Skip to main content

Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations

Abstract

In this paper, we use some fixed point theorems in Banach space for studying the existence and uniqueness results for Hilfer–Hadamard-type fractional differential equations

$$ {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+f\bigl(t,x(t)\bigr)=0 $$

on the interval \((1,e]\) with nonlinear boundary conditions

$$ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}),\qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}D^{1,1}x( \zeta _{i}). $$

Introduction

In this paper, we discuss the existence and uniqueness of the solutions for the n-point nonlinear boundary value problems for Hilfer–Hadamard-type fractional differential equations of the form

$$ \textstyle\begin{cases} {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+f(t,x(t))=0, \quad t\in J:=(1,e], \\ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}), \qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}D^{1,1}x(\zeta _{i}), \end{cases} $$
(1.1)

where \({}_{\mathrm{H}}D^{\alpha ,\beta }\) is the Hilfer–Hadamard fractional derivative of order \(1<\alpha \leq 2\) and type \(\beta \in [0,1]\), \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function, \(0<\epsilon <1\), \(\zeta _{i}\in (1,e)\), \(\nu _{i},\sigma _{i}\in \mathbb{R}\) for all \(i=1, 2,\dots ,n-2\), \(\zeta _{1}<\zeta _{2}<\cdots <\zeta _{n-2}\), and \({}_{\mathrm{H}}D^{1,1}=t\frac{d}{dt}\).

The fractional differential equations appear as more appropriate models for describing real world problems. Indeed, these problems cannot be described using the classical integer-order differential equations. In the past years, the theory of fractional differential equations has received much attention from the authors and has become an important field of investigation due to existing applications in engineering, biology, chemistry, economics, and numerous branches of physics [20, 27, 33, 40]. For example, the fractional differential equations are applied to describe the abundant phenomena such as flow in nonlinear electric circuits [15, 16, 20], properties of viscoelastic and dielectric materials [20, 21, 32], nonlinear oscillations of an earthquake [28], mechanics [35], aerodynamics, regular variations in thermodynamics [18], etc.

Fractional derivatives can be of several kinds, one of them is the Hadamard fractional derivative innovated by Hadamard in 1892 [17]. It differs from the preceding Riemann–Liouville- and Caputo-type fractional derivatives [33] in the sense that the kernel of the integral contains the logarithmic function of an arbitrary exponent. The properties of Hadamard fractional integral and derivative can be found in [26, 27]. Recently, scholars have studied the Hadamard-, Caputo–Hadamard- and Hilfer–Hadamard-type fractional derivatives by using the fixed point theorems with the boundary value problems and have given results of the existence and uniqueness of solutions, see [113, 2225, 30, 31, 34, 3639, 41, 4345] and the references mentioned therein.

In this paper, we find a variety of results for the boundary value problem (1.1) by using traditional fixed point theorems. The first result is Theorem 3.2, which depends on Banach contraction mapping principle and presents the existence and uniqueness result for the solution of problem (1.1). In Theorem 3.3, we prove the second result of the existence and uniqueness through a fixed point theorem and for nonlinear contractions due to Boyd and Wong. In Theorem 3.4, we prove the third existence result by using Krasnoselskii’s fixed point theorem. By using Leray–Schauder type of nonlinear alternative for single-valued maps, we prove the last result of existence, which is Theorem 3.5. Examples are included to illustrative our main results.

Preliminaries

In this section, we introduce some notations and definitions of Hilfer–Hadamard-type fractional calculus.

Definition 2.1

(Riemann–Liouville fractional integral, [27, 40])

The Riemann–Liouville fractional integral of order \(\alpha >0\) of a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined by

$$ \bigl(I^{\alpha }\varphi \bigr) (t)=\frac{1}{\Gamma (\alpha )} \int _{1}^{t} \frac{\varphi (\tau )\,d\tau }{(t-\tau )^{1-\alpha }} \quad (t>1). $$

Here, \(\Gamma (\alpha )\) is the Euler’s Gamma function defined by

$$ \Gamma (\alpha )= \int _{0}^{\infty }\tau ^{\alpha -1}e^{-\tau }\,d\tau . $$

Definition 2.2

(Riemann–Liouville fractional derivative, [27, 40])

The Riemann–Liouville fractional derivative of order \(\alpha >0\) of a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined by

$$\begin{aligned} \bigl(D^{\alpha }\varphi \bigr) (t)&:=\biggl(\frac{d}{dt} \biggr)^{n}\bigl(I^{n-\alpha }\varphi \bigr) (t) \\ &=\frac{1}{\Gamma (n-\alpha )}\frac{d^{n}}{dt^{n}} \int _{1}^{t} \frac{\varphi (\tau )\,d\tau }{(t-\tau )^{\alpha -n+1}}\quad \bigl(n=[ \alpha ]+1; t>1\bigr), \end{aligned}$$

where \([\alpha ]\) is the integer part of α.

Definition 2.3

(Hadamard fractional integral, [27])

The Hadamard fractional integral of order \(\alpha \in \mathbb{R}^{+}\) for a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined as

$$ {}_{\mathrm{H}}I^{\alpha }\varphi (t)=\frac{1}{\Gamma (\alpha )} \int _{1}^{t}\biggl( \log \frac{t}{\tau } \biggr)^{\alpha -1} \frac{\varphi (\tau )}{\tau }\,d\tau\quad (t>1), $$

where \(\log (\cdot )=\log _{e}(\cdot )\).

Definition 2.4

(Hadamard fractional derivative, [27])

The Hadamard fractional derivative of order α applied to the function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined as

$$ {}_{\mathrm{H}}D^{\alpha }\varphi (t)=\delta ^{n} \bigl({}_{\mathrm{H}}I^{n-\alpha }\varphi (t)\bigr),\quad n-1< \alpha < n, n=[ \alpha ]+1, $$

where \(\delta ^{n}=(t\frac{d}{dt})^{n}\) and \([\alpha ]\) denotes the integer part of the real number α.

Definition 2.5

(Caputo–Hadamard fractional derivative, [17])

The Caputo–Hadamard fractional derivative of order α applied to the function \(\varphi \in AC_{\delta }^{n}[a,b]\) is defined as

$$ _{\mathrm{HC}}D^{\alpha }\varphi (t)=\bigl({}_{\mathrm{H}}I^{n-\alpha } \delta ^{n}\varphi \bigr) (t), \quad n=[\alpha ]+1, $$

where \(\varphi \in AC_{\delta }^{n}[a,b]= \{\varphi :[a,b]\rightarrow \mathbb{C}:\delta ^{(n-1)}\varphi \in AC[a,b],\delta =t\frac{d}{dt}\}\).

Definition 2.6

(Hilfer fractional derivative, [20, 22])

Let \(n-1<\alpha <n\), \(0\leq \beta \leq 1\), \(\varphi \in L^{1}(a,b)\). The Hilfer fractional derivative \(D^{\alpha ,\beta }\) of order α and type β of φ is defined as

$$\begin{aligned} \bigl(D^{\alpha ,\beta }\varphi \bigr) (t)&= \biggl(I^{\beta (n-\alpha )}\biggl( \frac{d}{dt}\biggr)^{n} I^{(n-\alpha )(1-\beta )}\varphi \biggr) (t) \\ &= \biggl(I^{\beta (n-\alpha )}\biggl(\frac{d}{dt}\biggr)^{n}I^{n-\gamma } \varphi \biggr) (t); \quad \gamma =\alpha +n\beta -\alpha \beta \\ &= \bigl(I^{\beta (n-\alpha )}D^{\gamma }\varphi \bigr) (t), \end{aligned}$$

where \(I^{(\cdot )}\) and \(D^{(\cdot )}\) are the Riemann–Liouville fractional integral and derivative defined by Definitions 2.1 and 2.2, respectively.

In particular, if \(0<\alpha <1\), then

$$\begin{aligned} \bigl(D^{\alpha ,\beta }\varphi \bigr) (t)&= \biggl(I^{\beta (1-\alpha )} \frac{d}{dt} I^{(1-\alpha )(1-\beta )}\varphi \biggr) (t) \\ &= \biggl(I^{\beta (1-\alpha )}\frac{d}{dt}I^{1-\gamma }\varphi \biggr) (t);\quad \gamma =\alpha +\beta -\alpha \beta \\ &= \bigl(I^{\beta (1-\alpha )}D^{\gamma }\varphi \bigr) (t). \end{aligned}$$

Proposition 2.7

([22, 34])

Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +\beta -\alpha \beta \), and \(\varphi \in L^{1}(a,b)\). If \(D^{\gamma }\varphi \) exists and is in \(L^{1}(a,b)\), then

$$ I_{a+}^{\alpha } \bigl(D_{a+}^{\alpha ,\beta } \varphi \bigr) (t)=I_{a+}^{\gamma } \bigl(D_{a+}^{ \gamma } \varphi \bigr) (t)= \varphi (t)- \frac{(I_{a+}^{1-\gamma }\varphi )(a)}{\Gamma (\gamma )}(t-a)^{\gamma -1}. $$

Definition 2.8

(Hilfer–Hadamard fractional derivative, [12, 21])

Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\varphi \in L^{1}(a,b)\). The Hilfer–Hadamard fractional derivative \({}_{\mathrm{H}}D^{\alpha ,\beta }\) of order α and type β of φ is defined as

$$\begin{aligned} \bigl({}_{\mathrm{H}}D^{\alpha ,\beta }\varphi \bigr) (t)&= \bigl({}_{\mathrm{H}}I^{\beta (1-\alpha )} \delta\, {}_{\mathrm{H}}I^{(1-\alpha )(1-\beta )} \varphi \bigr) (t) \\ &= \bigl({}_{\mathrm{H}}I^{\beta (1-\alpha )}\delta\, {}_{\mathrm{H}}I^{1-\gamma } \varphi \bigr) (t); \quad \gamma =\alpha +\beta -\alpha \beta \\ &= \bigl({}_{\mathrm{H}}I^{\beta (1-\alpha )}\, {}_{\mathrm{H}}D^{\gamma } \varphi \bigr) (t), \end{aligned}$$

where \({}_{\mathrm{H}}I^{(\cdot )}\) and \({}_{\mathrm{H}}D^{(\cdot )}\) are the Hadamard fractional integral and derivative defined by Definitions 2.3 and 2.4, respectively.

Theorem 2.9

([17, 27])

Let \(\Re (\alpha )>0\), \(n=[\Re (\alpha )]+1\), and \(0< a< b<\infty \). If \(\varphi \in L^{1}(a,b)\) and \(({}_{\mathrm{H}}I_{a+}^{n-\alpha }\varphi )(t)\in AC_{\delta }^{n}[a,b]\), then

$$ \bigl({}_{\mathrm{H}}I_{a+}^{\alpha }\, {}_{\mathrm{H}}D_{a+}^{\alpha }\varphi \bigr) (t)= \varphi (t)- \sum_{j=0}^{n-1} \frac{(\delta ^{(n-j-1)}({}_{\mathrm{H}}I_{a+}^{n-\alpha }\varphi ))(a)}{\Gamma (\alpha -j)}\biggl( \log \frac{t}{a}\biggr)^{\alpha -j-1}. $$

Theorem 2.10

([17])

Let \(\varphi (t)\in AC_{\delta }^{n}[a,b]\) or \(\varphi (t)\in C_{\delta }^{n}[a,b]\), and \(\alpha \in \mathbb{C}\), then

$$ \bigl({}_{\mathrm{H}}I_{a+}^{\alpha } {}_{\mathrm{HC}}D_{a+}^{\alpha } \varphi \bigr) (t)= \varphi (t)- \sum_{K=0}^{n-1} \frac{\delta ^{K}\varphi (a)}{\Gamma (K+1)}\biggl(\log \frac{t}{a}\biggr)^{K}. $$

Definition 2.11

([45])

Let E be a Banach space and let \(F :E\rightarrow E \) be a mapping. Then F is said to be a nonlinear contraction if there exists a continuous nondecreasing function \(\psi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) such that \(\psi (0) = 0\) and \(\psi (\phi ) < \phi \) for all \(\phi > 0\) with the property

$$ \Vert Fx-Fy \Vert \leq \psi \bigl( \Vert x-y \Vert \bigr),\quad x,y\in E. $$

Lemma 2.12

([14])

Let E be a Banach space and let \(F :E\rightarrow E \) be a nonlinear contraction. Then, F has a unique fixed point in E.

Theorem 2.13

(Krasnoselskii’s fixed point theorem, [29])

Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that

  1. (a)

    \(Ax+By\in M\), whenever \(x, y\in M\);

  2. (b)

    A is compact and continuous;

  3. (c)

    B is a contraction mapping.

Then, there exists \(z\in M\) such that \(z=Az+Bz\).

Theorem 2.14

(Nonlinear alternative for single-valued maps, [19, 42])

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and \(0\in U\). Suppose that \(F : \overline{U}\rightarrow C\) is a continuous, compact (i.e., \(F(\overline{U})\) is a relatively compact subset of C) map. Then, either

  1. (i)

    F has a fixed point in or

  2. (ii)

    there is a \(u\in \partial U\) (the boundary of U in C) and \(\bar{\lambda }\in (0,1)\) with \(u =\bar{\lambda } F(u)\).

Definition 2.15

(Hilfer–Hadamard fractional derivative, [38])

Let \(n-1<\alpha<n\), \(0\leq\beta\leq 1\), \(\varphi\in L^{1}(a,b)\). The Hilfer–Hadamard fractional derivative \({}_{\mathrm{H}}D^{\alpha ,\beta }\) of order α and type β of φ is defined as

$$\begin{aligned} \bigl({}_{\mathrm{H}}D^{\alpha ,\beta }\varphi \bigr) (t)&= \bigl({}_{\mathrm{H}}I^{\beta (n-\alpha )} (\delta)^{n}\, {}_{\mathrm{H}}I^{(n-\alpha )(1-\beta )} \varphi \bigr) (t) \\ &= \bigl({}_{\mathrm{H}}I^{\beta (n-\alpha )}(\delta)^{n}\, {}_{\mathrm{H}}I^{n-\gamma } \varphi \bigr) (t); \quad \gamma =\alpha +n\beta -\alpha \beta \\ &= \bigl({}_{\mathrm{H}}I^{\beta (n-\alpha )}\, {}_{\mathrm{H}}D^{\gamma } \varphi \bigr) (t), \end{aligned}$$

where \({}_{\mathrm{H}}I^{(\cdot )}\) and \({}_{\mathrm{H}}D^{(\cdot )}\) are the Hadamard fractional integral and derivative defined by Definitions 2.3 and 2.4, respectively.

Lemma 2.16

([38])

Let \(\Re (\alpha )>0\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +n \beta -\alpha \beta \), \(n-1<\gamma \leq n\), \(n=[\Re (\alpha )]+1\), and \(0< a< b<\infty \). If \(\varphi \in L^{1}(a,b)\) and \(({}_{\mathrm{H}}I_{a+}^{n-\gamma }\varphi )(t)\in AC_{\delta }^{n}[a,b]\), then

$$ \begin{aligned} {}_{\mathrm{H}}I_{a+}^{\alpha } \bigl({}_{\mathrm{H}}D_{a+}^{\alpha ,\beta }\varphi \bigr) (t)&={}_{\mathrm{H}}I_{a+}^{ \gamma } \bigl({}_{\mathrm{H}}D_{a+}^{\gamma } \varphi \bigr) (t) \\ &= \varphi (t)-\sum_{j=0}^{n-1} \frac{(\delta ^{(n-j-1)}({}_{\mathrm{H}}I_{a+}^{n-\gamma }\varphi ))(a)}{\Gamma (\gamma -j)}\biggl(\log \frac{t}{a}\biggr)^{\gamma -j-1}. \end{aligned} $$

From this lemma, we notice that if \(\beta =0\) then the equation reduces to the equation in Theorem 2.9, and if the \(\beta =1\) then the equation reduces to the equation in Theorem 2.10.

Main results

Lemma 3.1

For \(1<\alpha \leq 2\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +2\beta -\alpha \beta \), \(\gamma \in (1,2]\), and \(\varphi \in C([1,e],\mathbb{R})\), the problem

$$ \textstyle\begin{cases} {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+\varphi (t)=0, \quad t\in J, 1< \alpha \leq 2, 0\leq \beta \leq 1, \\ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}), \qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i} \, {}_{\mathrm{H}}D^{1,1}x(\zeta _{i}), \end{cases} $$
(3.1)

has a unique solution given by

$$\begin{aligned} x(t) =&- {}_{\mathrm{H}}I^{\alpha }\varphi (t)+ \frac{(\gamma -1)\delta _{1} (\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \\ &{}\times\Biggl[ {}_{\mathrm{H}}I^{\alpha }\varphi (1+\epsilon )-\sum _{i=1}^{n-2}\nu _{i}\, {}_{\mathrm{H}}I^{ \alpha }\varphi (\zeta _{i}) \Biggr] \\ &{} + \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}\varphi (e)-\sum_{i=1}^{n-2}\sigma _{i}\, {}_{\mathrm{H}}I^{ \alpha -1}\varphi (\zeta _{i}) \Biggr],\quad t\in J, \end{aligned}$$

where

$$\begin{aligned} &\lambda =(\gamma -1)\delta _{1}\mu _{2}-(\gamma -2) \delta _{2}\mu _{1},\quad \textit{with } \lambda \neq 0, \\ & \mu _{1}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -1}-\sum _{i=1}^{n-2}\nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -1}, \\ & \mu _{2}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -2}-\sum _{i=1}^{n-2}\nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -2}, \\ & \delta _{1}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -2}, \\ & \delta _{2}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -3}. \end{aligned}$$

Proof

In view of Lemma 2.16, the solution of the Hilfer–Hadamard differential equation (3.1) can be written as

$$ x(t)=- {}_{\mathrm{H}}I^{\alpha }\varphi (t)+c_{0}(\log t)^{\gamma -1}+c_{1}( \log t)^{\gamma -2}, $$
(3.2)

and

$$ {}_{\mathrm{H}}D^{1,1}x(t)=- {}_{\mathrm{H}}I^{\alpha -1} \varphi (t)+(\gamma -1)c_{0}(\log t)^{ \gamma -2}+(\gamma -2) c_{1}(\log t)^{\gamma -3}. $$
(3.3)

The boundary condition \(x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i})\) gives

$$ c_{1}=\frac{1}{\mu _{2}} \Biggl[ {}_{\mathrm{H}}I^{\alpha }\varphi (1+\epsilon )- \sum _{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha }\varphi (\zeta _{i})-c_{0} \mu _{1} \Biggr], $$
(3.4)

where

$$ \mu _{1}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -1}-\sum _{i=1}^{n-2}\nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -1}, \qquad \mu _{2}= \bigl(\log (1+\epsilon )\bigr)^{ \gamma -2}-\sum_{i=1}^{n-2} \nu _{i}\bigl(\log (\zeta _{i})\bigr)^{\gamma -2}. $$

In view of the boundary condition \({}_{\mathrm{H}}D^{1,1}x(e)=\sum_{i=1}^{n-2}\sigma _{i} \, {}_{\mathrm{H}}D^{1,1}x(\zeta _{i})\) and from equations (3.3) and (3.4), we have

$$ c_{0}=\frac{1}{(\gamma -1)\delta _{1}} \Biggl[-(\gamma -2)c_{1}\delta _{2}+ {}_{\mathrm{H}}I^{ \alpha -1} \varphi (e)- \sum_{i=1}^{n-2}\sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1} \varphi (\zeta _{i}) \Biggr], $$
(3.5)

where

$$ \delta _{1}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -2},\qquad \delta _{2}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -3}. $$

By using (3.5) in equation (3.4), we have

$$\begin{aligned} c_{1} =&\frac{1}{\lambda } \Biggl[(\gamma -1)\delta _{1} \Biggl[ {}_{\mathrm{H}}I^{ \alpha }\varphi (1+ \epsilon )-\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha } \varphi (\zeta _{i}) \Biggr] \\ &{} - \mu _{1} \Biggl[ {}_{\mathrm{H}}I^{ \alpha -1}\varphi (e)- \sum_{i=1}^{n-2}\sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1} \varphi (\zeta _{i}) \Biggr] \Biggr], \end{aligned}$$

where

$$ \lambda =(\gamma -1)\delta _{1}\mu _{2}-(\gamma -2) \delta _{2}\mu _{1}, \quad \text{with } \lambda \neq 0. $$

By substituting the value of \(c_{1}\) into (3.5), we have

$$\begin{aligned} c_{0}={}&\frac{1}{\lambda } \Biggl[-(\gamma -2)\delta _{2} \Biggl[ {}_{\mathrm{H}}I^{ \alpha }\varphi (1+ \epsilon )-\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha } \varphi (\zeta _{i}) \Biggr] \\ &{} + \mu _{2} \Biggl[ {}_{\mathrm{H}}I^{ \alpha -1}\varphi (e)- \sum_{i=1}^{n-2}\sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1} \varphi (\zeta _{i}) \Biggr] \Biggr]. \end{aligned}$$

Now, substituting the values of \(c_{0}\) and \(c_{1}\) in (3.2), we obtain the solution of problem (3.1). □

Next, we present the existence and uniqueness of solutions for Hilfer–Hadamard-type fractional differential equation (1.1). For that, suppose that

$$ K=C\bigl([1,e],\mathbb{R}\bigr) $$
(3.6)

is a Banach space of all continuous functions from \([1,e]\) into \(\mathbb{R}\) equipped with the norm \(\| x\|=\sup_{t\in J}|x(t)|\). From Lemma 3.1, we get an operator \(\rho :K\rightarrow K\) defined as

$$\begin{aligned} (\rho x) (t)={}&{-} {}_{\mathrm{H}}I^{\alpha }f \bigl(\tau ,x(\tau )\bigr) (t) \\ &{} + \frac{(\gamma -1)\delta _{1}(\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha }f \bigl(\tau ,x(\tau )\bigr) (1+\epsilon ) \\ &{} -\sum_{i=1}^{n-2}\nu _{i}\, {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau ) \bigr) ( \zeta _{i}) \Biggr] \\ &{} + \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f \bigl(\tau ,x(\tau )\bigr) (e) \\ &{} -\sum_{i=1}^{n-2} \sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr], \quad \text{with } \lambda \neq 0. \end{aligned}$$
(3.7)

It must be noticed that problem (1.1) has a solution if and only if operator ρ has a fixed point. The results of existence and uniqueness are based on the Banach contraction mapping principle.

Theorem 3.2

Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption

(\(Q_{1}\)):

there exists a constant \(C > 0 \) such that \(| f(t,x)-f(t,y)|\leq C| x-y|\) for each \(t\in J\) and \(x,y\in \mathbb{R}\). If Φ is such that \(C\Phi <1\), where

$$\begin{aligned} \Phi =& \Biggl\{ \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert + ( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &{}\times \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i})\bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr\} , \end{aligned}$$
(3.8)

then the boundary value problem (1.1) has a unique solution on J.

Proof

We are using Banach contraction mapping principle to transform the boundary value problem (1.1) into a fixed point problem \(x=\rho x\), where the operator ρ is defined by (3.7). We will show that ρ has a fixed point, which is a unique solution of problem (1.1).

We put \(\sup |f(\tau ,0)|= p <\infty \) and choose

$$ r\geq \frac{\Phi P}{1-C\Phi }. $$
(3.9)

Now, assume that \(B_{r}=\{x\in K:| x|\leq r\}\). We will show that \(\rho B_{r}\subset B_{r}\).

For any \(x\in B_{r}\), we have

$$\begin{aligned} \Vert \rho x \Vert =&\sup_{t\in J} \Biggl\{ \Biggl\vert - {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x( \tau )\bigr) (t) \\ &{} + \frac{(\gamma -1)\delta _{1}(\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha }f \bigl(\tau ,x(\tau )\bigr) (1+\epsilon ) \\ &{} -\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha }f\bigl( \tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \\ &{} + \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f \bigl(\tau ,x(\tau )\bigr) (e) \\ &{} -\sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1}f\bigl( \tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \Biggr\vert \Biggr\} \\ \leq &\sup_{t\in J} \Biggl\{ {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl( \tau ,x(\tau )\bigr) \bigr\vert (t) \\ &{} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert (\log t)^{\gamma -2}+( \vert \gamma -2 \vert ) \vert \delta _{2} \vert (\log t)^{\gamma -1}}{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (1+\epsilon ) \\ &{} +\sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \bigl\vert f \bigl( \tau ,x(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert (\log t)^{\gamma -1}+ \vert \mu _{1} \vert (\log t)^{\gamma -2}}{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (e) \\ &{} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \Biggr\} \\ \leq& {}_{\mathrm{H}}I^{\alpha } \bigl( \bigl\vert f\bigl( \tau ,x(\tau )\bigr)-f( \tau ,0) \bigr\vert + \bigl\vert f(\tau ,0) \bigr\vert \bigr) (e) \\ &{} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl( \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f(\tau ,0) \bigr\vert + \bigl\vert f(\tau ,0) \bigr\vert \bigr) (1+\epsilon ) \\ &{} +\sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \bigl( \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f(\tau ,0) \bigr\vert + \bigl\vert f(\tau ,0) \bigr\vert \bigr) (\zeta _{i}) \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl( \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f(\tau ,0) \bigr\vert + \bigl\vert f(\tau ,0) \bigr\vert \bigr) (e) \\ &{} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha -1} \bigl( \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f(\tau ,0) \bigr\vert + \bigl\vert f( \tau ,0) \bigr\vert \bigr) ( \zeta _{i}) \Biggr] \\ \leq& (Cr+P) \Biggl\{ \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &{}\times \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i})\bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr\} \\ =&(Cr+P)\Phi \leq r. \end{aligned}$$
(3.10)

Thus, we have shown that \(\rho B_{r}\subset B_{r}\).

Now, for \(x,y\in K\) and \(t\in J\), we have

$$\begin{aligned}& \bigl\vert (\rho x) (t)-(\rho y) (t)\bigr\vert \\& \quad = \Biggl\vert - {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl( \tau ,x(\tau )\bigr)-f\bigl( \tau ,y(\tau )\bigr) \bigr) (t) \\& \qquad {} + \frac{(\gamma -1)\delta _{1}(\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (1+ \epsilon ) \\& \qquad {} -\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl(\tau ,x( \tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl(f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (e) \\& \qquad {} -\sum_{i=1}^{n-2} \sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1} \bigl(f\bigl(\tau ,x( \tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (\zeta _{i}) \Biggr] \Biggr\vert \\& \quad \leq {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl( \tau ,y(\tau )\bigr) \bigr\vert (t) \\& \qquad {} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert (\log t)^{\gamma -2} +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert (\log t)^{\gamma -1}}{ \vert \lambda \vert } \\& \qquad {}\times \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (1+ \epsilon ) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \bigl\vert f \bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{ \vert \mu _{2} \vert (\log t)^{\gamma -1}+ \vert \mu _{1} \vert (\log t)^{\gamma -2}}{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (e) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert ( \zeta _{i}) \Biggr] \\& \quad \leq C \Vert x-y \Vert \Biggl\{ \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert + ( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\& \qquad {}\times\Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i})\bigr)^{\alpha } \Biggr] \\& \qquad {} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr\} \\& \quad =C \Vert x-y \Vert \Phi . \end{aligned}$$
(3.11)

Therefore, it has been shown that \(\|(\rho x)(t)-(\rho y)(t)\|\leq C \Phi \| x-y\|\), where \(C \Phi <1\). Hence, ρ is a contraction. Thus, by Banach contraction mapping principle, problem (1.1) has a unique solution. □

Theorem 3.3

Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption

(\(Q_{2}\)):

\(|f(t,x)-f(t,y) |\leq \varphi (t) (|x-y|/(P^{*}+|x-y|) )\), \(t\in J\), \(x,y\geq 0\), where \(\varphi :J\rightarrow \mathbb{R}^{+}\) is continuous and a constant \(P^{*}\) is defined by

$$\begin{aligned} P^{*} =& {}_{\mathrm{H}}I^{\alpha } \varphi (e)+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \varphi (1+\epsilon )+ \sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha } \varphi (\zeta _{i}) \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \varphi (e)+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{\alpha -1}\varphi ( \zeta _{i}) \Biggr]. \end{aligned}$$
(3.12)
Then, the boundary value problem (1.1) has a unique solution on J.

Proof

We have the operator \(\rho :K\rightarrow K\) defined by (3.7) and by applying Definition 2.11, we can define a continuous nondecreasing function \(\Psi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) by

$$ \Psi (\phi )=\frac{P^{*}\phi }{P^{*}+\phi },\quad \text{for } \phi \geq 0, $$
(3.13)

where the function Ψ satisfies \(\Psi (0) = 0\) and \(\Psi (\phi )<\phi \) for all \(\phi > 0\).

For any \(x,y \in K\) and for each \(t\in J\), we have

$$\begin{aligned}& \bigl| (\rho x) (t)-(\rho y) (t)\bigr| \\& \quad = \Biggl\vert - {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl( \tau ,x(\tau )\bigr)-f\bigl( \tau ,y(\tau )\bigr) \bigr) (t) \\& \qquad {} + \frac{(\gamma -1)\delta _{1}(\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (1+ \epsilon ) \\& \qquad {} -\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha } \bigl(f\bigl(\tau ,x( \tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl(f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (e) \\& \qquad {} -\sum_{i=1}^{n-2} \sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1} \bigl(f\bigl(\tau ,x( \tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr) (\zeta _{i}) \Biggr] \Biggr\vert \\& \quad \leq {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl( \tau ,y(\tau )\bigr) \bigr\vert (t) \\& \qquad {} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert (\log t)^{\gamma -2}+( \vert \gamma -2 \vert ) \vert \delta _{2} \vert (\log t)^{\gamma -1}}{ \vert \lambda \vert } \\& \qquad {}\times\Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (1+ \epsilon ) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \bigl\vert f \bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{ \vert \mu _{2} \vert (\log t)^{\gamma -1}+ \vert \mu _{1} \vert (\log t)^{\gamma -2}}{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert (e) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr)-f\bigl(\tau ,y(\tau )\bigr) \bigr\vert ( \zeta _{i}) \Biggr] \\& \quad \leq {}_{\mathrm{H}}I^{\alpha } \biggl(\varphi (\tau ) \frac{ \vert x(\tau )-y(\tau ) \vert }{P^{*}+ \vert x(\tau )-y(\tau ) \vert } \biggr) (e) \\& \qquad {} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \biggl(\varphi (\tau ) \frac{ \vert x(\tau )-y(\tau ) \vert }{P^{*}+ \vert x(\tau )-y(\tau ) \vert } \biggr) (1+ \epsilon ) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \biggl(\varphi (\tau ) \frac{ \vert x(\tau )-y(\tau ) \vert }{P^{*}+ \vert x(\tau )-y(\tau ) \vert } \biggr) (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \biggl(\varphi (\tau ) \frac{ \vert x(\tau )-y(\tau ) \vert }{P^{*}+ \vert x(\tau )-y(\tau ) \vert } \biggr) (e) \\& \qquad {} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha -1} \biggl( \varphi (\tau ) \frac{ \vert x(\tau )-y(\tau ) \vert }{P^{*}+ \vert x(\tau )-y(\tau ) \vert } \biggr) (\zeta _{i}) \Biggr] \\& \quad \leq \frac{\Psi ( \Vert x-y \Vert )}{P^{*}} \Biggl\{ {}_{\mathrm{H}}I^{ \alpha } \varphi (e)+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \\& \qquad {}\times\Biggl[ {}_{\mathrm{H}}I^{\alpha } \varphi (1+\epsilon )+ \sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha } \varphi (\zeta _{i}) \Biggr] \\& \qquad {} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \varphi (e)+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{\alpha -1}\varphi ( \zeta _{i}) \Biggr] \Biggr\} \\& \quad =\Psi \bigl( \Vert x-y \Vert \bigr), \end{aligned}$$
(3.14)

which implies that \(\|\rho x-\rho y\|\leq \Psi (\|x-y\|)\). Then, the operator ρ is a nonlinear contraction. Thus, by Lemma 2.12 (Banach contraction mapping principle) the operator ρ has a unique fixed point, which is the unique solution of problem (1.1). □

Next, we will give the existence result by using Theorem 2.13 (Krasnoselskii’s fixed point theorem).

Theorem 3.4

Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption \((Q_{1})\). In addition, assume that

$$ (Q_{3}) \quad \bigl\vert f(t,x) \bigr\vert \leq g(t),\quad \textit{for } (t,x)\in J\times \mathbb{R} \textit{ and } g\in C\bigl([1,e], \mathbb{R}^{+}\bigr). $$

If

$$ {C} {\Gamma (\alpha +1)}< 1, $$
(3.15)

then the boundary value problem (1.1) has at least one solution on J.

Proof

We put \(\sup_{t\in J}|g(t)|=\|g\|\) and choose a suitable constant such that

$$ \hat{r}\geq \Vert g \Vert \Phi , $$
(3.16)

where Φ is defined by (3.8). Moreover, we consider the operators \(\mathscr{F}\) and \(\mathscr{G}\) on \(B_{\hat{r}}=\{x\in K:\|x\|\leq \hat{r}\}\) defined as

$$\begin{aligned}& \begin{aligned} (\mathscr{F}x) (t)={}& \frac{(\gamma -1)\delta _{1}(\log t)^{\gamma -2}-(\gamma -2)\delta _{2}(\log t)^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau )\bigr) (1+\epsilon ) \\ &{}-\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{ \alpha }f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \\ &{}+ \frac{\mu _{2}(\log t)^{\gamma -1}-\mu _{1}(\log t)^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f \bigl(\tau ,x(\tau )\bigr) (e) \\ &{}-\sum_{i=1}^{n-2} \sigma _{i} \, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr],\quad t\in J; \end{aligned} \\& (\mathscr{G}x) (t)= - {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau ) \bigr) (t),\quad t\in J. \end{aligned}$$
(3.17)

For any \(x,y\in B_{\hat{r}}\), we have

$$\begin{aligned} \Vert \mathscr{F}x+\mathscr{G}x \Vert \leq& \Vert g \Vert \Biggl( \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &{}\times \Biggl[ \bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum_{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i}) \bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr) \\ =& \Vert g \Vert \Phi \leq \hat{r}, \end{aligned}$$
(3.18)

which implies that \(\mathscr{F}x+\mathscr{G}x\in B_{\hat{r}}\). It follows from assumption \((Q_{1})\), together with (3.15), that \(\mathscr{G}\) is a contraction. Furthermore, it is easy to show that the operator \(\mathscr{F}\) is continuous. Moreover,

$$\begin{aligned} \Vert \mathscr{F}x \Vert \leq& \Vert g \Vert \Biggl( \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+ \sum_{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i}) \bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr). \end{aligned}$$
(3.19)

Hence, \(\mathscr{F}\) is uniformly bounded on \(B_{\hat{r}}\).

Next, we prove that the operator \(\mathscr{F}\) is compact. For that, we put \(\sup_{(t,x)\in J\times B_{\hat{r}}}|f(t,x)|=\bar{p}<\infty \).

Consequently, for \(t_{1},t_{2}\in J\), we get

$$\begin{aligned} & \bigl\vert (\mathscr{F}x) (t_{1})-(\mathscr{F}x) (t_{2}) \bigr\vert \\ &\quad = \Biggl\vert \Biggl\{ \frac{(\gamma -1)\delta _{1}(\log t_{1})^{\gamma -2}-(\gamma -2)\delta _{2}(\log t_{1})^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha }f \bigl(\tau ,x(\tau )\bigr) (1+\epsilon ) \\ &\qquad {} -\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x( \tau )\bigr) (\zeta _{i}) \Biggr] \\ &\qquad {} + \frac{\mu _{2}(\log t_{1})^{\gamma -1}-\mu _{1}(\log t_{1})^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f\bigl( \tau ,x(\tau )\bigr) (e) \\ &\qquad {} -\sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \Biggr\} \\ &\qquad {} - \Biggl\{ \frac{(\gamma -1)\delta _{1}(\log t_{2})^{\gamma -2}-(\gamma -2)\delta _{2}(\log t_{2})^{\gamma -1}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha }f \bigl(\tau ,x(\tau )\bigr) (1+\epsilon ) \\ &\qquad {} -\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{\alpha }f\bigl( \tau ,x(\tau )\bigr) (\zeta _{i}) \Biggr] \\ &\qquad {} + \frac{\mu _{2}(\log t_{2})^{\gamma -1}-\mu _{1}(\log t_{2})^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f\bigl( \tau ,x(\tau )\bigr) (e) \\ &\qquad {} -\sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \Biggr\} \Biggr\vert \\ &\quad \leq \bar{p} \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert \vert (\log t_{2})^{\gamma -2}-\log t_{1})^{\gamma -2} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert \vert (\log t_{2})^{\gamma -1}-(\log t_{1})^{\gamma -1} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &\qquad {} \times \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl( \log (\zeta _{i})\bigr)^{\alpha } \Biggr] \\ &\qquad {} +\bar{p} \frac{ \vert \mu _{2} \vert \vert (\log t_{2})^{\gamma -1}-\log t_{1})^{\gamma -1} \vert + \vert \mu _{1} \vert \vert (\log t_{2})^{\gamma -2}-(\log t_{1})^{\gamma -2} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \\ & \qquad {}\times \Biggl[1+\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl( \log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr], \end{aligned}$$

which is independent of x and tends to zero as \(t_{2}\rightarrow t_{1}\). Thus, \(\mathscr{F}\) is equicontinuous. Hence, \(\mathscr{F}\) is relatively compact on \(B_{\hat{r}}\). Therefore, by the Arzelà–Ascoli theorem, \(\mathscr{F}\) is compact on \(B_{\hat{r}}\). Thus, by Theorem 2.13, the boundary value problem (1.1) has at least one solution on J. □

Now, the final existence result is based on Theorem 2.14 (nonlinear alternative for single-valued maps).

Theorem 3.5

Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function, and assume that:

(\(Q_{4}\)):

there exists a continuous nondecreasing function \(\vartheta : \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\backslash \{0 \}\) such that

$$ \bigl\vert f(t,x) \bigr\vert \leq q(t)\vartheta \bigl( \vert x \vert \bigr)\quad \textit{for each } (t,x)\in J \times \mathbb{R}, $$
(3.20)

where \(q\in C([1,e],\mathbb{R}^{+})\) is a function;

(\(Q_{5}\)):

there exists a constant \(L>0\) such that

$$ \frac{L}{ \Vert q \Vert \vartheta (L)\Phi }>1, $$
(3.21)

where Φ is defined by (3.8). Then, the boundary value problem (1.1) has at least one solution on J.

Proof

We have the operator ρ defined by (3.7). Firstly, we will show that ρ maps bounded sets (balls) into bounded sets in K. For that, let be a positive number, and \(B_{\bar{r}}=\{x\in K :\|x\|\leq \bar{r}\}\) be a bounded ball in K, where K is defined by (3.6). For \(t\in J\), we have

$$\begin{aligned} \bigl\vert \rho x(t) \bigr\vert \leq& {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (e) \\ &{} + \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (1+\epsilon ) \\ &{} + \sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{\alpha } \bigl\vert f \bigl(\tau ,x(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \bigl\vert f\bigl( \tau ,x(\tau )\bigr) \bigr\vert (e) \\ &{} +\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{\alpha -1} \bigl\vert f\bigl(\tau ,x(\tau )\bigr) \bigr\vert (\zeta _{i}) \Biggr] \\ \leq& \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr) \frac{1}{\Gamma (\alpha +1)} \\ &{} + \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr) \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+ \sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl( \log (\zeta _{i})\bigr)^{\alpha } \Biggr] \\ &{} + \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr) \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \\ \leq& \Vert q \Vert \vartheta (\bar{r}) \Biggl\{ \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &{}\times\Biggl[\bigl(\log (1+\epsilon ) \bigr)^{\alpha }+\sum_{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i}) \bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr\} \\ :=&C_{1}, \end{aligned}$$
(3.22)

which implies that \(\|\rho x\|\leq C_{1}\).

Now, we will show that ρ maps bounded sets into equicontinuous sets of K. For that, let \(\sup_{(t,x)\in J\times B_{\bar{r}}}|f(t,x)|=p^{\star }<\infty \), where \(\omega _{1}, \omega _{2}\in J\), with \(\omega _{1}<\omega _{2}\) and \(x\in B_{\bar{r}}\). Hence, we have

$$\begin{aligned}& \bigl\vert (\rho x) (\omega _{1})-(\rho x) (\omega _{2}) \bigr\vert \\ & \quad = \Biggl\vert \Biggl\{ - {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x( \tau )\bigr) (\omega _{1})+ \frac{(\gamma -1)\delta _{1}(\log \omega _{1})^{\gamma -2}-(\gamma -2) \delta _{2}(\log \omega _{1})^{\gamma -1}}{\lambda } \\ & \qquad {} \times \Biggl[ {}_{\mathrm{H}}I^{ \alpha }f\bigl(\tau ,x(\tau ) \bigr) (1+\epsilon )-\sum_{i=1}^{n-2}\nu _{i} \, {}_{\mathrm{H}}I^{ \alpha }f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \\ & \qquad {} + \frac{\mu _{2}(\log \omega _{1})^{\gamma -1}-\mu _{1}(\log \omega _{1})^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f\bigl( \tau ,x(\tau )\bigr) (e) \\ & \qquad {} -\sum_{i=1}^{n-2}\sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x( \tau ) \bigr) (\zeta _{i}) \Biggr] \Biggr\} \\ & \qquad {} - \Biggl\{ - {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau )\bigr) ( \omega _{2})+ \frac{(\gamma -1)\delta _{1}(\log \omega _{2})^{\gamma -2}-(\gamma -2)\delta _{2}(\log \omega _{2})^{\gamma -1}}{\lambda } \\ & \qquad {} \times \Biggl[ {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau )\bigr) (1+\epsilon )- \sum_{i=1}^{n-2}\nu _{i}\, {}_{\mathrm{H}}I^{\alpha }f\bigl(\tau ,x(\tau ) \bigr) (\zeta _{i}) \Biggr] \\ & \qquad {} + \frac{\mu _{2}(\log t_{2})^{\gamma -1}-\mu _{1}(\log t_{2})^{\gamma -2}}{\lambda } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1}f\bigl( \tau ,x(\tau )\bigr) (e) \\ & \qquad {} -\sum_{i=1}^{n-2}\sigma _{i}\, {}_{\mathrm{H}}I^{\alpha -1}f\bigl(\tau ,x(\tau ) \bigr) ( \zeta _{i}) \Biggr] \Biggr\} \Biggr\vert \\ & \quad \leq p^{\star } \frac{ \vert (\log \omega _{2})^{\alpha }-\log \omega _{1})^{\alpha } \vert }{\Gamma (\alpha +1)} \\ & \qquad {} + p^{\star } \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert \vert (\log \omega _{2})^{\gamma -2}-\log \omega _{1})^{\gamma -2} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert \vert (\log \omega _{2})^{\gamma -1}-(\log \omega _{1})^{\gamma -1} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ & \qquad {} \times \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log (\zeta _{i})\bigr)^{\alpha } \Biggr] \\ & \qquad {} +p^{\star } \frac{ \vert \mu _{2} \vert \vert (\log \omega _{2})^{\gamma -1}-\log \omega _{1})^{\gamma -1} \vert + \vert \mu _{1} \vert \vert (\log \omega _{2})^{\gamma -2}-(\log \omega _{1})^{\gamma -2} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \\ & \qquad {} \times \Biggl[1+\sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr]. \end{aligned}$$

Clearly, as \(\omega _{2}\rightarrow \omega _{1}\), the right-hand side of the latter inequality tends to zero, which happens independently of \(x\in B_{\bar{r}}\). Thus, by the Arzelà–Ascoli theorem, it follows that \(\rho : K\rightarrow K\) is completely continuous.

Finally, let x be a solution. So, for \(t\in J\), following similar computations as in the first step, we have

$$\begin{aligned} \Vert x \Vert \leq& \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr)\frac{1}{\Gamma (\alpha +1)} \\ &{} + \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr) \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+ \sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl( \log (\zeta _{i})\bigr)^{\alpha } \Biggr] \\ &{} + \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr) \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \\ =& \Vert q \Vert \vartheta \bigl( \Vert x \Vert \bigr)\Phi . \end{aligned}$$

Thus, we have

$$ \frac{ \Vert x \Vert }{ \Vert q \Vert \vartheta ( \Vert x \Vert )\Phi }\leq 1. $$

In view of \((Q_{5})\), there exists L such that \(\| x\|\neq L\). Let us set

$$ V=\bigl\{ x \in K: \Vert x \Vert < L\bigr\} . $$

Note that the operator \(\rho :\overline{V}\rightarrow K\) is continuous and completely continuous. From the choice of V, there is no \(x\in \partial V\) such that \(x=\bar{\lambda }\rho x\) for some \(\bar{\lambda }\in (0,1)\). Thus, by Theorem 2.14, the operator ρ has a fixed point in , which is a solution of the boundary value problem (1.1). □

Example

Example 4.1

Consider the following boundary value problem for Hilfer–Hadamard-type fractional differential equation:

$$ \textstyle\begin{cases} {}_{\mathrm{H}}D^{3/2,1/2}x(t)+f(t,x(t))=0, \quad t\in J:=(1,e], \\ x(1.3)=\frac{1}{2}x(3/2)-\frac{3}{4}x(7/4), \\ {}_{\mathrm{H}}D^{1,1}x(e)=\frac{2}{3}\, {}_{\mathrm{H}}D^{1,1}x(3/2)+\frac{4}{3} \, {}_{\mathrm{H}}D^{1,1}x(7/4). \end{cases} $$
(4.1)

Here, \(\alpha =3/2\), \(\beta =1/2\), \(\gamma =7/4\), \(\nu _{1}=1/2\), \(\nu _{2}=-3/4 \), \(\sigma _{1}=2/3\), \(\sigma _{2}=4/3\), \(\zeta _{1}=3/2\), \(\zeta _{2}=7/4\), \(\epsilon =0.3\), \(1+\epsilon =1.3\), and

$$ f\bigl(t,x(t)\bigr)=\frac{(\sqrt{t}+\log t^{2})}{2e^{t}(3+t)^{2}} \biggl( \frac{ \vert x(t) \vert }{2+ \vert x(t) \vert } \biggr). $$

Clearly,

$$ \bigl\vert f(t,x)-f(t,y) \bigr\vert \leq \frac{3}{64e}\bigl( \vert x-y \vert \bigr). $$

Hence, (\(Q_{1}\)) is satisfied with \(C=\frac{3}{64e}\). We can show that

$$\begin{aligned} & \mu _{1}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -1}-\sum _{i=1}^{n-2} \nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -1}\approx 0.59779, \\ & \mu _{2}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -2}-\sum _{i=1}^{n-2} \nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -2}\approx 1.63780, \\ & \delta _{1}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -2} \approx -1.37703, \\ & \delta _{2}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -3} \approx -3.81518, \\ & \lambda =(\gamma -1)\delta _{1}\mu _{2}-(\gamma -2) \delta _{2}\mu _{1}\approx -2.26164, \\ & \begin{aligned} \Phi ={}&\frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \Biggl[\bigl(\log (1+ \epsilon )\bigr)^{\alpha }+\sum_{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i}) \bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \\ \approx{}& 3.835201, \end{aligned} \\ & C\Phi =\frac{3}{64e}(3.835201)\approx 0.06613554378< 1. \end{aligned}$$

Therefore, by Theorem 3.2, the boundary value problem (4.1) has a unique solution on J.

Example 4.2

Consider the following boundary value problem for Hilfer–Hadamard-type fractional differential equation:

$$ \textstyle\begin{cases} {}_{\mathrm{H}}D^{3/2,2/3}x(t)+f(t,x(t))=0, \quad t\in J:=(1,e], \\ x(1.5)=2x(4/3)-\frac{1}{2}x(2)+\frac{5}{3}x(9/7), \\ {}_{\mathrm{H}}D^{1,1}x(e)=- {}_{\mathrm{H}}D^{1,1}x(4/3)+3D^{1,1}x(2)-\frac{11}{3} \, {}_{\mathrm{H}}D^{1,1}x(9/7). \end{cases} $$
(4.2)

Here, \(\alpha =3/2\), \(\beta =2/3\), \(\gamma =11/6\), \(\nu _{1}=2\), \(\nu _{2}=-1/2\), \(\nu _{3}=5/3\), \(\sigma _{1}=-1\), \(\sigma _{2}=3\), \(\sigma _{3}=-11/3\), \(\zeta _{1}=4/3\), \(\zeta _{2}=2\), \(\zeta _{2}=9/7\), \(\epsilon =0.5\), \(1+ \epsilon =1.5\), and

$$ f\bigl(t,x(t)\bigr)=\frac{(1+\log t)}{(t+1)^{2}} \biggl( \frac{ \vert x(t) \vert +1}{3+ \vert x(t) \vert } \biggr). $$

Clearly,

$$\begin{aligned} \bigl\vert f(t,x) \bigr\vert &\leq \biggl\vert \frac{(1+\log t)}{(t+1)^{2}} \biggl( \frac{ \vert x(t) \vert +1}{3+ \vert x(t) \vert } \biggr) \biggr\vert \\ &\leq \biggl\vert (1+\log t) \biggl(\frac{ \vert x(t) \vert +1}{12} \biggr) \biggr\vert . \end{aligned}$$

We choose \(q(t)=1+\log t\) and \(\vartheta (|x|)=(| x(t)|+1)/12\). Then, we can show that

$$\begin{aligned} & \mu _{1}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -1}-\sum _{i=1}^{n-2} \nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -1}\approx -0.395713, \\ & \mu _{2}=\bigl(\log (1+\epsilon )\bigr)^{\gamma -2}-\sum _{i=1}^{n-2} \nu _{i}\bigl( \log (\zeta _{i})\bigr)^{\gamma -2}\approx -2.865742, \\ & \delta _{1}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -2} \approx 3.65750, \\ & \delta _{2}=1-\sum_{i=1}^{n-2} \sigma _{i}\bigl(\log (\zeta _{i})\bigr)^{ \gamma -3} \approx 19.04369, \\ & \lambda =(\gamma -1)\delta _{1}\mu _{2}-(\gamma -2) \delta _{2}\mu _{1}\approx -9.990516, \\ & \begin{aligned} \Phi ={}&\frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \Biggl[\bigl(\log (1+ \epsilon )\bigr)^{\alpha }+\sum_{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i}) \bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \\ \approx{}& 3.414437455. \end{aligned} \end{aligned}$$

Now, by (\(Q_{5}\)) we have

$$ \frac{L}{(2)((L+1)/12)(3.414437455)}> 1. $$

Hence, \(L>1.320578171\). Therefore, by Theorem 3.5, the boundary value problem (4.2) has at least one solution on J.

Availability of data and materials

No data were used to support this study.

References

  1. 1.

    Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11(5), 686 (2019)

    MATH  Article  Google Scholar 

  2. 2.

    Afshari, H., Atapour, M., Karapinar, E.: A discussion on a generalized Geraghty multi-valued mappings and applications. Adv. Differ. Equ. 2020, 356 (2020). https://doi.org/10.1186/s13662-020-02819-2

    MathSciNet  Article  Google Scholar 

  3. 3.

    Afshari, H., Jarad, F., Abdeljawad, T.: On a new fixed point theorem with an application on a coupled system of fractional differential equations. Adv. Differ. Equ. 2020, 461 (2020). https://doi.org/10.1186/s13662-020-02926-0

    MathSciNet  Article  Google Scholar 

  4. 4.

    Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 286 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Afshari, H., Karapinar, E.: A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces. Adv. Differ. Equ. 2020, 616 (2020). https://doi.org/10.1186/s13662-020-03076-z

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ahmad, B., Ntouyas, S.K.: An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. Abstr. Appl. Anal. 2014, Article ID 705809 (2014)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Ahmad, B., Ntouyas, S.K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 348–360 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Ahmad, B., Ntouyas, S.K.: On Hadamard fractional integro-differential boundary value problems. J. Appl. Math. Comput. 47, 119–131 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Ahmad, B., Ntouyas, S.K., Alsaedi, A.: A study of nonlinear fractional differential equations of arbitrary order with Riemann–Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ahmad, B., Ntouyas, S.K., Alsaedi, A.: New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions. Bound. Value Probl. 2013, 275 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019)

    Article  Google Scholar 

  12. 12.

    Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., Theory Methods Appl. 72(2), 916–924 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 71(7–8), 2391–2396 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Caponetto, R., Dongola, G., Fortuna, I., Petras, I.: Fractional Order Systems. Modeling and Control Applications. World Scientific Series on Nonlinear Science Series A, vol. 72. World Scientific, Singapore (2010)

    Google Scholar 

  16. 16.

    Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Academic Press, San Diego (1973)

    Google Scholar 

  17. 17.

    Gambo, Y.Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014(1), 10 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Vienna (1997)

    Google Scholar 

  19. 19.

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

    Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 21.

    Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284(1–2), 399–408 (2002)

    Article  Google Scholar 

  22. 22.

    Hilfer, R.: Threefold introduction to fractional derivatives. In: Anomalous Transport: Foundations and Applications, pp. 17–73 (2008)

    Google Scholar 

  23. 23.

    Karapinar, E., Binh, H.D., Luc, N.H., Can, N.H.: On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems. Adv. Differ. Equ. 2021(1), 70 (2021)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(5), 444 (2019)

    Article  Google Scholar 

  25. 25.

    Keyantuo, V., Lizama, C., Warma, M.: Asymptotic behavior of fractional order semilinear evolution equations. Differ. Integral Equ. 26(7/8), 757–780 (2013)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  28. 28.

    Kiryakova, V.S.: Generalized Fractional Calculus and Applications. CRC Press, Boca Raton (1993)

    Google Scholar 

  29. 29.

    Krasnosel’skii, M.A.: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10(1), 123–127 (1955)

    MathSciNet  Google Scholar 

  30. 30.

    Li, C.-G., Kostic, M., Li, M., Piskarev, S.: On a class of time-fractional differential equations. Fract. Calc. Appl. Anal. 15(4), 639–668 (2012)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Lizama, C.: Solutions of two-term fractional order differential equations with nonlocal initial conditions. Electron. J. Qual. Theory Differ. Equ. 2012, 82 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010)

    Google Scholar 

  33. 33.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  34. 34.

    Qassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, Article ID 391062 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581 (1997)

    MathSciNet  Google Scholar 

  36. 36.

    Salamooni, A.Y.A., Pawar, D.D.: Unique positive solution for nonlinear Caputo-type fractional q-difference equations with nonlocal and Stieltjes integral boundary conditions. Fract. Differ. Calc. 9(2), 295–307 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Salamooni, A.Y.A., Pawar, D.D.: Existence and uniqueness of generalised fractional Cauchy-type problem. Univers. J. Math. Appl. 3(3), 121–128 (2020)

    Google Scholar 

  38. 38.

    Salamooni, A.Y.A., Pawar, D.D.: Existence and uniqueness of boundary value problems for Hilfer–Hadamard-type fractional differential equations. Ganita 70(2), 1–16 (2020)

    MathSciNet  Google Scholar 

  39. 39.

    Salamooni, A.Y.A., Pawar, D.D.: Existence and stability results for Hilfer–Katugampola-type fractional implicit differential equations with nonlocal conditions. J. Nonlinear Sci. Appl. 14(3), 124–138 (2021)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993). Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)

    Google Scholar 

  41. 41.

    Sevinik Adigüzel, R., Aksoy, Ü., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652

    Article  Google Scholar 

  42. 42.

    Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    Google Scholar 

  43. 43.

    Tariboon, J., Ntouyas, S.K., Sudsutad, W.: Nonlocal Hadamard fractional integral conditions for nonlinear Riemann–Liouville fractional differential equations. Bound. Value Probl. 2014(1), 253 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Thabet, S.T.M., Dhakne, M.B.: On boundary value problems of higher order abstract fractional integro-differential equations. Int. J. Nonlinear Anal. Appl. 7(2), 165–184 (2016)

    MATH  Google Scholar 

  45. 45.

    Thiramanus, P., Ntouyas, S.K., Tariboon, J.: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions. Abstr. Appl. Anal. 2014, Article ID 902054 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editor and reviewers for their important remarks and suggestions.

Funding

We declare that funding is not applicable for our paper.

Author information

Affiliations

Authors

Contributions

The authors equally contributed to this manuscript and approved the final version of this paper.

Corresponding author

Correspondence to Ahmad Y. A. Salamooni.

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

Salamooni, A.Y.A., Pawar, D.D. Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations. Adv Differ Equ 2021, 198 (2021). https://doi.org/10.1186/s13662-021-03358-0

Download citation

MSC

  • 34A08
  • 35R11

Keywords

  • Existence
  • Uniqueness
  • Nonlinear boundary value problems
  • Hilfer–Hadamard type
  • Fractional differential equation and fractional calculus
\