The Existence and Uniqueness of Solutions for Variable-Order Fractional Differential Equations with Antiperiodic Fractional Boundary Conditions

Wang, Fang; Liu, Lei

doi:https://doi.org/10.1155/2022/7663192

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7663192 | https://doi.org/10.1155/2022/7663192

The Existence and Uniqueness of Solutions for Variable-Order Fractional Differential Equations with Antiperiodic Fractional Boundary Conditions

Fang Wang¹and Lei Liu¹

Academic Editor: Gennaro Infante

Received26 May 2022

Revised23 Jul 2022

Accepted29 Jul 2022

Published05 Sept 2022

Abstract

In this paper, we discuss the existence and uniqueness of solutions for nonlinear fractional differential equations of variable order with fractional antiperiodic boundary conditions. The main results are obtained by using fixed point theorem.

1. Introduction

Fractional calculus has become one of the important tools for the development of modern society; the fractional differential equation with variable order has gained lots of interest [1–4]. Some researchers have investigated the physical background and numerical analysis of fractional differential equations of variable order [5–8]. In [9], Bushnaq et al. used Bernstein polynomials with nonorthogonal basis to establish operational matrices for variable-order integration and differentiation which convert the considered problem to some algebraic type matrix equations and obtained numerical solution to variable-order fractional differential equations by numerical simulation. In [10], Shah et al. proposed a new algorithm for numerical solutions to variable-order partial differential equations, used properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration, and got the numerical solution by numerical experiments.

In recent years, the antiperiodic boundary value problem of fractional differential equation has gradually become the focus of research, which have broad application in engineering and sciences such as physics, mechanics, chemistry, economics, and biology [11–17]. In [18], Ahmad and Nieto considered the following antiperiodic fractional boundary value problems: where denotes the Caputo fractional derivative of order and is a given continuous function.

The problems related to the antiperiodic boundary value condition have been considered in [19–26], but the antiperiodic boundary value problem of fractional differential equation with variable order is almost not considered. In this paper, we investigate the existence of solutions for an antiperiodic fractional boundary value problem given by where denotes the Caputo fractional derivative of order , , denotes the Caputo fractional derivative of variable order , , is a positive constant, and is a given continuous function.

2. Preliminary Knowledge

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

Definition 1 (see [27]). The Riemann-Liouville fractional integral of order for a continuous function : is defined as provided the integral exists.

Definition 2 (see [27]). For times absolutely continuous function :, the Caputo derivative of fractional order is defined as where denotes the integer part of the real number .

Definition 3 (see [3]). The Riemann-Liouville fractional integral of variable order for a continuous function : is defined as provided that the right-hand side is pointwise defined.

Definition 4 (see [3]). For times absolutely continuous function : , the Caputo fractional derivative of variable order is defined as

Definition 5 (see [25]). Let , is called a generalized interval if it is either an interval, or or .
A finite set is called a partition of if each in lies in exactly one of the generalized intervals in .
A function : is called piecewise constant with respect to partition of if for any , is constant on .

Theorem 6 (see [27]). Let be a closed, convex, and nonempty subset of a Banach space ; let : be a continuous mapping such that is a relatively compact subset of . Then, has at least one fixed point in .

Lemma 7 (see [27]). Let and let or then

Lemma 8 (see [27]). Let for for if or then

3. Main Results

Let . Denote be the Banach space of all continuous functions with the norm and introduce the following assumption.

Let be an integer, be a partition of the interval , and be a piecewise constant function with respect to with the following forms: where are constants, and is the indicator of the interval , (with , ), for , and for elsewhere.

Let be the Banach space of all continuous functions with the norm , (with , ).

The Caputo fractional derivative of variable order for the function could be presented as a sum of Caputo fractional derivatives of constant orders by Definition 4, :

Thus, according to (9), problem (2) can be written in the following form:

Definition 9. The problem (2) has a solution, if there are functions , so that , satisfy (10) and .
Let the function be such that on , then consider (2) as the following form:

Proposition 10. For any , is a unique solution of problem (11) if and only if satisfy the integral equation: where is Green’s function given by

Proof. If is a solution of problem (11), applying on both sides of (11), according to Lemma 8, we get according to the facts that , , and initial condition of problem (11), we get Thus, the solution of problem (11) is Green’s function can be written as It implies that is the solution to the integral equation (12). In turn, if is the solution to the integral equation (12), according to Lemma 7, we deduce that is the solution of the problem (11). Hence, we complete this proof.

Theorem 11. Assume that and there exists a positive constant such that , for any . Then, problem (11) has at least a solution.

Proof. According to Proposition 10, problem (11) is equivalent to the following integral equation: Define operator by ,where , observe that is a closed, bounded, and convex subset of Banach space . For any , we have It implies is well defined.
Now, we consider the continuity of operator . Since , given an arbitrary , for any ,we can findsuch that . When for , for any , we have We get the operator is continuous.
For each , we prove that if , and , then : By the mean value theorem, we have Therefore, . According to the previous analysis, we know that is equicontinuous and uniformly bounded. We know by the Arzela-Ascoli theorem that is compact on , so the operator is completely continuous. So, Theorem 11 implies that the antiperiodic boundary value problem of variable order (11) has at least a solution on . This completes the proof.

Theorem 12. Assume that such that , for any , and if , then problem (11) has a unique solution.

Proof. Let , , where For any , we have It implies that .
For any , It follows that is a contraction mapping. Thus, the Banach fixed point theorem yields that has a unique fixed point which is the unique solution of the antiperiodic boundary value problem (11).

Theorem 13. Assume that , hold for all =1, 2, , , then the problem (2) has at least a solution in .

Proof. According to Theorem 11, problem (11) has at least a solution . Define the function Thus, the function satify (10) and . Then, the function is a solution of problem (2) in .

Theorem 14. Assume that , hold for all , then problem (2) has a unique solution in .

The proof of Theorem 14 is similar to Theorem 13.

4. Conclusion

This paper is devoted to considering the existence of solutions to the antiperiodic fractional boundary value problems for nonlinear differential equations of variable order, which is a piecewise constant function based on the essential difference about the variable order. Based on the fixed point theory, the results are obtained. It is also worth considering fractional differential equations of variable-order problems related to thermodynamics, fluid mechanics, resonance, etc.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is financially supported by the National Natural Science Foundation of China (No. 12001064), the Education Department of Hunan Province Project (No. 20B006), and the Graduate Research Innovation Project of Changsha University of Science and Technology (No. CX2021SS88).

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Copyright © 2022 Fang Wang and Lei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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