Parameterized Anti-Periodic Problems: Existence and Ulam-Hyers Stability for Fractional p(t)-Laplacian Langevin Equations

Abstract

This paper investigates a novel class of fractional Langevin equations, which introduces a time-varying p(t)-Laplacian operator and parameterized anti-periodic boundary conditions. This approach overcomes the limitations of traditional models characterized by constant diffusion exponents and fixed boundary locations. Under non-compactness conditions, the existence of solutions is established by applying Schaefer’s fixed-point theorem, which significantly relaxes the conventional constraints on the nonlinear term. Moreover, by imposing a Lipschitz condition on the nonlinear term, a Ulam–Hyers-type stability criterion for the coupled system is derived. This work not only extends the relevant stability theory but also provides a rigorous theoretical foundation for error control in practical applications. The effectiveness of the theoretical results is validated through numerical examples.

1. Introduction

Fractional calculus transcends the limitations of traditional integer-order calculus, with its core advantage lying in non-locality. This enables precise characterization of complex systems exhibiting memory and historical dependencies, a property that demonstrates unique value in fields such as economics, control science and materials science. Consequently, it has propelled the development of fractional differential equations, establishing itself as a potent tool for interdisciplinary modelling. The Langevin equation [1] effectively captures the fluctuation characteristics of Brownian motion by incorporating stochastic forces and viscous damping. However, traditional models based on integer-order derivatives implicitly assume instantaneous response, making it challenging to accurately describe complex systems exhibiting memory effects or non-exponential relaxation properties (such as viscoelastic materials and porous media). To overcome this limitation, fractional-order Langevin equations were proposed. By leveraging the “long-memory” properties of fractional calculus, these equations more precisely characterise dynamical behaviours such as anomalous diffusion and non-Markovian noise [2,3,4,5], advancing the study of stochastic dynamics from “no-memory” to “long-memory” frameworks.

Although the fractional Langevin equation provides a more universal framework for modeling the dynamics of complex systems, its applications in practical physical scenarios (e.g., particle transport in confined environments, microscale boundary control, etc.) often need to be solved in conjunction with specific boundary conditions, and such a need has driven scholars to study the boundary value problems of the fractional Langevin equation [6,7,8,9]. For example, Ahmad et al. [10] discussed a three-point boundary value problem for fractional Langevin equation. The authors employed Krasnoselskii’s fixed-point theorem and Banach’s contraction mapping theorem to derive sufficient conditions for the existence and uniqueness of solutions to the equation.

The above studies focus on the properties of the solutions of the fractional Langevin equation under classical boundary conditions, providing theoretical support for the dynamical behavior of complex systems. However, in practical applications, many physical phenomena often involve stronger nonlinear features that need to be represented by introducing nonlinear differential operators such as p-Laplacian operators. Meanwhile, in scenarios such as quantum regulation and thermodynamic cycles, the boundary conditions of the system may exhibit anti-periodic characteristics, so many scholars have combined the fractional Langevin equation with the p-Laplacian operator and considered the anti-periodic boundary value problem. For example, Zhou et al. [11] coupled the above two types of nonlinear factors to construct the following boundary value problem

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\beta }{\varphi }_p\left[(D_{0^{+}}^{\alpha }+\lambda )x\left(t\right)\right]=f(t,x\left(t\right),D_{0^{+}}^{\alpha }x\left(t\right)),t\in [0,1],\hfill \\ x\left(0\right)=-x\left(1\right),D_{0^{+}}^{\alpha }x\left(0\right)=-D_{0^{+}}^{\alpha }x\left(1\right),\hfill \end{array}\right.$$

(1)

where $D_{0^{+}}^{\rho }$ is the Caputo fractional derivative operator of order $\rho (\rho =\alpha ,\beta )$ with $0<\alpha ⩽1$ and $0<\beta ⩽1$, $\lambda ⩾0$, ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$, For $p>1$, the p-Laplacian operator is considered, with the function $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ satisfying the Carathéodory conditions. The authors employ the Leray-Schaefer fixed-point theorem to establish sufficient conditions for the existence of solutions.

However, in the aforementioned studies as well as in most existing literature, the exponent p in the considered p-Laplace operator is treated as a constant. This implies that the nonlinear characteristics of the model do not vary over time, making it difficult to accurately describe diffusion processes under time-varying external fields or non-stationary environments. To overcome this limitation, the time-varying p(t)-Laplace operator has been introduced into the study of fractional differential equations. Its exponent $p\left(t\right)$ can vary continuously with time, enabling the description of physical phenomena with “point-by-point distinct characteristics”. Such operators originate from fields such as image restoration and elasticity theory [12]. Compared with the traditional p-Laplace operator, the p(t)-Laplace operator exhibits more complex nonlinear behavior. For instance, Shen et al. [13] investigates the existence of solutions for fractional integral boundary value problems featuring the p(t)-Laplacian operator. Xue et al. [14] examine the existence of solutions for fractional Sturm-Liouville boundary value problems incorporating the p(t)-Laplacian operator. Tang et al. [15] investigated the existence of solutions for mixed fractional resonance boundary value problems concerning the p(t)-Laplacian operator. Shen et al. [16] explored the solvability of the Erdélyi-Kober fractional integral boundary value problem with the p(t)-Laplacian operator during resonance.

In recent years, the Schaefer fixed point theorem has been widely applied in the study of the existence of solutions to nonlinear fractional differential equations. For example, based on this theorem, Shah et al. [17] investigated a class of fractional Langevin equations with anti-periodic boundary conditions and have obtained results on the existence, uniqueness, and stability of solutions. Zhang et al. [18] discussed the anti-periodic boundary value problem of fractional Langevin equations with p(t)-Laplacian operators

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

(2)

where ${}^C{D}_{0^{+}}^{\alpha }$ and ${}^C{D}_{0^{+}}^{\beta }$ be Caputo fractional derivative operator with $0<\alpha ,\beta ⩽1$ and $1<\alpha +\beta ⩽2$, $f\in ([0,1]\times {\mathbb{R}}^2,\mathbb{R})$, Let ${\phi }_{p\left(t\right)}(·)$ be the p(t) -Laplacian operator, satisfying $p\left(0\right)=p\left(1\right)$ and $p\left(t\right)>1$. The author employs the Schaefer fixed-point theorem to derive sufficient conditions for the existence of solutions. This work provides a theoretical foundation for combining Schaefer fixed-point theorem with the p(t)-Laplacian operator to address more generalised fractional Langevin equation anti-periodic boundary value problems.

Most existing studies are confined to constant-exponent p-Laplacian operators and standard endpoint boundary conditions. Although reference [18] introduced the time-varying p(t)-Laplacian operator into the anti-periodic boundary value problem for fractional Langevin equations, its model still retains the classical endpoint anti-periodic conditions. Consequently, it cannot describe real physical scenarios where boundary feedback or symmetry constraints act on interior points of the domain, and it also lacks a systematic analysis of the Ulam–Hyers stability of the solutions under perturbations.

To overcome these limitations, this paper extends the model in [18]. Regarding the choice of fractional derivative, the Caputo–Hadamard fractional derivative with the logarithmic kernel $ln(t/s)$ is adopted. Compared with the commonly used Caputo or Riemann-Liouville derivatives, this operator is more suitable for describing dynamical processes that evolve on a logarithmic time scale and can be coupled harmoniously with the time-varying p(t)-Laplacian operator, thereby better capturing systems with memory effects and time-dependent diffusion properties. Concerning the description of boundary conditions, we note that traditional anti-periodic conditions usually assume that constraints act at the endpoints of the interval. However, in practical problems, symmetric or anti-periodic relations of the system state may occur at specific interior points. To characterize such “non-endpoint” anti-periodic phenomena, this paper introduces parameterized anti-periodic boundary conditions. By means of a parameter $a\in [1,e]$ that specifies the location where the anti-periodicity is imposed, the classical endpoint condition is generalized to any interior point of the interval, which enhances the model’s ability to describe realistic complex boundary-feedback scenarios.

Based on the above, this paper combines the fractional Langevin equation, the time-varying p(t)-Laplacian operator and the parameterized anti-periodic boundary conditions to study the following boundary value problem:

$$\left\{\begin{array}{c}{}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]=f(t,x\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(t\right)),t\in [1,e],\hfill \\ x\left(a\right)=-x\left(e\right),{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=a}=-{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=e},a\in [1,e],\hfill \end{array}\right.$$

(3)

where ${}_H{}^C{D}_{1^{+}}^{\alpha }$ and ${}_H{}^C{D}_{1^{+}}^{\beta }$ are Caputo–Hadamard fractional derivatives (with $0<\alpha ,\beta ⩽1$). $f:[1,e]\times {\mathbb{R}}^2\to \mathbb{R}$. The time-varying p(t)-Laplacian operator ${\phi }_{p\left(t\right)}(·)$ satisfies $p\left(t\right)>1$ and $p\left(a\right)=p\left(e\right)$, and can be used to model physical processes with time-dependent diffusion coefficients. The parameter $\lambda ⩾0$ is the classical linear damping coefficient. The parameter $a\in [1,e]$ in the boundary condition constitutes a generalization of the classical anti-periodic problem: when a is an endpoint, the model reduces to the classical anti-periodic boundary value problem; when a lies in the interior of the interval, it can describe physical scenarios where the system is subject to feedback or symmetric constraints at an internal point, thereby enhancing the model’s capability to capture the dynamics of complex real-world systems.

This paper establishes a class of Langevin equation models within the Caputo–Hadamard fractional derivative framework, integrating p(t)-Laplacian operators with parameterised anti-periodic boundary conditions. The model simultaneously encompasses both dynamic nonlinearity (time-varying p(t)) and asymmetric boundaries (parameter $a\in [1,e]$), thereby extending the theoretical scope of existing models. Methodologically, we employ Schaefer’s fixed point theorem in place of traditional contraction mappings to establish the existence of solutions under non-compactness conditions, thereby relaxing the structural restrictions on the nonlinear term. Furthermore, a Ulam-Hyers-type stability criterion is established for this system, filling a gap in stability theory for such models and providing a theoretical foundation for error control and robustness analysis in practical applications.

The structure of this paper is as follows: Section 2 introduces necessary definitions and preliminary lemmas; Section 3 proves the existence of solutions to the boundary value problem via Schaefer’s fixed point theorem; Section 4 conducts Ulam-Hyers stability analysis; Section 5 verifies the validity of the theoretical results through a numerical example; Section 6 summarises the entire paper and discusses future research directions.

2. Preliminaries

In this section, we introduce some definitions and preliminary theorems which will be used later on. Let $X=\{x:x,{}_H{}^C{D}_{1^{+}}^{\alpha }x\in C\left([1,e]\right)\}$ be endowed with the norm

$${\parallel x\parallel }_X={\parallel x\parallel }_{\infty }+{\parallel {}_H{}^C{D}_{1^{+}}^{\alpha }x\parallel }_{\infty },$$

where ${\parallel ·\parallel }_X=\underset{t\in [1,e]}{max}|·|$. Then $(X,\parallel ·{\parallel }_X)$ is the Banach space.

Definition 1

([19]). The Hadamard integral of order $\alpha >0$ for a given functionis defined as

$${}^H{I}_{1^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\alpha -1}f\left(s\right){\displaystyle \frac{ds}{s}}.$$

Definition 2

([19]). The Caputo derivative of order $\alpha >0$ for a given functionis defined as

$${}^C{D}_{0^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds,$$

where $n=\left[\alpha \right]+1$.

Definition 3

([19]). The Caputo-Hadamard derivative is defined as

$${}_H{}^C{D}_{1^{+}}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{n-\alpha -1}{\left(\delta \right)}^{n}f\left(s\right){\displaystyle \frac{ds}{s}},$$

where $\delta =\left(t{\displaystyle \frac{d}{dt}}\right),n=\left[\alpha \right]+1$.

Definition 4

([20]). Let X be a metric space and $A\subset X$. A is called a compact set if every open cover of A has a finite subcover. If the closure $\overline{A}$ of A is compact, then A is called a relatively compact set.

Definition 5

([20]). An operator A is continuous at $x_0$ if, for a point $x_0$ in a subset D of a Banach space X, for every $\epsilon >0$, there exists $\delta =\delta (x_0,\epsilon )>0$ such that whenever $x\in D$ and $∥x-x_0∥<\delta$, it holds that $∥Ax-A{x}_0∥<\epsilon$. If A is continuous at every point of D, then is A said to be continuous on D. If the above δ depends only on ε and not on $x_0\in D$, then A is called uniformly continuous on D.

Lemma 1

([21]). Let $u\in A{C}_{\delta }^n[a,b]$ or $C_{\delta }^n[a,b]$ and $\alpha >0$, where $X_{\delta }^n[a,b]=u:[a,b]\to \mathbb{R}:{\delta }^{n-1}$ $u\left(t\right)\in X[a,b].$ Then,

$${}^H{I}_{1^{+}}^{\alpha }{}_H{}^C{D}_{1^{+}}^{\alpha }u\left(t\right)=u\left(t\right)-\sum _{k=0}^{n-1}{c}_k{(lnt)}^k,$$

where $c_k\in \mathbb{R}$, $k=0,1,2,\dots ,n-1$,$(n=[\alpha ]+1)$.

Lemma 2

([22]). For any $(t,x)\in [0,1]\times \mathbb{R},{\phi }_{p\left(t\right)}\left(x\right)=\mid x{\mid }^{p\left(t\right)-2}x$ is a homeomorphism from $\mathbb{R}$ to $\mathbb{R}$ and strictly monotone increasing for any fixed t. Moreover, its inverse operator ${\phi }_{p\left(t\right)}^{-1}(·)$ is defined by

$$\left\{\begin{array}{c}{\phi }_{p\left(t\right)}^{-1}\left(x\right)={\mid x\mid }^{{\displaystyle \frac{2-p\left(t\right)}{p\left(t\right)-1}}}x,x\in \mathbb{R}\setminus \left\{0\right\},\hfill \\ {\phi }_{p\left(t\right)}^{-1}\left(x\right)=0,x=0,\hfill \end{array}\right.$$

which is continuous and sends bounded sets into bounded sets.

Theorem 1

([23] Arzelà–Ascoli theorem). A subset $A\subset C[a,b]$ is relatively compact if and only if the following two conditions hold:

(1) 

There exists a constant $M>0$ such that for all $x\in A$ and all $t\in [a,b]$, have

$$\left|x\left(t\right)\right|⩽M,$$

that is, A is uniformly bounded.

(2) 

For every $\epsilon >0$, there exists $\delta \left(\epsilon \right)>0$ such that for all $x\in A$ and for any $t_1,t_2\in [a,b]$, if $\left|t_1-t_2\right|<\delta$, then

$$\left|x\left(t_1\right)-x\left(t_2\right)\right|<\epsilon ,$$

that is, A is equicontinuous.

Theorem 2

([24] Schaefer fixed-point theorem). Let X be a Banach space and assume that $T:X\to X$ is a completely continuous operator. If the set

$$\mathrm{\Omega }=\{x\in X\mid x=\mu Tx,\mu \in (0,1\left)\right\}$$

is bounded, then T has a fixed point in X.

3. Existence of Solutions

Let X be the Banach space defined in Section 2 with the norm ${\parallel x\parallel }_X={\parallel x\parallel }_{\infty }+{\parallel {}_H{}^C{D}_{1^{+}}^{\alpha }x\parallel }_{\infty },$ Let $N:C[1,e]\to C[1,e]$ as the Nemytskii operator $Nx\left(t\right)=f(t,x\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(t\right)),t\in [1,e]$. Under the assumption that $f:[1,e]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous and satisfies the growth condition (9) in Theorem 3, the operator N is well-defined and continuous in the Banach space X.

Lemma 3.

Let the fractional Langevin equation be given by

$${}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]=Nx\left(t\right),t\in [1,e],$$

(4)

under the boundary conditions

$$x\left(a\right)=-x\left(e\right),{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=a}=-{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=e}.$$

Then there exists a solution of the following form

$$\begin{array}{cc}\hfill x\left(t\right)=& {}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]+\mathcal{B}Nx-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{(ln{\displaystyle \frac{s}{\tau }})}^{\beta -1}Nx\left(\tau \right){\displaystyle \frac{d\tau }{\tau }}+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\hfill \\ & +\mathcal{B}Nx-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathcal{A}Nx=& -{\displaystyle \frac{1}{2}}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\beta }Nx\left(t\right){\mid }_{t=e})\hfill \\ \hfill =& -{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}\left({\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\beta -1}Nx\left(s\right){\displaystyle \frac{ds}{s}}+{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\beta -1}Nx\left(s\right){\displaystyle \frac{ds}{s}}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{B}Nx\left(t\right)=& -{\displaystyle \frac{1}{2}}[{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=a}+{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)\hfill \\ & +\mathcal{A}Nx){\mid }_{t=e}]+{\displaystyle \frac{\lambda }{2}}\left({}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=e}\right)\hfill \\ \hfill =& -{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{(ln{\displaystyle \frac{s}{\tau }})}^{\beta -1}Nx\left(\tau \right){\displaystyle \frac{d\tau }{\tau }}+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\hfill \\ & -{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{(ln{\displaystyle \frac{s}{\tau }})}^{\beta -1}Nx\left(\tau \right){\displaystyle \frac{d\tau }{\tau }}+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{\lambda }{2\Gamma \left(\alpha \right)}}\left({\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}+{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}\right).\hfill \end{array}$$

Proof. 

By applying the fractional integral operators ${}^H{I}_{1^{+}}^{\beta }$ to (4), we obtain

$${\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{=}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+c_0,c_0\in \mathbb{R},$$

(5)

by the boundary condition ${\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=a}=-{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{\mid }_{t=e}$, we conclude that

$${\phi }_{p\left(a\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(a\right)\right]{=}^H{I}_{1^{+}}^{\beta }Nx\left(a\right)+c_0,$$

and

$${\phi }_{p\left(e\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(e\right)\right]{=}^H{I}_{1^{+}}^{\beta }Nx\left(e\right)+c_0,$$

yield

$$c_0=\mathcal{A}Nx=-{\displaystyle \frac{1}{2}}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\beta }Nx\left(t\right){\mid }_{t=e}),$$

substituting the value of $c_0$ in (5), we obtain

$${\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]{=}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx,$$

and then applying Lemma 2, we get

$$\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )x\left(t\right)\right]={\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx],$$

(6)

by applying the fractional integral operators ${}^H{I}_{1^{+}}^{\alpha }$ to (6), we obtain

$$x\left(t\right){=}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]+c_1-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right),c_1\in \mathbb{R},$$

(7)

on the other hand, $x\left(a\right)=-x\left(e\right)$, combining with

$$x\left(a\right){=}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(a\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(a\right)+\mathcal{A}Nx]+c_1-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(a\right),$$

and

$$x\left(e\right){=}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(e\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(e\right)+\mathcal{A}Nx]+c_1-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(e\right),$$

yield

$$\begin{array}{cc}\hfill c_1=BNx\left(t\right)=& -{\displaystyle \frac{1}{2}}{[}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=a}\hfill \\ \hfill & {+}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=e}]\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{2}}{(}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=e}).\hfill \end{array}$$

(8)

Substitute Equation (8) into Equation (7) to prove the conclusion. □

Based on Lemma 3, the operator $T:C[1,e]\to C[1,e]$ is defined as follows

$$\begin{array}{cc}\hfill Tx\left(t\right)& ={}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx\left(t\right)\right]+\mathcal{B}Nx\left(t\right)-\lambda {}^H{I}_{1^{+}}^{\alpha }x\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^t{\left(ln{\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}[{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{\left(ln{\displaystyle \frac{s}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}({\int }_1^a{\left(ln{\displaystyle \frac{a}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & +{\int }_1^e{\left(ln{\displaystyle \frac{e}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\left)\right]{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^a{\left(ln{\displaystyle \frac{a}{s}}\right)}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}[{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{\left(ln{\displaystyle \frac{s}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}({\int }_1^a{\left(ln{\displaystyle \frac{a}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & +{\int }_1^e{\left(ln{\displaystyle \frac{e}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\left)\right]{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^e{\left(ln{\displaystyle \frac{e}{s}}\right)}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}[{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^s{\left(ln{\displaystyle \frac{s}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}({\int }_1^a{\left(ln{\displaystyle \frac{a}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \hfill & +{\int }_1^e{\left(ln{\displaystyle \frac{e}{\tau }}\right)}^{\beta -1}f(\tau ,x\left(\tau \right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(\tau \right)){\displaystyle \frac{d\tau }{\tau }}\left)\right]{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{2\Gamma \left(\alpha \right)}}\left({\int }_1^a{\left(ln{\displaystyle \frac{a}{s}}\right)}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}+{\int }_1^e{\left(ln{\displaystyle \frac{e}{s}}\right)}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}\right)\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_1^t{\left(ln{\displaystyle \frac{t}{s}}\right)}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Thus, the existence of a solution to the boundary value problem (3) is equivalent to proving the existence of a fixed point for the operator T.

The main results of this study are presented based on the Schaefer fixed-point theorem. For the sake of clarity, let us denote

$$p_m=\underset{t\in [1,e]}{min}p\left(t\right),p_M=\underset{t\in [1,e]}{max}p\left(t\right),Q={\displaystyle \frac{{[6+2{(lna)}^{\beta }]}^{\frac{1}{p_m-1}}}{2\Gamma \left(\alpha \right){(2\Gamma (\beta +1))}^{\frac{1}{p_M-1}}}}.$$

Theorem 3.

Assume that $f:[1,e]\times {\mathbb{R}}^2\to \mathbb{R}$ is a continuous function that satisfies the condition: there exist non-negative functions $\xi ,\phi ,\eta \in C[1,e]$, such that

$$\mid f(t,u,v)\mid ⩽\xi \left(t\right)+\phi \left(t\right)\mid u{\mid }^{r-1}+\eta \left(t\right)\mid v{\mid }^{r-1},(u,v)\in {\mathbb{R}}^2,1<r⩽p_m,t\in [1,e],$$

(9)

when

$$\begin{array}{c}\hfill l\left({\displaystyle \frac{3+{(lna)}^{\alpha }}{\alpha }}Q+2\Gamma \left(\alpha \right)Q\right)+{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}+\lambda <1,\end{array}$$

(10)

holds, the boundary value problem (3) has at least one solution in X, where

$$l:=max\left\{{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p_m-1}}},{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p_M-1}}}\right\}.$$

Proof. 

we will proceed in two steps.

Step 1: To demonstrate that T is a completely continuous operator. let $k>0$ be arbitrary and define a bounded open set $\mathbb{B}=\left\{x\in {X:\Vert x\Vert }_X<k\right\}$ in X. By the continuity of f and ${\phi }_{p\left(t\right)}^{-1}(·)$, it follows that T is continuous on $[1,e]$, and there exists a constant $H>0$, such that

$$\left|{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx)\right|⩽H,x\left(t\right)\in \overline{\mathbb{B}},t\in [1,e],$$

we have

$$\begin{array}{cc}\hfill \left|\mathcal{B}Nx\right|=& |-{\displaystyle \frac{1}{2}}\left[{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=e}\right]\hfill \\ & +{\displaystyle \frac{\lambda }{2}}\left({}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=e}\right)|\hfill \\ \hfill ⩽& |-{\displaystyle \frac{1}{2}}{[}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=e}]|\hfill \\ & +\left|{\displaystyle \frac{\lambda }{2}}{(}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=e})\right|\hfill \\ \hfill ⩽& ({\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}{|}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx|{\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}{|}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx\left|{\displaystyle \frac{ds}{s}}\right)\hfill \\ & +{\displaystyle \frac{\lambda }{2\Gamma \left(\alpha \right)}}\left({\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}|x\left(s\right)|{\displaystyle \frac{ds}{s}}+{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}\left|x\left(s\right)\right|{\displaystyle \frac{ds}{s}}\right)\hfill \\ \hfill ⩽& {\displaystyle \frac{[{(lna)}^{\alpha }+1](\lambda k+H)}{2\Gamma (\alpha +1)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Vert Tx\Vert }_{\infty }=& \underset{t\in [1,e]}{max}{|}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]+\mathcal{B}Nx-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right)|\hfill \\ \hfill ⩽& \underset{t\in [1,e]}{max}{|}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]|+\underset{t\in [1,e]}{max}|\mathcal{B}Nx|+\underset{t\in [1,e]}{max}|{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right)|\hfill \\ \hfill ⩽& {\displaystyle \frac{H}{\Gamma (\alpha +1)}}+{\displaystyle \frac{[{(lna)}^{\alpha }+1](\lambda k+H)}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{\lambda k}{\Gamma (\alpha +1)}}\hfill \\ \hfill =& {\displaystyle \frac{[{(lna)}^{\alpha }+3](\lambda k+H)}{2\Gamma (\alpha +1)}}.\hfill \end{array}$$

Noting that ${}_H{}^C{D}_{1^{+}}^{\alpha }Tx\left(t\right)={\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]-\lambda x\left(t\right)$, it follows that

$$\begin{array}{cc}\hfill \Vert {}_H{}^C{D}_{1^{+}}^{\alpha }{Tx\left(t\right)\Vert }_{\infty }=& \underset{t\in [1,e]}{max}|{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]-\lambda x\left(t\right)|\hfill \\ \hfill ⩽& \underset{t\in [1,e]}{max}\left|{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]\right|+\lambda \underset{t\in [1,e]}{max}\left|x\left(t\right)\right|\hfill \\ \hfill ⩽& H+\lambda k<\infty .\hfill \end{array}$$

Therefore, T is uniformly bounded on $\overline{\mathbb{B}}$. To prove that T is equicontinuous on $\overline{\mathbb{B}}$, for any $x\left(t\right)\in \overline{\mathbb{B}}$ and $1⩽t_1<t_2⩽e$, we have

$$\begin{array}{cc}\hfill \Vert Tx\left(t_2\right)-Tx\left(t_1\right)\Vert =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}|{\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right){\phi }_{p\left(s\right)}^{-1}\left({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\hfill \\ & +{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}\left({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\hfill \\ & -\lambda {\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right)x\left(s\right){\displaystyle \frac{ds}{s}}+{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left|{\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right){\phi }_{p\left(s\right)}^{-1}\left({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\right|\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left|{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}\left({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx\right){\displaystyle \frac{ds}{s}}\right|\hfill \\ & +{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}|{\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right)x\left(s\right){\displaystyle \frac{ds}{s}}+{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}x\left(s\right){\displaystyle \frac{ds}{s}}|\hfill \\ \hfill ⩽& {\displaystyle \frac{H}{\Gamma \left(\alpha \right)}}\left({\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right){\displaystyle \frac{ds}{s}}+{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}{\displaystyle \frac{ds}{s}}\right)\hfill \\ & +{\displaystyle \frac{\lambda k}{\Gamma \left(\alpha \right)}}\left({\int }_1^{t_1}\left({(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}-{(ln{\displaystyle \frac{t_1}{s}})}^{\alpha -1}\right){\displaystyle \frac{ds}{s}}+{\int }_{t_1}^{t_2}{(ln{\displaystyle \frac{t_2}{s}})}^{\alpha -1}{\displaystyle \frac{ds}{s}}\right)\hfill \\ \hfill ⩽& {\displaystyle \frac{H+\lambda k}{\Gamma (\alpha +1)}}\left({(ln{t}_2)}^{\alpha }-{(ln{t}_1)}^{\alpha }\right)\to 0,t_1\to t_2.\hfill \end{array}$$

Let $G_x\left(t\right){=}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx,W=\underset{x\in \overline{\mathbb{B}},t\in [1,e]}{max}\left|G_x\left(t\right)\right|$, we have

$\Vert {}_H{}^C{D}_{1^{+}}^{\alpha }Tx\left(t_2\right)-{}_H{}^C{D}_{1^{+}}^{\alpha }Tx\left(t_1\right)\Vert$

$$\begin{array}{cc}\hfill =& |{\phi }_{p\left(t_2\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t_2\right)+\mathcal{A}Nx]-\lambda x\left(t_2\right)-{\phi }_{p\left(t_1\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t_1\right)+\mathcal{A}Nx]+\lambda x\left(t_1\right)|\hfill \\ \hfill ⩽& \left|{\phi }_{p\left(t_2\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t_2\right)+\mathcal{A}Nx]-{\phi }_{p\left(t_1\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t_1\right)+\mathcal{A}Nx]\right|+\lambda \left|x\left(t_2\right)-x\left(t_1\right)\right|\hfill \\ \hfill =& \left|{\phi }_{p\left(t_2\right)}^{-1}{G}_x\left(t_2\right)-{\phi }_{p\left(t_1\right)}^{-1}{G}_x\left(t_1\right)\right|+\lambda \left|x\left(t_2\right)-x\left(t_1\right)\right|,\hfill \end{array}$$

since $G_x\left(t\right)$ is equicontinuous, ${\phi }_{p\left(t\right)}^{-1}$ is uniformly continuous on $[1,e]\times [-W,W]$ and $x\left(t\right)$ is equicontinuous, therefore, both terms uniformly tend to zero as $t_1\to t_2$, proving equicontinuous of ${}_H{}^C{D}_{1^{+}}^{\alpha }Tx$. Combined with the previously established uniform boundedness and applying Arzelà-Ascoli theorem, both $Tx$ and ${}_H{}^C{D}_{1^{+}}^{\alpha }Tx$ are relatively compact in $C\left(\right[1,e\left]\right)$. Hence T is completely continuous.

Step 2: Proving the existence of a fixed point for the operator T in the space X. Let us define the set $\mathrm{\Omega }=\{x\in X\mid x=\mu Tx,\mu \in (0,1\left)\right\}$. According to the Theorem 2, it suffices to demonstrate that the set $\mathrm{\Omega }$ is bounded. For any $\mathrm{\Omega }$, from Equation (9) it follows that

$$\begin{array}{cc}\hfill \mid \mathcal{A}Nx\mid =& |-{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}({\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\beta -1}f(s,x\left(s\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(s\right)){\displaystyle \frac{ds}{s}}+\hfill \\ & {\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\beta -1}f(s,x\left(s\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(s\right)){\displaystyle \frac{ds}{s}}\left)\right|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\beta -1}\left(\left(\xi \left(s\right)+\phi \left(s\right)\right)|x\left(s\right){|}^{r-1}+\eta \left(s\right)|{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(s\right){|}^{r-1}\right){\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1}{2\Gamma \left(\beta \right)}}{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\beta -1}\left(\left(\xi \left(s\right)+\phi \left(s\right)\right)|x\left(s\right){|}^{r-1}+\eta \left(s\right)|{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(s\right){|}^{r-1}\right){\displaystyle \frac{ds}{s}}\hfill \\ \hfill ⩽& {\displaystyle \frac{1+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\left(\parallel \xi {\parallel }_{\infty }+\parallel \phi {\parallel }_{\infty }\parallel x{\parallel }_{\infty }^{r-1}+\parallel \eta {\parallel }_{\infty }\parallel {}_H{}^C{D}_{1^{+}}^{\alpha }x{\parallel }_{\infty }^{r-1}\right)\hfill \\ \hfill ⩽& {\displaystyle \frac{1+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\left(\parallel \xi {\parallel }_{\infty }+(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })\parallel x{\parallel }_X^{r-1}\right),\hfill \end{array}$$

which leads to

$$\begin{array}{cc}\hfill \mid {}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx\mid ⩽& {|}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)|+|\mathcal{A}Nx|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\beta -1}\left|Nx\left(s\right)\right|{\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\left(\parallel \xi {\parallel }_{\infty }+(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })\parallel x{\parallel }_X^{r-1}\right)\hfill \\ \hfill ⩽& {\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\left(\parallel \xi {\parallel }_{\infty }+(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })\parallel x{\parallel }_X^{r-1}\right).\hfill \end{array}$$

Based on the inequalities ${(u+v)}^q⩽2^q(u^q+v^q)(u,v,q>0)$ and $x^r⩽x+1,r\in [0,1],x⩾0$, it follows that for any $t\in [1,e]$, we have

$$\begin{array}{cc}\hfill \left|\mathcal{B}Nx\right|=& |-{\displaystyle \frac{1}{2}}{[}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{(}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){\mid }_{t=e}]\hfill \\ & +{\displaystyle \frac{\lambda }{2}}{(}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=a}{+}^H{I}_{1^{+}}^{\alpha }x\left(t\right){\mid }_{t=e})|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}{|}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx|{\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}{\phi }_{p\left(s\right)}^{-1}{|}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx|{\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{\lambda }{2\Gamma \left(\alpha \right)}}\left({\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}|x\left(s\right)|{\displaystyle \frac{ds}{s}}+{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}\left|x\left(s\right)\right|{\displaystyle \frac{ds}{s}}\right)\hfill \\ \hfill ⩽& Q{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}\left({\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(s\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(s\right)-1}}}(\parallel x{\parallel }_X+1)\right){\displaystyle \frac{ds}{s}}\hfill \\ \hfill ⩽& Q{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}\left({\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(s\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(s\right)-1}}}(\parallel x{\parallel }_X+1)\right){\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{\lambda +{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}\parallel x{\parallel }_X,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|x\right(t\left)\right|=& {|}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]+\mathcal{B}Nx-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right)|\hfill \\ \hfill ⩽& {|}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]|+|\mathcal{B}Nx|+\lambda {|}^H{I}_{1^{+}}^{\alpha }x\left(t\right)|\hfill \\ \hfill ⩽& 2Q{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\alpha -1}\left({\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(s\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(s\right)-1}}}(\parallel x{\parallel }_X+1)\right){\displaystyle \frac{ds}{s}}\hfill \\ & +Q{\int }_1^a{(ln{\displaystyle \frac{a}{s}})}^{\alpha -1}\left({\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(s\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(s\right)-1}}}(\parallel x{\parallel }_X+1)\right){\displaystyle \frac{ds}{s}}\hfill \\ & +Q{\int }_1^e{(ln{\displaystyle \frac{e}{s}})}^{\alpha -1}\left(\parallel \xi {\parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(s\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(s\right)-1}}}(\parallel x{\parallel }_X+1)\right){\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{\lambda (1+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}\parallel x{\parallel }_X+{\displaystyle \frac{\lambda }{\Gamma (\alpha +1)}}{\parallel x\parallel }_X\hfill \\ \hfill ⩽& {\displaystyle \frac{3+{(lna)}^{\alpha }}{\alpha }}Q\left(C_{\xi }+l(\parallel x{\parallel }_X+1)\right)+{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}\parallel x{\parallel }_X.\hfill \end{array}$$

where $C_{\xi }:max\left\{{\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(m\right)-1}}},{\parallel \xi \parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(M\right)-1}}}\right\}$, and given that

$$\begin{array}{cc}\hfill \left|{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(t\right)\right|=& |{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]-\lambda x\left(t\right)|\hfill \\ \hfill ⩽& \left|{\phi }_{p\left(t\right)}^{-1}{[}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]\right|+\lambda \left|x\left(t\right)\right|\hfill \\ \hfill ⩽& {\left({\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\left(\parallel \xi {\parallel }_{\infty }+(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })\parallel x{\parallel }_X^{r-1}\right)\right)}^{{\displaystyle \frac{1}{p\left(t\right)-1}}}+\lambda {\parallel x\parallel }_X\hfill \\ \hfill ⩽& 2\Gamma \left(\alpha \right)Q\left(\parallel \xi {\parallel }_{\infty }^{{\displaystyle \frac{1}{p\left(t\right)-1}}}+{(\parallel \phi {\parallel }_{\infty }+\parallel \eta {\parallel }_{\infty })}^{{\displaystyle \frac{1}{p\left(t\right)-1}}}(\parallel x{\parallel }_X+1)\right)+\lambda {\parallel x\parallel }_X\hfill \\ \hfill ⩽& 2\Gamma \left(\alpha \right)Q\left(C_{\xi }+l(\parallel x{\parallel }_X+1)\right)+\lambda {\parallel x\parallel }_X.\hfill \end{array}$$

It follows from the above equation that

$$\begin{array}{cc}\hfill \parallel x{\parallel }_X⩽& {\displaystyle \frac{3+{(lna)}^{\alpha }}{\alpha }}Q\left(C_{\xi }+l(\parallel x{\parallel }_X+1)\right)\hfill \\ & +2\Gamma \left(\alpha \right)Q\left(C_{\xi }+l(\parallel x{\parallel }_X+1)\right)+\lambda \parallel x{\parallel }_X+{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}\parallel x{\parallel }_X.\hfill \end{array}$$

It follows from the conditions that there exists a constant $M>0$ such that $\parallel x{\parallel }_X⩽M$. According to the Theorem 2, the operator T has a fixed point in $\mathrm{\Omega }$, which implies that the boundary value problem (3) has at least one solution. □

4. Ulam-Hyers Stability Analysis

In this section, we investigate the stability of solutions to the boundary value problem (3) in the sense of Ulam-Hyers. The stability analysis ensures that small perturbations in the equation or boundary conditions do not lead to large deviations in the solutions, which is crucial for the robustness of the model in practical applications.

We begin by revising the definition of Ulam-Hyers stability to include the requirement that the perturbed solution satisfies the boundary conditions. This ensures that the error function satisfies homogeneous boundary conditions, allowing us to apply the integral representation from Lemma 3.

Definition 6.

The boundary value problem (3) is said to be Ulam-Hyers stable if there exists a real number $C>0$ such that for every $\epsilon >0$ and for every function $y\in X$ satisfying the boundary conditions of (3)

$$y\left(a\right)=-y\left(e\right),{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]{\mid }_{t=a}=-{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]{\mid }_{t=e}.$$

and the inequality

$$\left|{}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]-f(t,y\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }y\left(t\right))\right|⩽\epsilon ,t\in [1,e],$$

there exists a solution $x\in X$ of (3) such that

$${∥y-x∥}_X⩽C\epsilon .$$

To establish these stability results, we impose the following Lipschitz condition on the nonlinear function f:

Hypothesis 1.

There exists a constant $L>0$ such that for all $t\in [1,e]$ and $u_1,u_2,v_1,v_2\in \mathbb{R}$,

$$\left|f(t,u_1,v_1)-f(t,u_2,v_2)\right|⩽L(\left|u_1-u_2\right|+\left|v_1-v_2\right|).$$

Additionally, we introduce the following assumption on the inverse p(t)-Laplacian operator.

Hypothesis 2.

There exists a constant $L_{\phi }>0$ and a bounded set $\mathbb{B}\in \mathbb{R}$ such that for all $t\in [1,e]$ and all $u,v\in \mathbb{B}$,

$$\left|{\phi }_{p\left(t\right)}^{-1}\left(u\right)-{\phi }_{p\left(t\right)}^{-1}\left(v\right)\right|⩽L_{\phi }\left|u-v\right|.$$

We now state and prove the main stability theorems.

Theorem 4

(Ulam-Hyers Stability). Under the assumptions of Theorem 3 and Hypotheses 1 and 2, if the following inequality holds

$$\mathrm{\Lambda }:=({\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}+1)(L_{\phi }L{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}+\lambda )<1,$$

then the boundary value problem (3) is Ulam-Hyers stable.

Proof. 

Let $\epsilon >0$ and let $y\in X$ be a function satisfying the boundary conditions of (3) and the inequality

$$\left|{}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]-f(t,y\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }y\left(t\right))\right|⩽\epsilon ,t\in [1,e].$$

Define the perturbation function $h\left(t\right)$ by

$$h\left(t\right)={}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]-f(t,y\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }y\left(t\right)),$$

so that ${∥h\left(t\right)∥}_{\infty }⩽\epsilon .$ Then $y\left(t\right)$ satisfies the perturbed equation

$${}_H{}^C{D}_{1^{+}}^{\beta }{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )y\left(t\right)\right]=N_{h}y\left(t\right).$$

(11)

where $N_{h}y\left(t\right)=f(t,y\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }y\left(t\right))+h\left(t\right)$. By Theorem 3, there exists a solution $x\left(t\right)\in X$ of the boundary value problem (3). Define the error function $z\left(t\right)=y\left(t\right)-x\left(t\right)$. Since both $y\left(t\right)$ and $x\left(t\right)$ satisfy the boundary conditions, it follows that $z\left(t\right)$ satisfies the homogeneous boundary conditions

$$z\left(a\right)=-z\left(e\right),{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )z\left(t\right)\right]{\mid }_{t=a}=-{\phi }_{p\left(t\right)}\left[({}_H{}^C{D}_{1^{+}}^{\alpha }+\lambda )z\left(t\right)\right]{\mid }_{t=e}.$$

Now, we derive the integral representation for both $y\left(t\right)$ and $x\left(t\right)$. For $y\left(t\right)$, applying Lemma 3 to the perturbed Equation (11), we have

$$y\left(t\right)={}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y\right]+\mathcal{B}{N}_{h}y-{\lambda }^H{I}_{1^{+}}^{\alpha }y\left(t\right),$$

where $\mathcal{A}{N}_{h}y$ and $\mathcal{B}{N}_{h}y$ are defined similarly to $\mathcal{A}Nx$ and $\mathcal{B}Nx$ in Lemma 3, but with $Nx\left(t\right)$ replaced by $N_{h}y\left(t\right)=f(\tau ,y,{}_H{}^C{D}_{1^{+}}^{\alpha }y)+h\left(\tau \right).$

• For $x\left(t\right)$, we have the standard representation

$$x\left(t\right)={}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx\right]+\mathcal{B}Nx-{\lambda }^H{I}_{1^{+}}^{\alpha }x\left(t\right).$$

Subtracting these two representations, we obtain

$$\begin{array}{cc}\hfill z\left(t\right)=& {}^H{I}_{1^{+}}^{\alpha }\left({\phi }_{p\left(t\right)}^{-1}[{}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y]-{\phi }_{p\left(t\right)}^{-1}[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]\right)\hfill \\ \hfill +& (\mathcal{B}{N}_{h}y-\mathcal{B}Nx)-{\lambda }^H{I}_{1^{+}}^{\alpha }z\left(t\right).\hfill \end{array}$$

(12)

By Theorem 3, the solutions $x\left(t\right)$ and the perturbed function $y\left(t\right)$ are bounded, which implies that the arguments of ${\phi }_{p\left(t\right)}^{-1}$ in the above expressions are confined to the bounded set $\mathbb{B}$ in assumption (Hypothesis 2). Therefore, we can apply (Hypothesis 2) to estimate the difference

$$\begin{array}{cc}\hfill & \left|{\phi }_{p\left(t\right)}^{-1}[{}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y]-{\phi }_{p\left(t\right)}^{-1}[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx]\right|\hfill \\ \hfill ⩽& L_{\phi }\left|{}^H{I}_{1^{+}}^{\beta }[f(\tau ,y,{}_H{}^C{D}_{1^{+}}^{\alpha }y)-f(\tau ,x,{}_H{}^C{D}_{1^{+}}^{\alpha }x)+h\left(\tau \right)]+(\mathcal{A}{N}_{h}y-\mathcal{A}Nx)\right|.\hfill \end{array}$$

Using the Lipschitz condition (Hypothesis 1) and the bound on $h\left(t\right)$, we have

$$\begin{array}{cc}\hfill & \left|{}^H{I}_{1^{+}}^{\beta }[f(\tau ,y,{}_H{}^C{D}_{1^{+}}^{\alpha }y)-f(\tau ,x,{}_H{}^C{D}_{1^{+}}^{\alpha }x)+h\left(\tau \right)]\right|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\beta -1}[L(\left|z\left(s\right)\right|+\left|{}_H{}^C{D}_{1^{+}}^{\alpha }z\left(s\right)\right|+\epsilon ]{\displaystyle \frac{ds}{s}}\hfill \\ \hfill ⩽& {\displaystyle \frac{L{∥z∥}_X+\epsilon }{\Gamma (\beta +1)}}.\hfill \end{array}$$

Similarly, we estimate

$$\left|\mathcal{A}{N}_{h}y-\mathcal{A}Nx\right|⩽{\displaystyle \frac{1+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon ).$$

Combining these estimates

$$\begin{array}{cc}\hfill & \left|{}^H{I}_{1^{+}}^{\beta }[f(\tau ,y,{}_H{}^C{D}_{1^{+}}^{\alpha }y)-f(\tau ,x,{}_H{}^C{D}_{1^{+}}^{\alpha }x)+h\left(\tau \right)]+(\mathcal{A}{N}_{h}y-\mathcal{A}Nx)\right|\hfill \\ \hfill ⩽& {\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon ).\hfill \end{array}$$

Therefore, using assumption (Hypothesis 2)

$$\begin{array}{cc}\hfill & \left|{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }(f(\tau ,y,{}_H{}^C{D}_{1^{+}}^{\alpha }y)+h\left(\tau \right))+\mathcal{A}{N}_{h}y\right]-{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }\left(f(\tau ,x,{}_H{}^C{D}_{1^{+}}^{\alpha }x)\right)+\mathcal{A}Nx\right]\right|\hfill \\ \hfill ⩽& L_{\phi }\left[{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon )\right].\hfill \end{array}$$

Similarly, we estimate

$$\begin{array}{cc}\hfill & |\mathcal{B}{N}_{h}y-\mathcal{B}Nx|\hfill \\ \hfill =& \left|-{\displaystyle \frac{1}{2}}\left[{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y){{|}_{t=a}+{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y)|}_{t=e}\right]\right.\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{2}}\left({}^H{I}_{1^{+}}^{\alpha }y\left(t\right){{|}_{t=a}+{}^H{I}_{1^{+}}^{\alpha }y\left(t\right)|}_{t=e}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left[{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx){{|}_{t=a}+{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx)|}_{t=e}\right]\hfill \\ \hfill & \left.-{\displaystyle \frac{\lambda }{2}}\left({}^H{I}_{1^{+}}^{\alpha }x\left(t\right){{|}_{t=a}+{}^H{I}_{1^{+}}^{\alpha }x\left(t\right)|}_{t=e}\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^a{\left(ln{\displaystyle \frac{a}{s}}\right)}^{\alpha -1}\left|{\phi }_{p\left(s\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(s\right)+\mathcal{A}{N}_{h}y)-{\phi }_{p\left(s\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2\Gamma \left(\alpha \right)}}{\int }_1^e{\left(ln{\displaystyle \frac{e}{s}}\right)}^{\alpha -1}\left|{\phi }_{p\left(s\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(s\right)+\mathcal{A}{N}_{h}y)-{\phi }_{p\left(s\right)}^{-1}({}^H{I}_{1^{+}}^{\beta }Nx\left(s\right)+\mathcal{A}Nx)\right|{\displaystyle \frac{ds}{s}}\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{2\Gamma \left(\alpha \right)}}\left({\int }_1^a{\left(ln{\displaystyle \frac{a}{s}}\right)}^{\alpha -1}\left|z\left(s\right)\right|{\displaystyle \frac{ds}{s}}+{\int }_1^e{\left(ln{\displaystyle \frac{e}{s}}\right)}^{\alpha -1}\left|z\left(s\right)\right|{\displaystyle \frac{ds}{s}}\right).\hfill \end{array}$$

Using the same estimate as above for the difference of ${\phi }_{p\left(t\right)}^{-1}$, and noting that the integrals are bounded, we obtain

$$|\mathcal{B}{N}_{h}y-\mathcal{B}Nx|⩽{\displaystyle \frac{1+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{L}_{\phi }(L{∥z∥}_X+\epsilon )+{\displaystyle \frac{\lambda (1+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}{∥z∥}_X.$$

Now, we estimate ${∥z∥}_{\infty }$. From Equation (12), we have

$$\begin{array}{cc}\hfill \left|z\left(t\right)\right|⩽& \left|{}^H{I}_{1^{+}}^{\alpha }{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y\right]-{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx\right]\right|\hfill \\ & +|(\mathcal{B}{N}_{h}y-\mathcal{B}Nx){|-\lambda |}^H{I}_{1^{+}}^{\alpha }z\left(t\right)|\hfill \\ \hfill ⩽& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^t{(ln{\displaystyle \frac{t}{s}})}^{\alpha -1}{L}_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon ){\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{L}_{\phi }(L{∥z∥}_X+\epsilon )\hfill \\ & +{\displaystyle \frac{\lambda (1+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}{∥z∥}_X+{\displaystyle \frac{\lambda }{\Gamma (\alpha +1)}}{∥z∥}_X\hfill \\ \hfill ⩽& L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon )({\displaystyle \frac{1}{\Gamma (\alpha +1)}}\hfill \\ & +{\displaystyle \frac{1+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}})+{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}{∥z∥}_X.\hfill \end{array}$$

we get

$$\begin{array}{cc}\hfill {∥z\left(t\right)∥}_{\infty }⩽& L_{\phi }{\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon )+{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}{∥z∥}_X.\hfill \end{array}$$

(13)

Now, for the fractional derivative term, we have

$$\begin{array}{cc}\hfill {\left|{}_H{}^C{D}_{1^{+}}^{\alpha }z\left(t\right)\right|}_{\infty }=& \left|{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }{N}_{h}y\left(t\right)+\mathcal{A}{N}_{h}y\right]-{\phi }_{p\left(t\right)}^{-1}\left[{}^H{I}_{1^{+}}^{\beta }Nx\left(t\right)+\mathcal{A}Nx\right]-\lambda z\left(t\right)\right|\hfill \\ \hfill ⩽& L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon )+\lambda {∥z∥}_X.\hfill \end{array}$$

Taking the supremum

$${∥{}_H{}^C{D}_{1^{+}}^{\alpha }z\left(t\right)∥}_{\infty }⩽L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}(L{∥z∥}_X+\epsilon )+\lambda {∥z∥}_X.$$

(14)

Combining (13) and (14), we get

$$\begin{array}{cc}\hfill {∥z∥}_X=& {∥z∥}_{\infty }+{∥{}_H{}^C{D}_{1^{+}}^{\alpha }z\left(t\right)∥}_{\infty }\hfill \\ \hfill ⩽& L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}L{∥z∥}_X+L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}\epsilon \hfill \\ & +{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}{∥z∥}_X+L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}L{∥z∥}_X+L_{\phi }{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}\epsilon +\lambda {∥z∥}_X\hfill \\ \hfill ⩽& [{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}]L_{\phi }L{∥z∥}_X\hfill \\ & +[{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}{\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}}]L_{\phi }\epsilon +[{\displaystyle \frac{\lambda (3+{(lna)}^{\alpha })}{2\Gamma (\alpha +1)}}+\lambda ]{∥z∥}_X.\hfill \end{array}$$

Simplify the expressions by defining

$${\mathrm{\Theta }}_1={\displaystyle \frac{3+{(lna)}^{\beta }}{2\Gamma (\beta +1)}},{\mathrm{\Theta }}_2={\displaystyle \frac{3+{(lna)}^{\alpha }}{2\Gamma (\alpha +1)}}.$$

Then

$${∥z∥}_X⩽({\mathrm{\Theta }}_2+1)(L_{\phi }L{\mathrm{\Theta }}_1+\lambda ){∥z∥}_X+L_{\phi }{\mathrm{\Theta }}_1({\mathrm{\Theta }}_2+1)\epsilon .$$

Let $\mathrm{\Lambda }=({\mathrm{\Theta }}_2+1)(L_{\phi }L{\mathrm{\Theta }}_1+\lambda )$, then

$${∥z∥}_X⩽\mathrm{\Lambda }{∥z∥}_X+L_{\phi }{\mathrm{\Theta }}_1({\mathrm{\Theta }}_2+1)\epsilon .$$

Since $\mathrm{\Lambda }<1$ by assumption, we have

$${∥z∥}_X⩽{\displaystyle \frac{L_{\phi }{\mathrm{\Theta }}_1({\mathrm{\Theta }}_2+1)}{1-\mathrm{\Lambda }}}\epsilon =C\epsilon ,$$

where $C={\displaystyle \frac{L_{\phi }{\mathrm{\Theta }}_1({\mathrm{\Theta }}_2+1)}{1-\mathrm{\Lambda }}}$. This completes the proof of Ulam-Hyers stability. □

5. Illustrative Examples

In this section, we present an example that satisfies boundary value problem (3), verifying the conditions of Theorems 3 and 4. Here, a quadratic function is chosen for $p\left(t\right)$ to ensure that the basic assumptions $p\left(t\right)>1$ and $p\left(a\right)=p\left(e\right)$ are met, which facilitates explicit computation of $p_m$ and $p_M$ to verify key constants in the theorems. Extremely small coefficients are selected for the nonlinear term f so that both the growth condition (9) required by Theorem 3 and the Lipschitz condition required by Theorem 4 are satisfied simultaneously, thereby verifying both the existence of solutions and Ulam–Hyers stability. The parameter a is intentionally chosen inside the interval to demonstrate the generalized case where the boundary parameter is not an endpoint, highlighting the broader applicability of the model studied in this paper compared to traditional endpoint anti-periodic problems.

Example 1.

Consider the boundary value problem

$$\left\{\begin{array}{c}{}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{2}{3}}}{\phi }_{(-t^2+({\displaystyle \frac{3}{2}}+e)t+2)}\left[({}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}+{\displaystyle \frac{1}{50}})x\left(t\right)\right]=sint+{\displaystyle \frac{1}{10000}}x+{\displaystyle \frac{1}{50000}}{}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}x,t\in [1,e],\hfill \\ x\left({\displaystyle \frac{3}{2}}\right)=-x\left(e\right),\hfill \\ {\phi }_{(-t^2+({\displaystyle \frac{3}{2}}+e)t+2)}({}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}+{\displaystyle \frac{1}{50}})x\left(t\right){|}_{t={\displaystyle \frac{3}{2}}}=-{\phi }_{(-t^2+({\displaystyle \frac{3}{2}}+e)t+2)}({}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}+{\displaystyle \frac{1}{50}})x\left(t\right){|}_{t=e},\hfill \end{array}\right.$$

(15)

where

$$f(t,x\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(t\right))=sint+{\displaystyle \frac{1}{10000}}x+{\displaystyle \frac{1}{50000}}{}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}x,$$

$$p\left(t\right)=-t^2+({\displaystyle \frac{3}{2}}+e)t+2,$$

$$\alpha ={\displaystyle \frac{3}{4}},\beta ={\displaystyle \frac{2}{3}},\lambda ={\displaystyle \frac{1}{50}},a={\displaystyle \frac{3}{2}},L_{\phi }=1,$$

Under the condition $r=2,\xi \left(t\right)=1,\phi \left(t\right)={\displaystyle \frac{1}{10000}},\eta \left(t\right)={\displaystyle \frac{1}{50000}}$, the nonlinear function f satisfies the growth condition:

$$\left|f(t,x\left(t\right),{}_H{}^C{D}_{1^{+}}^{\alpha }x\left(t\right))\right|⩽1+{\displaystyle \frac{1}{10000}}|x\left(t\right)|+{\displaystyle \frac{1}{50000}}\left|{}_H{}^C{D}_{1^{+}}^{{\displaystyle \frac{3}{4}}}x\left(t\right)\right|.$$

$$p_m=p\left(1\right)=-1^2+({\displaystyle \frac{3}{2}}+e)\times 1+2={\displaystyle \frac{5}{2}}+e\approx 5.2183,$$

$$p_M=p({\displaystyle \frac{3}{4}}+{\displaystyle \frac{e}{2}})=-{({\displaystyle \frac{3}{4}}+{\displaystyle \frac{e}{2}})}^2+({\displaystyle \frac{3}{2}}+e)\times ({\displaystyle \frac{3}{4}}+{\displaystyle \frac{e}{2}})+2={\displaystyle \frac{41}{16}}+{\displaystyle \frac{3e}{4}}+{\displaystyle \frac{e^2}{4}}\approx 6.4485,$$

$${\displaystyle \frac{1}{p_m-1}}\approx 0.2371,{\displaystyle \frac{1}{p_M-1}}\approx 0.1835,$$

$$Q={\displaystyle \frac{{[6+2{(ln\frac{3}{2})}^{\frac{2}{3}}]}^{\frac{1}{4.2183}}}{2\Gamma \left(\frac{3}{4}\right){(2\Gamma \left(\frac{5}{3}\right))}^{\frac{1}{5.4485}}}}\approx 0.5826,L={\displaystyle \frac{1}{10000}}+{\displaystyle \frac{1}{50000}}={\displaystyle \frac{3}{25000}},l={\left({\displaystyle \frac{3}{25000}}\right)}^{0.1835},$$

$${\left({\displaystyle \frac{3}{25000}}\right)}^{0.1835}\left({\displaystyle \frac{3+ln{\left(\frac{3}{2}\right)}^{\frac{3}{4}}}{\frac{3}{4}}}\times 0.5826+2\Gamma \left({\displaystyle \frac{3}{4}}\right)\times 0.5826\right)+{\displaystyle \frac{\frac{1}{50}(3+ln{\left(\frac{3}{2}\right)}^{\frac{3}{4}})}{2\Gamma (\frac{3}{4}+1)}}+{\displaystyle \frac{1}{50}}\approx 0.8504<1.$$

According to Theorem 3, the boundary value problem (15) has at least one solution.

$${\mathrm{\Theta }}_1={\displaystyle \frac{3+{(ln\frac{3}{2})}^{\frac{2}{3}}}{2\Gamma \left(\frac{5}{3}\right)}}\approx {\displaystyle \frac{3+0.5477}{2\times 0.9027}}\approx 1.965,$$

$${\mathrm{\Theta }}_2={\displaystyle \frac{3+{(ln\frac{3}{2})}^{\frac{3}{4}}}{2\Gamma \left(\frac{7}{4}\right)}}\approx {\displaystyle \frac{3+0.5028}{2\times 0.9191}}\approx 1.906,$$

then

$$\mathrm{\Lambda }:=({\displaystyle \frac{3+{(ln\frac{3}{2})}^{\frac{3}{4}}}{2\Gamma \left(\frac{7}{4}\right)}}+1)(1\times {\displaystyle \frac{3}{25000}}{\displaystyle \frac{3+{(ln\frac{3}{2})}^{\beta }}{2\Gamma \left(\frac{5}{3}\right)}}+{\displaystyle \frac{1}{50}})\approx 0.0588<1.$$

Since $\mathrm{\Lambda }<1$, the Ulam-Hyers stability condition is satisfied. The stability constant C is given by:

$$C={\displaystyle \frac{1\times 1.965\times (1.906+1)}{1-0.0588}}\approx 6.066.$$

This means that for any $\epsilon >0$ and for any function y satisfying the boundary conditions and the inequality

$$\left|{}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{2}{3}}}{\phi }_{-t^2+({\displaystyle \frac{3}{2}}+e)t+2}[({}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}+{\displaystyle \frac{1}{50}})y\left(t\right)]-f(t,y\left(t\right),{}_H{}^{C}D_{1^{+}}^{{\displaystyle \frac{3}{4}}}y\left(t\right))\right|⩽\epsilon ,$$

there exists a solution x of (15) such that

$${∥y-x∥}_X⩽6.066\epsilon .$$

According to Theorem 4, the boundary value problem (15) satisfies the Ulam-Hyers stability condition.

6. Conclusions

This paper studied a class of boundary value problems for fractional Langevin equations involving a time-varying p(t)-Laplacian operator and parametric anti-periodic boundary conditions (with $a\in [1,e]$). By analyzing this model that combines dynamic nonlinearity and asymmetric boundaries, the limitations of the constant-exponent p-Laplacian in describing non-stationary diffusion processes are overcome.

Methodologically, the Schaefer fixed point theorem was employed instead of traditional contraction mappings to establish the existence of solutions under non-compactness conditions, thereby relaxing the structural constraints on the nonlinear term. Furthermore, under the assumption of Lipschitz continuity, the system was proven to possess Ulam–Hyers stability, and explicit error constants were provided, offering a theoretical basis for perturbation analysis and error estimation in numerical computations and practical control.

This study extends the theoretical framework for fractional Langevin equations, and the established general results can provide a theoretical foundation for research in areas such as viscoelastic material mechanics, anomalous diffusion in heterogeneous media, and transport processes with memory effects. Future work may focus on theoretical extensions to more general variable-order operators or non-local boundary conditions, further investigation of qualitative properties of solutions such as asymptotic behavior and regularity, development of efficient and stable numerical algorithms with simulation verification, and exploration of applications in specific physical models such as anomalous transport and control of viscoelastic systems.