Existence of Mild Solutions for the Generalized Anti-Periodic Boundary Value Problem to the Fractional Hybird Differential Equations with p(t)-Laplacian Operator

Abstract

The main objective of this work is to discuss the generalized anti-periodic boundary conditions of the generalized Caputo fractional differential equations with $p\left(t\right)$-Laplacian operators. By applying the Schaefer fixed point theorem, the existence of mild solutions to this problem is obtained, which generalizes and enriches the anti-periodic boundary value problem of Caputo fractional hybrid differential equations. Finally, a numerical example is given to verify our main results. The anti-periodic boundary value condition imparts a form of symmetric inversion with respect to the original state, so it exhibits an anti-symmetric structural feature.

1. Introduction

This paper is concerned with the following generalized anti-periodic boundary value problem (BVP) of generalized Caputo fractional differential equations (FDEs) involving the $p\left(t\right)$-Laplacian operator.

$$\left\{\begin{array}{c}{}^C\mathfrak{D}_{0^{+}}^{\beta ,\chi }{\varphi }_{p\left(t\right)}\left({}^C\mathfrak{D}_{0^{+}}^{\alpha ,\chi }{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}\left(t,x\left(t\right)\right)}}\right)=\mathfrak{g}(t,x\left(t\right)),t\in \left(0,1\right),\hfill \\ \\ {\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=0}=-{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=1},{}^C\mathfrak{D}_{0+}^{\alpha ,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=0}=-{}^C\mathfrak{D}_{0+}^{\alpha ,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=1},\hfill \end{array}\right.$$

(1)

where ${}^C\mathfrak{D}_{0+}^{\varrho ,\chi }$ is the generalized Caputo fractional derivatives of order $\varrho ,\varrho =\alpha$ or $\beta ,$$0<\alpha <1,0<\beta <1,1<\alpha +\beta <2$, $\mathfrak{f}\in C([0,1]\times \mathbb{R},\mathbb{R}\setminus \left\{0\right\})$, $\mathfrak{g}\in C\left(\right[0,1]\times \mathbb{R},\mathbb{R})$, ${\varphi }_{p\left(t\right)}\left(x\right)={\left|x\right|}^{p\left(t\right)-2}x$ is a $p\left(t\right)$-Laplacian operator with $p\in C\left(\right[0,1],\mathbb{R})$, $p\left(t\right)>1$, and $p\left(0\right)=p\left(1\right)$.

Fractional calculus, as an extension of classical calculus theory, has aroused growing interest recently because of its effective application advantages and theoretical value in many fields such as physics, engineering, and biomedicine (see [1,2,3,4]). In these disciplines, the Caputo FDEs are important modeling tools for many practical problems. The Caputo fractional derivative not only preserves the fundamental characteristics of classical derivatives but also brings in the concept of fractional order, enabling the model to reflect the time dependence and historical effects of the system more precisely, which helps to promote the development of fractional calculus theory and provides new ideas and methods for dealing with practical problems in complex systems (see [5,6]).

The BVP of FDEs is one of the central parts of the qualitative theory of fractional differential equations. Since the Langevin equation, which was established by Langevin in 1908 based on Newton’s laws, has strong physical significance, the BVP of Caputor fractional Langevin equations have been widely studied by scholars (see [7,8,9,10,11]). For example, Ahmad, Ntouyas, and Tariboon [7] studied the following BVP of mixed fractional order integral differential equation.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C\mathfrak{D}^{\alpha }\left[{\displaystyle \frac{x\left(t\right)-{\displaystyle \sum _{i=1}^m}{\mathfrak{I}}^{{\beta }_i}{\mathfrak{h}}_i\left(t,x\left(t\right)\right)}{\mathfrak{f}\left(t,x\left(t\right)\right)}}\right]=\mathfrak{g}\left(t,x\left(t\right)\right),t\in \left[0,1\right],\hfill \\ \\ x\left(0\right)=\mu \left(x\right),x\left(1\right)=A,\hfill \end{array}\right.\end{array}$$

(2)

where $1<\alpha \le 2,\mathfrak{f}\in C(\left[0,1\right]\times \mathbb{R},\mathbb{R}\setminus \left\{0\right\}),\mathfrak{g}\in C(\left[0,1\right]\times \mathbb{R},\mathbb{R}),{\mathfrak{h}}_i\in C(\left[0,1\right]\times \mathbb{R},\mathbb{R}),$${\beta }_i>0,i=1, 2, \cdots , m$, $\mu :C\left([0,1],\mathbb{R}\right)\to \mathbb{R}$, $A\in \mathbb{R}$, ${}^C\mathfrak{D}^{\alpha }$ is the Caputo fractional derivatives and ${\mathfrak{I}}^{{\beta }_i}$ are the Riemann-Liouville fractional integral of order ${\beta }_i$. The existence result of problem (2) was proved by employing the fixed point theorem. Moreover, Fazli and Nieto [8] discussed the existence of solutions to the following anti-periodic BVP for the Langevin equations with two different fractional orders by applying certain fixed point theorems.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathfrak{D}}^{\beta }\left({\mathfrak{D}}^{\alpha }+\lambda \right)x\left(t\right)=\mathfrak{f}\left(t,x\left(t\right)\right),t\in (0,1),\hfill \\ x\left(0\right)+x\left(1\right)=0,{\mathfrak{D}}^{\alpha }x\left(0\right)+{\mathfrak{D}}^{\alpha }x\left(1\right)=0,\hfill \\ {\mathfrak{D}}^{2\alpha }x\left(0\right)+{\mathfrak{D}}^{2\alpha }x\left(1\right)=0,\hfill \end{array}\right.\end{array}$$

(3)

where $0<\alpha \le 1 ,1<\beta \le 2,$ and ${\mathfrak{D}}^{\alpha }$ is the Caputo fractional derivatives of order $\alpha ,\lambda \in \mathbb{R}$, $f\in C(\left[0,1\right]\times \mathbb{R},\mathbb{R})$.

On the other hand, with the development of the BVP of FDEs, the case with a variable coefficient Laplacian operator has been paid more attention (see [12,13,14,15,16,17]). The $p\left(t\right)$-Laplacian operator can be used to describe physical phenomena with “point-by-point distinct characteristics”, and it stems from image restoration, elasticity theory, and so on (see [12]). Compared with the conventional p-Laplacian, the $p\left(t\right)$-Laplacian operator has more complex nonlinear characteristics. For example, Tang, Wang, Wang, and Ouyang [14] investigated the following mixed fractional order resonant BVP with $p\left(t\right)$-Laplacian operator.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C\mathfrak{D}_{0^{+}}^{\beta }{\phi }_{p\left(t\right)}\left({\mathfrak{D}}_{0^{+}}^{\alpha }u\left(t\right)\right)=\mathfrak{f}\left(t,u\left(t\right),{\mathfrak{D}}_{0^{+}}^{\alpha }u\left(t\right)\right),t\in [0,T],\hfill \\ \\ {\left.t^{1-\alpha }u\left(t\right)\right|}_{t=0}=0,{\mathfrak{D}}_{0^{+}}^{\alpha }{u\left(t\right)|}_{t=0}={\mathfrak{D}}_{0^{+}}^{\alpha }u\left(T\right),\hfill \end{array}\right.\end{array}$$

(4)

where ${\phi }_{p\left(t\right)}(·)$ is a $p\left(t\right)$–Lapiacian operator with $p\in C^1([0,T],\mathbb{R}),p\left(t\right)>1,p\left(0\right)=p\left(T\right)$, ${}^C\mathfrak{D}_{0^{+}}^{\beta }$ is a Caputo fractional derivative, ${\mathfrak{D}}_{0^{+}}^{\alpha }$ is a Riemann–Liouville fractional derivative. Through the use of the coincidence degree theory, the existence of solutions to problem (4) was established. Although the classical Caputo fractional derivative plays a vital role in solving mathematical problems, it has certain limitations in some models, so the generalized Caputo fractional derivative was founded. The generalized Caputo fractional derivative is an extension of the classical Caputo fractional derivative (see [18,19]). Its main advantage lies in the introduction of a tunable function, which extends the integral kernel of the derivative from a fixed power function kernel ${(t-s)}^{\alpha }$ to a tunable kernel function ${(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }$ that depends on the independent variable. When the kernel function is chosen as $\psi \left(t\right)=t$ or $\psi \left(t\right)=lnt$, the generalized Caputo derivative reduces to the Riemann–Liouville and Hadamard fractional derivatives, respectively. Therefore, the generalized Caputo derivative naturally encompasses and transcends the traditional definition. In the context of solving differential equations, the generalized Caputo derivative offers greater flexibility in the solution space, making it applicable to a broader class of systems. Consequently, it exhibits stronger adaptability and descriptive power in both theoretical analysis and practical applications compared to the classical Caputo derivative (see [18,19]). Therefore, generalized Caputo FDEs have gradually become a research hotspot for scholars (see [20,21,22,23]). Subsequently, Ali, Khan, Kamran, Aloqaily, and Mlaiki [20] dealt with the following BVP to the nonlinear mixed $\psi$–Caputo fractional order Langevin equation.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C\mathfrak{D}_{c^{+}}^{\sigma ,\psi }\left[{}^C\mathfrak{D}_{c^{+}}^{\varsigma ,\psi }\left[{\displaystyle \frac{u\left(\zeta \right)}{\mathfrak{g}(\zeta ,u\left(\zeta \right){,}^C{\mathfrak{D}}_{c^{+}}^{\varsigma ,\psi }u\left(\zeta \right))}}\right]-\mu u\left(\zeta \right)\right]=\mathfrak{f}(\zeta ,u\left(\zeta \right){,}^C{\mathfrak{D}}_{c^{+}}^{\varsigma ,\psi }u\left(\zeta \right)),\zeta \in J=[c,d],\hfill \\ \\ u\left(c\right)=0,{\left.{}^C\mathfrak{D}_{c^{+}}^{\varsigma ,\psi }{\displaystyle \frac{u\left(\zeta \right)}{\mathfrak{g}(\zeta ,u\left(\zeta \right){,}^C{\mathfrak{D}}_{c^{+}}^{\varsigma ,\psi }u\left(\zeta \right))}}\right|}_{\zeta =a}=0,u\left(d\right)=\xi u\left(\delta \right),\delta \in (c,d),\hfill \end{array}\right.\end{array}$$

(5)

where ${}^C\mathfrak{D}_{c^{+}}^{\varpi ,\psi }$ stands for $\psi$–Caputo fractional derivatives of order $\varpi ,\varpi =\sigma$ or $\varsigma ,$$\sigma \in (0,1],$$\varsigma \in (1,2]$, $\mathfrak{f}\in C(J\times \mathbb{R},\times \mathbb{R}),\mathfrak{g}\in C(J\times \mathbb{R}\times \mathbb{R},\mathbb{R}\setminus \left\{0\right\})$. By using some fixed point theorems, the existence and uniqueness of the solutions to the problem (5) were obtained. Moreover, the stability of Ulam–Hyer was also proved.

Inspired by the above works, a natural idea is to consider the existence of solutions for the generalized anti-periodic BVP to the fractional hybrid differential equations with $p\left(t\right)$-Laplacian operator (1). Let us present the innovative points of this article:

•

When $p\left(t\right)=p$, the $p\left(t\right)$-Laplacian operator under study becomes the classical p-Laplacian operator with constant coefficients. Similarly, when $\chi \left(t\right)=t$, the generalized Caputo fractional derivative becomes the classical Caputo derivative. Therefore, the existence results obtained in this paper extend the conclusions corresponding to those in reference [24].

•

Since the $p\left(t\right)$-Laplacian operator is an operator with non-standard growth, when we apply the Schaefer fixed point theorem to prove the existence of mild solutions, we need to construct a bounded convex closed set, which is somewhat difficult.

•

As far as we know, in the study of the BVP of fractional mixed differential equations with $p\left(t\right)$-Laplacian operators, the fractional derivatives usually adopted are Caputo derivatives (see [15]) or Riemann–Liouville derivatives (see [16]), while the application of generalized Caputo derivatives is still rare. Therefore, in a certain sense, this article enriches the research results of this type of issue.

The structure of this paper is organized as follows: Section 1 provides the introduction, which presents the research background and significance. Section 2 summarizes the relevant theorems and lemmas to establish the theoretical foundation for the subsequent analysis. Section 3 first presents the expression of the solution to the problem, and then proves the existence of solutions by applying the Schaefer fixed point theorem. Section 4 verifies the theoretical results through specific examples and includes two illustrative figures to enhance intuitive understanding. Section 5 presents the conclusion, providing a concise summary of the entire work.

2. Materials and Methods

For the sake of the reader’s convenience, we describe some related concepts, lemmas, and symbols that will be used to prove the main conclusions of this paper. Let $\mathcal{C}=C\left(\right[0,1],\mathbb{R})$, whose norm is ${∥u∥}_{\infty }=\underset{\varsigma \in [0,1]}{max}\left|u\left(\varsigma \right)\right|.$ The generalized Caputo fractional integrals and derivatives are shown as follows.

Definition 1 

([18,19]). For every $\varsigma \in [0,1],{\chi }^{\prime }\left(\varsigma \right)\ne 0$, let $\chi :[0,1]\to \mathbb{R}$ be an increasing function. The left-sided generalized Riemann–Liouville fractional integral of order $\alpha >0$ of a function $\mathfrak{f}$ on $[0,1]$ is defined as follows with respect to χ:

$${\mathfrak{I}}_{0^{+}}^{\alpha ,\chi }\mathfrak{f}\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{\chi }^{\prime }\left(s\right){\left(\chi \right(t)-\chi (s\left)\right)}^{\alpha -1}\mathfrak{f}\left(s\right)ds,t\in [0,1].$$

It should be pointed out that the Hadamard fractional integral and the Riemann–Liouville integral can be derived when $\chi \left(t\right)=lnt$ and $\chi \left(t\right)=t$.

Definition 2 

([18,19]). Let $\alpha >0,n=\left[\alpha \right]+1,$ and $n\in \mathbb{N}$. For any $t\in [0,1]$, the left-side generalized Caputo fractional derivative for a function $\mathfrak{f}\in C^n([0,1],\mathbb{R})$ is defined by

$${}^C\mathfrak{D}_{0^{+}}^{\alpha ,\chi }\mathfrak{f}\left(t\right)={\mathfrak{I}}_{0^{+}}^{n-\alpha ,\chi }{\left({\displaystyle \frac{1}{{\chi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n\mathfrak{f}\left(t\right),$$

where χ is a strictly increasing function.

Lemma 1 

([18,19]). Let $\alpha ,\beta >0,\mathfrak{f}\in {\mathcal{L}}^1([0,1],\mathbb{R})$, then

$${\mathfrak{I}}_{0^{+}}^{\alpha ,\chi }{\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathfrak{f}\left(t\right)={\mathfrak{I}}_{0^{+}}^{\alpha +\beta ,\chi }\mathfrak{f}\left(t\right),t\in [0,1].$$

Lemma 2 

([18,19]). Let $\alpha >0$. The following conclusions hold.

(1) If $\mathfrak{f}\in \mathcal{C}$, it follows

$${}^C\mathfrak{D}_{0^{+}}^{\alpha ,\chi }{\mathfrak{I}}_{0^{+}}^{\alpha ,\chi }\mathfrak{f}\left(t\right)=\mathfrak{f}\left(t\right),t\in [0,1].$$

(2) If $\alpha \in (n-1,n)$ and $\mathfrak{f}\in C^n([0,1],\mathbb{R})$, we have

$${\mathfrak{I}}_{0^{+}}^{\alpha ,\chi }{}^C\mathfrak{D}_{0^{+}}^{\alpha ,\chi }\mathfrak{f}\left(t\right)=\mathfrak{f}\left(t\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{{\left(\frac{1}{{\chi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^k\mathfrak{f}\left(t\right){\mid }_{t=0}}{k!}}{\left[\chi \left(t\right)-\chi \left(0\right)\right]}^k,t\in [0,1].$$

Definition 3 

([25]). For any $(t,x)\in [0,1]\times \mathbb{R},{\varphi }_{p\left(t\right)}\left(x\right)={\left|x\right|}^{p\left(t\right)-2}x$ is a homeomorphic mapping from $\mathbb{R}$ to $\mathbb{R}$ and is strictly monotonically increasing when t is fixed. Moreover, its inverse mapping is defined by ${\varphi }_{p\left(t\right)}^{-1}\left(x\right)=|x{|}^{\frac{2-p\left(t\right)}{p\left(t\right)-1}}x,x\in \mathbb{R}\setminus \left\{0\right\},{\varphi }_{p\left(t\right)}^{-1}\left(0\right)=0,$ which is a continuous mapping and send a bounded set to a bounded set.

Lemma 3 

([26,27]). (Schaefer fixed point theorem) Let X be a Banach space and the operator $\mathcal{T}:X\to X$ be a completely continuous operator, and if the set $\mathrm{\Omega }=\{x\in X|x=\mu \mathcal{T}x,\mu \in (0,1)\}$ is bounded, then the operator $\mathcal{T}$ has at least one fixed point in X.

3. Results

To derive the core conclusion, the following lemma must be proved.

Lemma 4. 

Letting $x\in C^1([0,1],\mathbb{R})$ and $f(t,x)\in C^1([0,1]\times \mathbb{R},\mathbb{R})$ satisfying ${\varphi }_{p\left(t\right)}({}^C\mathfrak{D}_{0+}^{\alpha ,\chi }({\displaystyle \frac{x}{\mathfrak{f}(t,x)}}))$$\in C^1(\mathbb{R},\mathbb{R})$ when t is fixed, if $\mathfrak{h}\in \mathcal{C}$ and x is a solution for (1), then x has the unique expression of solution to the following problem:

$$\begin{array}{c}\hfill {}^C\mathfrak{D}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}({}^C\mathfrak{D}_{0+}^{\alpha ,\chi }({\displaystyle \frac{x(t)}{\mathfrak{f}(t,x(t))}}))=\mathfrak{h}\left(t\right),t\in (0,1),\end{array}$$

(6)

with the boundary conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=0}=-{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=1},\hfill \\ \\ {}^C\mathfrak{D}_{0+}^{\alpha ,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=0}=-{}^C\mathfrak{D}_{0+}^{\alpha ,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=1},\hfill \end{array}\right.\end{array}$$

(7)

which takes the form

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \mathfrak{f}(t,x\left(t\right))J\mathfrak{h}\left(t\right)+\mathfrak{f}(t,x\left(t\right)){\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)+K\mathfrak{h}\left(t\right))\hfill \\ & =& \mathfrak{f}(t,x\left(t\right))J\mathfrak{h}\left(t\right)+{\displaystyle \frac{\mathfrak{f}(t,x(t\left)\right)}{\Gamma \left(\alpha \right)}}{\int }_0^t{\chi }^{\prime }\left(s\right){(\chi \left(t\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\hfill \\ & & \times {\int }_0^s{\chi }^{\prime }\left(\tau \right){(\chi \left(s\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{h}\left(\tau \right)d\tau +K\mathfrak{h}\left(s\right))ds,\hfill \end{array}$$

where the operators $K:\mathcal{C}\to \mathbb{R}$ and $J:\mathcal{C}\to \mathbb{R}$ are given by

$$\begin{array}{ccc}\hfill K\mathfrak{h}& =& {\left.-{\displaystyle \frac{1}{2}}{\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)\right|}_{t=1}=-{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\beta -1}\mathfrak{h}\left(s\right)ds,\hfill \\ \hfill J\mathfrak{h}& =& {\left.-{\displaystyle \frac{1}{2}}{\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)+K\mathfrak{h}\left(t\right))\right|}_{t=1}\hfill \\ & =& -{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\hfill \\ & & \times {\int }_0^s{\chi }^{\prime }\left(\tau \right){(\chi \left(s\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{h}\left(\tau \right)d\tau +K\mathfrak{h}\left(s\right))ds.\hfill \end{array}$$

Proof. 

Acting on both sides of (6) with the operator ${\mathfrak{I}}_{0+}^{\beta ,\chi }$ and combining with Lemma 2 and Definition 3, for the fixed t, we can deduce

$$\begin{array}{c}\hfill {}^C\mathfrak{D}_{0+}^{\alpha ,\chi }\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)={\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)+C_0),C_0\in \mathbb{R}.\end{array}$$

(8)

Form the boundary conditions, it follows

$$\begin{array}{c}\hfill C_0={\left.-{\displaystyle \frac{1}{2}}{\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)\right|}_{t=1}=K\mathfrak{h}.\end{array}$$

Letting the operator ${\mathfrak{I}}_{0+}^{\alpha ,\chi }$ act on both sides of (8), one has

$$\begin{array}{c}\hfill x\left(t\right)=\mathfrak{f}(t,x\left(t\right))C_1+\mathfrak{f}(t,x\left(t\right))[{\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)+K\mathfrak{h}\left(t\right))],C_1\in \mathbb{R}.\end{array}$$

(9)

Then, from the boundary conditions, we can derive

$$\begin{array}{c}\hfill C_1={\left.-{\displaystyle \frac{1}{2}}{\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathfrak{h}\left(t\right)+K\mathfrak{h}\left(t\right))\right|}_{t=1}=J\mathfrak{h}.\end{array}$$

(10)

Substituting $C_1$ into (9), we can obtain the desired conclusions. □

Definition 4. 

Let $\mathfrak{f}\in C([0,1]\times \mathbb{R},\mathbb{R}\setminus \left\{0\right\})$, $\mathfrak{g}\in C\left(\right[0,1]\times \mathbb{R};\mathbb{R})$. A function $x\in \mathcal{C}$ is said to be a mild solution of the boundary value problem (1) when x satisfies the integral equation

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\displaystyle \frac{\mathfrak{f}(t,x(t\left)\right)}{\Gamma \left(\alpha \right)}}{\int }_0^t{\chi }^{\prime }\left(s\right){(\chi \left(t\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left[{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\right.\hfill \\ & & \times {\int }_0^s{\chi }^{\prime }\left(\tau \right){(\chi \left(s\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{g}(\tau ,x\left(\tau \right))d\tau \hfill \\ & & -\left.{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(\tau \right){(\chi \left(1\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{g}(\tau ,x\left(\tau \right))d\tau \right]ds\hfill \\ & & -{\displaystyle \frac{\mathfrak{f}(t,x(t\left)\right)}{2\Gamma \left(\alpha \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left[{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\right.\hfill \\ & & \times {\int }_0^s{\chi }^{\prime }\left(\tau \right){(\chi \left(s\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{g}(\tau ,x\left(\tau \right))d\tau \hfill \\ & & -\left.{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(\tau \right){(\chi \left(1\right)-\chi \left(\tau \right))}^{\beta -1}\mathfrak{g}(\tau ,x\left(\tau \right))d\tau \right]ds.\hfill \end{array}$$

Based on Definition 4, define the operator $\mathcal{T}:\mathcal{C}\to \mathcal{C}$ by

$$\begin{array}{ccc}\hfill \mathcal{T}x\left(t\right)& =& \mathfrak{f}(t,x\left(t\right)){\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(t\right)+K\mathcal{N}x\left(t\right))+\mathfrak{f}(t,x\left(t\right))J\mathcal{N}x\left(t\right),\hfill \end{array}$$

where $\mathcal{N}:\mathcal{C}\to \mathcal{C}$ stands for the Nemytskii operator, which is given by

$$\begin{array}{c}\hfill \mathcal{N}x\left(t\right)=\mathfrak{g}(t,x(t\left)\right),\forall t\in [0,1].\end{array}$$

From Definition 4, it follows that the mild solution of the problem (1) is the fixed point of the operator $\mathcal{T}$. Next, the main results of this paper are given according to the Schaefer fixed point theorem. For computational convenience, we introduce some notations as follows.

$$\begin{array}{ccc}& & \mathfrak{G}=6^{1/P_m-1}{(\chi \left(1\right)-\chi \left(0\right))}^{\beta /P_m-1}/\Gamma \left(\alpha \right){(2\Gamma (\beta +1))}^{1/P_M-1},\hfill \\ & & P_m=\underset{t\in [0,1]}{min}p\left(t\right),\hfill \\ & & P_M=\underset{t\in [0,1]}{max}p\left(t\right).\hfill \end{array}$$

Theorem 1. 

Suppose that $\mathfrak{f}\in C([0,1]\times \mathbb{R},\mathbb{R}\setminus \left\{0\right\})$, $\mathfrak{g}\in C\left(\right[0,1]\times \mathbb{R},\mathbb{R})$ and satisfies conditions $\left({\mathcal{A}}_1\right)$ and $\left({\mathcal{A}}_2\right)$.

$\left({\mathcal{A}}_1\right)$ For $(t,u)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\delta ,\xi \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{g}(t,u)\right|\le \delta \left(t\right)+{\xi \left(t\right)\left|u\right|}^{r_1-1},\end{array}$$

where $1<r_1\le P_m$.

$\left({\mathcal{A}}_2\right)$ For $(t,v)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\theta ,\omega \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{f}(t,v)\right|\le \theta \left(t\right)+{\omega \left(t\right)\left|v\right|}^{r_2-1},\end{array}$$

where $1<r_2\le 2.$

Hence, there exists at least one mild solution to problem (1) in X, as long as

$$\begin{array}{c}\hfill {\displaystyle \frac{3\mathfrak{G}{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\alpha }}\left[(\gamma +\eta ){∥\omega ∥}_{\infty }+\eta {∥\theta ∥}_{\infty }\right]<1,\end{array}$$

(11)

where

$$\begin{array}{ccc}\hfill \gamma & =& max\left\{{∥\delta ∥}_{\infty }^{1/p_m-1},{∥\delta ∥}_{\infty }^{1/p_M-1}\right\},\hfill \\ \hfill \eta & =& max\left\{{∥\xi ∥}_{\infty }^{1/p_m-1},{∥\xi ∥}_{\infty }^{1/p_M-1}\right\}.\hfill \end{array}$$

Proof. 

The proof of the theorem is done in two steps. The first step is to prove that $\mathcal{T}$ is a completely continuous operator. In fact, for any $\rho >0$, define a bounded open set $\mathrm{\Omega }=\{x\in \mathcal{C}:\parallel x{\parallel }_{\infty }<\rho \}$ on $\mathcal{C}$. From the continuity of $\mathfrak{f}$ and ${\varphi }_{p\left(t\right)}^{-1}(·)$, it is easy to get that $\mathcal{T}$ is continuous and there is a constant $M>0$ such that

$$\begin{array}{c}\hfill \left|{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(t\right)+K\mathcal{N}x\left(t\right))\right|\le M,for(t,x)\in [0,1]\times \overline{\mathrm{\Omega }}.\end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \left|\mathcal{T}x\right(t\left)\right|& =& \left|\mathfrak{f}(t,x(t\left)\right)\left[{\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(t\right)+K\mathcal{N}x\left(t\right))+J\mathcal{N}x\left(t\right)\right]\right|\hfill \\ & \le & \left[{∥\theta ∥}_{\infty }+{∥\omega ∥}_{\infty }\left({∥x∥}_{\infty }+1\right)\right]\left[\left|{\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(t\right)+K\mathcal{N}x\left(t\right))\right|+\left|J\mathcal{N}x\left(t\right)\right|\right]\hfill \\ & \le & \left[{∥\theta ∥}_{\infty }+{∥\omega ∥}_{\infty }\left({∥x∥}_{\infty }+1\right)\right]\left({\displaystyle \frac{M{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}+{\displaystyle \frac{M{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\Gamma (\alpha +1)}}\right)\hfill \\ & =& \left[{∥\theta ∥}_{\infty }+{∥\omega ∥}_{\infty }\left({∥x∥}_{\infty }+1\right)\right]·{\displaystyle \frac{3M{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\Gamma (\alpha +1)}}<+\infty .\hfill \end{array}$$

Thus, $\mathcal{T}$ is uniformly bounded on $\overline{\mathrm{\Omega }}$. Next, we will prove that $\mathcal{T}$ is equicontinuous on $\overline{\mathrm{\Omega }}$. For any $x\in \overline{\mathrm{\Omega }},$ assuming that $0\le t_1\le t_2\le 1$, it follows

$$\begin{array}{ccc}& & |\mathcal{T}x\left(t_2\right)-\mathcal{T}x\left(t_1\right)|\hfill \\ & =& |{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & -{\displaystyle \frac{\mathfrak{f}(t_1,x\left(t_1\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & +\mathfrak{f}(t_2,x\left(t_2\right))J\mathcal{N}x\left(t_2\right)-\mathfrak{f}(t_1,x\left(t_1\right))J\mathcal{N}x\left(t_1\right)|\hfill \\ & =& \left|{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)\right.\mathrm{ds}\hfill \\ & & +{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))}{\Gamma \left(\alpha \right)}}{\int }_{t_1}^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & -{\displaystyle \frac{\mathfrak{f}(t_1,x\left(t_1\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & +\mathfrak{f}(t_2,x\left(t_2\right))J\mathcal{N}x\left(t_2\right)-\mathfrak{f}(t_1,x\left(t_1\right))J\mathcal{N}x\left(t_1\right)|\hfill \\ & =& |{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))}{\Gamma \left(\alpha \right)}}{\int }_{t_1}^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & -{\displaystyle \frac{\mathfrak{f}(t_1,x\left(t_1\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & +\left[\left(\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right)+\mathfrak{f}(t_1,x\left(t_1\right))\right]J\mathcal{N}x\left(t_2\right)-\mathfrak{f}(t_1,x\left(t_1\right))J\mathcal{N}x\left(t_1\right)\hfill \\ & & +{\displaystyle \frac{\left(\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right)+\mathfrak{f}(t_1,x\left(t_1\right)}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\hfill \\ & & \times {\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds|\hfill \\ & =& |\left[\left(\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right)+\mathfrak{f}(t_1,x\left(t_1\right))\right]J\mathcal{N}x\left(t_2\right)-\mathfrak{f}(t_1,x\left(t_1\right))J\mathcal{N}x\left(t_1\right)\hfill \\ & & +{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))}{\Gamma \left(\alpha \right)}}{\int }_{t_1}^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & +{\displaystyle \frac{\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\hfill \end{array}$$

$$\begin{array}{ccc}& & \times {\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds\hfill \\ & & +{\displaystyle \frac{\mathfrak{f}(t_1,x\left(t_1\right))}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right)\left[{(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}-{(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}\right]\hfill \\ & & \times {\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)ds|\hfill \\ & \le & {\displaystyle \frac{\left|\mathfrak{f}(t_2,x\left(t_2\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_{t_1}^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\left|{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)\right|ds\hfill \\ & & +{\displaystyle \frac{\left|\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right)\left[{(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}-{(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\right]\hfill \\ & & \times \left|{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)\right|ds+\left|\left(\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right)J\mathcal{N}x\left(t_2\right)\right|\hfill \\ & & +{\displaystyle \frac{\left|\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\hfill \\ & & \times \left|{\varphi }_{p\left(s\right)}^{-1}\left({\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right)\right|ds\hfill \\ & \le & {\displaystyle \frac{M\left|\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\mathrm{ds}\hfill \\ & & +{\displaystyle \frac{M\left|\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_0^{t_1}{\chi }^{\prime }\left(s\right)\left[{(\chi \left(t_1\right)-\chi \left(s\right))}^{\alpha -1}-{(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}\right]ds\hfill \\ & & +{\displaystyle \frac{M\left|\mathfrak{f}(t_2,x\left(t_2\right))\right|}{\Gamma \left(\alpha \right)}}{\int }_{t_1}^{t_2}{\chi }^{\prime }\left(s\right){(\chi \left(t_2\right)-\chi \left(s\right))}^{\alpha -1}ds\hfill \\ & & +\left|\left(\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right)J\mathcal{N}x\left(t_2\right)\right|\hfill \\ & \le & {\displaystyle \frac{M\left|\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma (\alpha +1)}}\left[{(\chi \left(t_2\right)-\chi \left(0\right))}^{\alpha }-{(\chi \left(t_2\right)-\chi \left(t_1\right))}^{\alpha }\right]\hfill \\ & & +{\displaystyle \frac{M\left|\mathfrak{f}(t_1,x\left(t_1\right))\right|}{\Gamma (\alpha +1)}}\left[{(\chi \left(t_1\right)-\chi \left(0\right))}^{\alpha }-{(\chi \left(t_2\right)-\chi \left(0\right))}^{\alpha }+{(\chi \left(t_2\right)-\chi \left(t_1\right))}^{\alpha }\right]\hfill \\ & & +{\displaystyle \frac{M\left|\mathfrak{f}(t_2,x\left(t_2\right))-\mathfrak{f}(t_1,x\left(t_1\right))\right|}{2\Gamma (\alpha +1)}}{(\chi \left(1\right)-\chi \left(0\right))}^{\alpha }\hfill \\ & & +{\displaystyle \frac{M\left|\mathfrak{f}(t_2,x\left(t_2\right))\right|}{\Gamma (\alpha +1)}}{(\chi \left(t_2\right)-\chi \left(t_1\right))}^{\alpha }.\hfill \end{array}$$

So, we conclude

$$\begin{array}{c}\hfill |\mathcal{T}x\left(t_2\right)-\mathcal{T}x\left(t_1\right)|\to 0as{t}_1\to t_2.\end{array}$$

Therefore, $\mathcal{T}$ is equicontinuous on $\overline{\mathrm{\Omega }}$. In summary, according to the Arzelà–Ascoli theorem, $\mathcal{T}$ is a completely continuous operator. The second step is to prove that the following set S that is defined on $\mathcal{C}$ is bounded

$$\begin{array}{c}\hfill S=\{x\in \mathcal{C}|x=\mu \mathcal{T}x,\mu \in (0,1)\}.\end{array}$$

In fact, for any $x\in S$, by $\left({\mathcal{A}}_1\right),\left({\mathcal{A}}_2\right)$, we can obtain

$$\begin{array}{ccc}\hfill \left|K\mathcal{N}x\right(t\left)\right|& =& \left|{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\beta -1}\mathfrak{g}(s,x\left(s\right))ds\right|\hfill \\ & \le & {\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\beta -1}\left|\mathfrak{g}(s,x(s\left)\right)\right|ds\hfill \\ & \le & {\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){(\chi \left(1\right)-\chi \left(s\right))}^{\beta -1}\left(\delta \left(s\right)+{\xi \left(s\right)\left|x\left(s\right)\right|}^{r_1-1}\right)ds\hfill \\ & \le & {\displaystyle \frac{{(\chi \left(1\right)-\chi \left(0\right))}^{\beta }}{2\Gamma (\beta +1)}}\left[{\parallel \delta \parallel }_{\infty }+{\parallel \xi \parallel }_{\infty }{\parallel x\parallel }_{\infty }^{r_1-1}\right].\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill \left|K\mathcal{N}x\left(t\right)+{\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(t\right)\right|& \le & \left|K\mathcal{N}x\left(t\right)\right|+\left|{\mathfrak{I}}_{0^{+}}^{\beta ,\chi }\mathcal{N}x\left(t\right)\right|\hfill \\ & \le & \left|K\mathcal{N}x\left(t\right)\right|+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^t{\chi }^{\prime }\left(s\right){\left(\chi \right(t)-\chi (s\left)\right)}^{\beta -1}\left|\mathcal{N}x\left(s\right)\right|ds\hfill \\ & \le & {\displaystyle \frac{3{\left(\chi \right(1)-\chi (0\left)\right)}^{\beta }}{2\Gamma (\beta +1)}}\left({∥\delta ∥}_{\infty }+{∥\xi ∥}_{\infty }{∥x∥}_{\infty }^{r_1-1}\right).\hfill \end{array}$$

Considering the inequality ${(x_1+x_2)}^p\le 2^p(x_1^p+x_2^p)(x_1,x_2,p>0)$, and $x^q\le x+1(q\in [0,1],x\ge 0),$ for any $t\in [0,1],$ it follows

$$\begin{array}{ccc}\hfill \left|J\mathcal{N}x\left(t\right)\right|& =& \left|{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){\left(\chi \right(1)-\chi (s\left)\right)}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right))ds\right|\hfill \\ & \le & {\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_0^1{\chi }^{\prime }\left(s\right){\left(\chi \right(1)-\chi (s\left)\right)}^{\alpha -1}{\varphi }_{p\left(s\right)}^{-1}\left(\left|{\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(s\right)+K\mathcal{N}x\left(s\right)\right|\right)ds\hfill \\ & \le & {\int }_0^1{\chi }^{\prime }\left(s\right){\left(\chi \right(1)-\chi (s\left)\right)}^{\alpha -1}\left({∥\delta ∥}_{\infty }^{1/p\left(s\right)-1}+{∥\xi ∥}_{\infty }^{1/p\left(s\right)-1}\left({∥x∥}_{\infty }+1\right)\right)ds\hfill \\ & & \times {\displaystyle \frac{6^{1/p_m-1}{\left(\chi \right(1)-\chi (0\left)\right)}^{\beta /p_m-1}}{2\Gamma \left(\alpha \right){\left(2\Gamma (\beta +1)\right)}^{1/p_M-1}}}\hfill \\ & =& {\displaystyle \frac{\mathfrak{G}}{2}}{\int }_0^1{\chi }^{\prime }\left(s\right){\left(\chi \right(1)-\chi (s\left)\right)}^{\alpha -1}\left({∥\delta ∥}_{\infty }^{1/p\left(s\right)-1}+{∥\xi ∥}_{\infty }^{1/p\left(s\right)-1}\left({∥x∥}_{\infty }+1\right)\right)ds.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \left|x\left(t\right)\right|& \le & \left|\mathfrak{f}(t,x\left(t\right)){\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}({\mathfrak{I}}_{0+}^{\beta ,\chi }\mathcal{N}x\left(t\right)+K\mathcal{N}x\left(t\right))\right|+\left|\mathfrak{f}(t,x(t\left)\right)J\mathcal{N}x\left(t\right)\right|\hfill \\ & \le & \left|\mathfrak{f}(t,x\left(t\right)){\mathfrak{I}}_{0+}^{\alpha ,\chi }{\varphi }_{p\left(t\right)}^{-1}\left({\displaystyle \frac{3{\left(\chi \right(1)-\chi (0\left)\right)}^{\beta }}{2\Gamma (\beta +1)}}\left({∥\delta ∥}_{\infty }+{∥\xi ∥}_{\infty }{∥x∥}_{\infty }^{r_1-1}\right)\right)\right|\hfill \\ & & +\left|\mathfrak{f}(t,x(t\left)\right)J\mathcal{N}x\left(t\right)\right|\hfill \\ & \le & \left|\mathfrak{G}{\int }_0^t{\chi }^{\prime }\left(s\right){\left(\chi \right(t)-\chi (s\left)\right)}^{\alpha -1}\left({∥\delta ∥}_{\infty }^{1/p\left(s\right)-1}+{∥\xi ∥}_{\infty }^{1/p\left(s\right)-1}{∥x∥}_{\infty }^{r_1-1/p\left(s\right)-1}\right)ds\right|\hfill \\ & & \times \left({∥\theta ∥}_{\infty }+{∥\omega ∥}_{\infty }{∥x∥}_{\infty }^{r_2-1}\right)+\left|\mathfrak{f}(t,x(t\left)\right)J\mathcal{N}x\left(t\right)\right|\hfill \\ & \le & {\displaystyle \frac{3\mathfrak{G}{\left(\chi \right(1)-\chi (0\left)\right)}^{\alpha }}{2\alpha }}\left({∥\theta ∥}_{\infty }+{∥\omega ∥}_{\infty }{∥x∥}_{\infty }^{r_2-1}\right)\left(\gamma +\eta {∥x∥}_{\infty }^{r_1-1/p_{\ell }-1}\right)\hfill \\ & =& \left(\gamma {∥\theta ∥}_{\infty }+\gamma {∥\omega ∥}_{\infty }{∥x∥}_{\infty }^{r_2-1}+{∥\theta ∥}_{\infty }\eta {∥x∥}_{\infty }^{r_1-1/p_{\ell }-1}\right.\hfill \\ & & +\left.\eta {∥\omega ∥}_{\infty }{∥x∥}_{\infty }^{\left(r_1-1\right)\left(r_2-1\right)/p_{\ell }-1}\right)\times {\displaystyle \frac{3\mathfrak{G}{\left(\chi \right(1)-\chi (0\left)\right)}^{\alpha }}{2\alpha }}\hfill \\ & \le & {\displaystyle \frac{3\mathfrak{G}{\left(\chi \right(1)-\chi (0\left)\right)}^{\alpha }}{2\alpha }}\left[\gamma {∥\theta ∥}_{\infty }+\left(\gamma {∥\omega ∥}_{\infty }+\eta {∥\theta ∥}_{\infty }+\eta {∥\omega ∥}_{\infty }\right)\left({∥x∥}_{\infty }+1\right)\right],\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {∥x∥}_{\infty }^{r_1-1/p_{\ell }-1}& :=& max\left\{{∥x∥}_{\infty }^{r_1-1/p_m-1},{∥x∥}_{\infty }^{r_1-1/p_M-1}\right\}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill {∥x∥}_{\infty }\le {\displaystyle \frac{3\mathfrak{G}{\left(\chi \right(1)-\chi (0\left)\right)}^{\alpha }}{2\alpha }}\left[\gamma {∥\theta ∥}_{\infty }+\left(\gamma {∥\omega ∥}_{\infty }+\eta {∥\theta ∥}_{\infty }+\eta {∥\omega ∥}_{\infty }\right)\left({∥x∥}_{\infty }+1\right)\right].\end{array}$$

(12)

By conditions (11) and (12), it can be obtained that there is a constant $\overline{M}>0$ such that ${∥x∥}_{\infty }\le \overline{M}$. Utilizing the Schaefer fixed point theorem, it follows that $\mathcal{T}$ admits at least one fixed point within S, thereby ensuring the existence of at least one mild solution to problem (1) in $\mathcal{C}$. □

Corollary 1. 

If $f(t,x)=1$, the boundary value conditions of (1) become the fractional-type anti-periodic boundary conditions

$$x\left(0\right)=-x\left(1\right),{}^C\mathfrak{D}_{0+}^{\alpha ,\chi }x\left(t\right){{|}_{t=0}=-{}^C\mathfrak{D}_{0+}^{\alpha ,\chi }x\left(t\right)|}_{t=1}.$$

Moreover, the condition of $\left({\mathcal{A}}_2\right)$ of Theorem 1 is still satisfied.

Corollary 2. 

If $\chi \left(t\right)=t$, the problem (1) becomes the following anti-periodic BVP of Caputo FDE involving the $p\left(t\right)$-Laplacian operator:

$$\left\{\begin{array}{c}{}^C\mathfrak{D}_{0^{+}}^{\beta }{\varphi }_{p\left(t\right)}\left({}^C\mathfrak{D}_{0^{+}}^{\alpha }{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}\left(t,x\left(t\right)\right)}}\right)=\mathfrak{g}(t,x\left(t\right)),t\in \left(0,1\right),\hfill \\ \\ {\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=0}=-{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=1},{}^C\mathfrak{D}_{0+}^{\alpha }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=0}=-{}^C\mathfrak{D}_{0+}^{\alpha }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=1}.\hfill \end{array}\right.$$

It still satisfies the conditions $\left({\mathcal{A}}_1\right)$ and $\left({\mathcal{A}}_2\right)$ in Theorem 1, and

$${\displaystyle \frac{3{\mathfrak{G}}_1}{2\alpha }}\left[(\gamma +\eta ){∥\omega ∥}_{\infty }+\eta {∥\theta ∥}_{\infty }\right]<1,$$

where ${\mathfrak{G}}_1=6^{1/P_m-1}/\Gamma \left(\alpha \right){(2\Gamma (\beta +1))}^{1/P_M-1}$.

Corollary 3. 

If $\chi \left(t\right)=lnt$, the problem (1) becomes the following anti-periodic BVP of Hadamard FDE involving the $p\left(t\right)$-Laplacian operator:

$$\left\{\begin{array}{c}{}^H\mathfrak{D}_{0^{+}}^{\beta }{\varphi }_{p\left(t\right)}\left({}^H\mathfrak{D}_{0^{+}}^{\alpha }{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}\left(t,x\left(t\right)\right)}}\right)=\mathfrak{g}(t,x\left(t\right)),t\in \left(0,1\right),\hfill \\ \\ {\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=0}=-{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=1},{}^H\mathfrak{D}_{0+}^{\alpha }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=0}=-{}^H\mathfrak{D}_{0+}^{\alpha }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=1}.\hfill \end{array}\right.$$

It still satisfies the conditions $\left({\mathcal{A}}_1\right)$ and $\left({\mathcal{A}}_2\right)$ in Theorem 1, and

$${\displaystyle \frac{3{\mathfrak{G}}_2{\left(ln\left(1\right)-ln\left(0\right)\right)}^{\alpha }}{2\alpha }}\left[(\gamma +\eta ){∥\omega ∥}_{\infty }+\eta {∥\theta ∥}_{\infty }\right]<1,$$

where ${\mathfrak{G}}_2=6^{1/P_m-1}{(ln\left(1\right)-ln\left(0\right))}^{\beta /P_m-1}/\Gamma \left(\alpha \right){(2\Gamma (\beta +1))}^{1/P_M-1}$.

Corollary 4. 

If $r_1=1$, then the condition in Theorem 1 becomes

$\left({\mathcal{A}}_1\right)$ For $(t,u)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\delta \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{g}\right(t,u\left)\right|\le \delta \left(t\right).\end{array}$$

$\left({\mathcal{A}}_2\right)$ For $(t,v)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\theta ,\omega \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{f}(t,v)\right|\le \theta \left(t\right)+{\omega \left(t\right)\left|v\right|}^{r_2-1},\end{array}$$

where $1<r_2\le 2.$

The conclusion still holds, as long as

$${\displaystyle \frac{3{\mathfrak{G}}_3{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\alpha }}\gamma {∥\omega ∥}_{\infty }<1,$$

where ${\mathfrak{G}}_3=3^{1/P_m-1}{(\chi \left(1\right)-\chi \left(0\right))}^{\beta /P_m-1}/\Gamma \left(\alpha \right){(2\Gamma (\beta +1))}^{1/P_M-1}.$

Corollary 5. 

If $r_2=1$, then the conditions in Theorem 1 becomes

$\left({\mathcal{A}}_1\right)$ For $(t,u)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\delta ,\xi \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{g}(t,u)\right|\le \delta \left(t\right)+{\xi \left(t\right)\left|u\right|}^{r_1-1},\end{array}$$

where $1<r_1\le P_m$.

$\left({\mathcal{A}}_2\right)$ For $(t,v)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\theta \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{f}\right(t,v\left)\right|\le \theta \left(t\right).\end{array}$$

The conclusion still holds, as long as

$${\displaystyle \frac{3\mathfrak{G}{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\alpha }}\eta {∥\theta ∥}_{\infty }<1.$$

Corollary 6. 

If $r_1,r_2=1$, then the condition in Theorem 1 becomes

$\left({\mathcal{A}}_1\right)$ For $(t,u)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\delta \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{g}\right(t,u\left)\right|\le \delta \left(t\right).\end{array}$$

$\left({\mathcal{A}}_2\right)$ For $(t,v)\in [0,1]\times \mathbb{R}$, there exist non-negative functions $\theta \in \mathcal{C}$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{f}\right(t,v\left)\right|\le \theta \left(t\right).\end{array}$$

The conclusion still holds, as long as

$${\displaystyle \frac{3{\mathfrak{G}}_3{\left(\chi \left(1\right)-\chi \left(0\right)\right)}^{\alpha }}{2\alpha }}\gamma {∥\theta ∥}_{\infty }<1.$$

4. An Example

Example 1. 

Let $\alpha ={\displaystyle \frac{2}{5}},\beta ={\displaystyle \frac{7}{10}}$. Consider the following BVP:

$$\left\{\begin{array}{c}{}^C\mathfrak{D}_{0^{+}}^{7/10,\chi }{\varphi }_2\left({}^C\mathfrak{D}_{0^{+}}^{2/5,\chi }\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}\left(t,x\left(t\right)\right)}}\right)\right)=e^t+{\displaystyle \frac{15}{100000}}t,\hfill \\ \\ {\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=0}=-{\left.{\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right|}_{t=1},{}^C\mathfrak{D}_{0+}^{2/5,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=0}=-{}^C\mathfrak{D}_{0+}^{2/5,\chi }{\left.\left({\displaystyle \frac{x\left(t\right)}{\mathfrak{f}(t,x(t\left)\right)}}\right)\right|}_{t=1},\hfill \end{array}\right.$$

(13)

where

$$\begin{array}{c}\hfill \mathfrak{g}\left(t,x\left(t\right)\right)=e^t+{\displaystyle \frac{15}{100000}}t,\mathfrak{f}\left(t,x\left(t\right)\right)=1000+{\displaystyle \frac{3}{100000}}t,p\left(t\right)=2.\end{array}$$

Choose

$$r_1=2,r_2=1.5,\delta \left(t\right)=e^t+{\displaystyle \frac{15}{100000}}t,\xi \left(t\right)=0,\theta \left(t\right)=1000+{\displaystyle \frac{3}{100000}}t,\omega \left(t\right)=0,\chi \left(\kappa \right)=\kappa .$$

Moreover, we can get $P_m=P_M=2.$ Thus, it follows that (11) is true. Therefore, by Theorem 1, we conclude that the problem (13) possesses at least one mild solution in $\mathcal{C}$. Next, we present the curve of the solution in Figure 1:

Furthermore, if we take different values for $\alpha$ and $\beta$, the image of $x\left(t\right)$ is shown in Figure 2:

5. Conclusions

This paper focuses on a class of differential equations involving generalized Caputo fractional derivatives and $p\left(t\right)$-Laplacian operators, and investigates the existence of mild solutions under generalized anti-periodic boundary value conditions. By applying Schaefer fixed point theorem, we prove the existence of mild solutions within an appropriate function space under certain assumptions. Secondly, compared with the research on the constant p-Laplacian operator in the existing literature, this work introduces the $p\left(t\right)$-Laplacian operator into the operator framework, making the nonlinear characteristics of the studied problem more complex. In addition, the generalized Caputo derivative is adopted to replace the traditional Caputo derivative, offering a theoretical basis for modeling a broader range of memory effects. Furthermore, the application of Schaefer fixed point theorem in this paper serves as a reference for its generalization under complex operators and boundary value structures. In addition, the method we adopted demonstrates certain applicability and theoretical value in dealing with problems where nonlinear operators with variable coefficients are coupled with generalized fractional derivatives. It is expected that this work may offer useful insights and references for subsequent theoretical research and model analysis. However, the results obtained in this paper are established under specific assumptions and primarily focus on the existence of mild solutions. Deeper aspects such as the uniqueness and stability of solutions have not yet been addressed, so the relevant issues still need to be further explored.