214-1

Download citation

MSC

  • 26A33
  • 34A08
  • 34A12
  • 34A34
  • 34B10
  • 34D20

Keywords

  • Existence
  • Uniqueness
  • ψ-Hilfer fractional derivative
  • Nonlocal boundary condition
  • Ulam–Hyers stability
\