Some Results on Fractional m-Point Boundary Value Problems

Zhu, Huijuan; Han, Baozhi; Shen, Jun

doi:https://doi.org/10.1155/2021/3152688

Journal of Function Spaces

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

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Nonlinear and Variational Analysis and their Applications 2021

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Research Article | Open Access

Volume 2021 | Article ID 3152688 | https://doi.org/10.1155/2021/3152688

Some Results on Fractional m-Point Boundary Value Problems

Huijuan Zhu,¹Baozhi Han,¹and Jun Shen²

Academic Editor: Zisen Mao

Received27 May 2021

Accepted19 Jul 2021

Published06 Aug 2021

Abstract

In this paper, we will apply some fixed-point theorems to discuss the existence of solutions for fractional m-point boundary value problems In addition, we also present Lyapunov’s inequality and Ulam-Hyers stability results for the given m-point boundary value problems.

1. Introduction

Mathematical models due to fractional differential equations can describe the natural phenomenon in physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, and control of dynamical systems (see [1–4]). Due to the nonlocal characteristics and the rapid development of the theory of fractional operators, some authors have investigated different aspects of fractional differential equations including existence of solutions, Lyapunov’s inequality, and Hyers-Ulam stability for fractional differential equations by different mathematical techniques. For example, first, many authors have discussed the existence of nontrivial solutions of fractional differential equations in nonsingular case as well as singular case. Usually, the proof is based on either the method of upper and lower solutions, fixed-point theorems, alternative principle of Leray-Schauder, topological degree theory, or critical point theory. We refer the readers to [5–20]. Second, Lyapunov, during his study of general theory of stability of motion in 1892, introduced the stability criterion for second-order differential equations, which yielded a counter inequality be called Lyapunov inequality (see [21, 22]). Since then, we can find considerable modifications of Lyapunov-type inequality of differential equations, such as linear differential-algebraic equations, fractional differential equations, extreme Pucci equations, and dynamic equations, which are applied to study the stability and disconjugacy or oscillatory criterion for the mentioned problems, and we refer the readers to [23–32]. Finally, the stability of functional equations was originally raised by Hyers in 1941 (see [33, 34]). Thereafter, the stability properties of all kinds of equations have attracted the attention of many mathematicians. To see more details on the Ulam-Hyers stability and Ulam-Hyers-Rassias of differential equations, we refer the readers to [35–38].

Inspired by the references, this paper is mainly concerned with the existence, Lyapunov’s inequality, and Ulam-Hyers stability results for the m-point boundary value problems. where , , , and satisfy the following assumptions:

(H1) and

(H2) is Lebesgue integral

(H3) is continuous

For these goals, we first convert problem (1) into an integral equation via Green function. Furthermore, we study the properties and estimates of the Green function. Then, on the basis of these properties, we apply some fixed-point theorems to establish some existence results of problem (1) under some suitable conditions. In addition, the Lyapunov inequality and Hyers-Ulam stability of the proposed problem are also considered.

2. Preliminaries

Before beginning the main results, we state some classic and modified definitions and lemmas from fractional calculus.

Definition 1 [4]. The fractional integral of order of a function is given by provided the right-hand side is pointwise defined on .

Definition 2 [4]. The fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined on .

Definition 3 [21]. Assume that , then for some , where is the smallest integer greater than or equal to .

Lemma 4. Assume that (H1) holds. Then, for any , the boundary value problem has a unique solution . Let , and we have (i)for , (ii)for (iii)for , , (iv)for , , (v)for (vi)for ,

Proof. From Definition 3, it follows that Since , it is clear that . Then, On one hand, taking the derivative of , we can get On the other hand, combining the boundary conditions , we have Furthermore, we have According to these above expressions, we have Then, from , it follows that which yields If , , we have ☐

In the similar way, we also can get the expression of on other intervals.

Lemma 5. Assume that (H1) holds. Then, satisfies the following properties: (I)Sign of (i)(ii)(II)The range of (1)For (2)For , (3)For

Proof. For , by the definition of , it is clear that is continuous and derivativable with respect to at . On one hand, if , we have On the other hand, if , we have Then, is nonincreasing on t, which yields that So for , , it concludes that For , we have For , we have ☐

Let where

From Lemma 5, it is clear that , where .

Lemma 6. Assume that (H1) holds and . Then, .

Proof. Let From (H1) and , we can verify that Also, we can verify that So, it concludes that , namely, .☐

3. Main Results

3.1. Existence Results

Theorem 7. Assume that (H1)-(H3) hold. In addition, there exists a positive constant such that

Then, problem (1) has a unique solution if .

Proof. Let is a Banach space with the norm . From Lemma 4, it is clear that solutions of (1) can be rewritten as fixed points of operator , which is defined by Now, we show that and is a contraction map, where with On one hand, for any , we have which implies that .
On the other hand, for any , , we have which implies that is a contraction map.☐

Therefore, by the Banach contraction mapping principle, it follows that the operator has a unique fixed point, which is the unique solution for problem (1).

Theorem 8. Assume that (H1)-(H3) hold. In addition, there exists a positive constant such that for u ∈ R. Then, problem (1) has at least one solution.

The proof is based on the following fixed-point theorem.

Lemma 9 [39]. Let be a Banach space, is a closed, convex subset of , an open subset of , and . Suppose that is completely continuous. Then, either (i) has a fixed point in , or(ii)there are (the boundary of in ) and with

Proof of Theorem 8. First, we show that the operator is uniformly bounded.
For any , we have which implies that is uniformly bounded.
Second, for , from Lemma 4, we have (i)if (ii)if which implies that is bounded for , . In the similar way, we know that there exists a such that for , .Furthermore, for , we have Therefore, applying the Arzela-Ascoli theorem [39], we can find that is relatively compact.
Third, we claim that is continuous. Assume that , which converges to u₀ uniformly on [0,1]. Since is uniformly bounded and equicontinuous on [0,1], from the Arzela-Ascoli theorem, it follows that there exists a uniformly convergent subsequence in . Let be a subsequence which converges to uniformly on [0,1]. Observe that Furthermore, by Lebesgue’s dominated convergence theorem and letting , we have namely, . This shows that each subsequence of uniformly converges to . Therefore, the sequence uniformly converges to . This means that is continuous at . So, is continuous on . Thus, is completely continuous.
Finally, let with . If is a solution of problem (1), then, for , , we have which yields a contradiction. Therefore, by Lemma 9, the operator has a fixed point in .

Theorem 10. Assume that (H1)-(H3) hold. In addition, satisfies the following assumptions:
(H4) There exists a nondecreasing function such that (H5) There exists a constant such that . Then, problem (1) has at least one solution.

Proof. Now we show that (ii) of Lemma 9 does not hold. If u is a solution of problem (1), then for , we obtain Let . From the above inequality and (H5), it yields a contradiction. Therefore, by Lemma 9, the operator has a fixed point in .☐

3.2. Lyapunov’s Inequality

Theorem 11. Assume that (H1)-(H3) hold. In addition, is a concave function on . Then, for any nontrivial solution of problem (1), we have where

Proof. If is a nontrivial solution of problem (1), then by Lemma 4, we have Furthermore, by Lemma 6, we have Since is continuous and concave, then from Jensen’s inequality, it follows that namely, ☐

3.3. Stability Analysis

Definition 12 [34]. Equation (1) is said to be Ulam-Hyers-Rassias stability with respect to if there exists a nonzero positive real number such that for every and each solution of the inequality there exists a solution of problem (1) such that , .

Theorem 13. Assume that (H1)-(H3) hold. In addition, there exists a positive constant such that

Then, problem (1) is Ulam-Hyers-Rassias stability if .

Proof. Let be the solution of the inequality (55); then, Thus, for , we get By Theorem 7, problem (1) has a solution satisfies Then, for , we have which yields Therefore, problem (1) is Ulam-Hyers-Rassias stability.☐

4. Examples

Now we give some examples to illustrate our main results.

Example 1. We consider the following problem: where and . It is obvious that (H1)-(H3) hold. Via some computations, we have Since the function satisfies the condition Furthermore, we can verify that . Therefore, by Theorem 7 and Theorem 13, problem (62) has a solution , which is Ulam-Hyers-Rassias stability.

Example 2. Let us consider the following problem: where and It is obvious that (H1)-(H4) hold. By computations of Example 2, we have

Furthermore, for , the inequality holds, Therefore, by Theorem 10, problem (66) has at least one solution.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors were supported by the Fundamental Research Funds for the Central Universities (No. B200202003 and No. 2632020PY02). The authors are grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced the presentation of the manuscript.

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Copyright © 2021 Huijuan Zhu et al. 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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