Solvability of Some Two-Point Fractional Boundary Value Problems under Barrier Strip Conditions

Abstract

Topological techniques are used to establish existence results for a class of fractional differential equations , with one of the following boundary value conditions: and or and where is a real number, is the conformable fractional derivative, and is continuous. The main conditions on the nonlinear term are sign conditions (i.e., the barrier strip conditions). The topological arguments are based on the topological transversality theorem.

1. Introduction

Recently, boundary value problems of nonlinear fractional differential equations have been addressed by several researchers. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, fitting of experimental data, and so forth. For example, in 2006, by using the fixed point theorem in cones, the existence and multiplicity of solutions to the following problems are obtained [1]:where is a real number, is Caputo fractional derivative, and is a continuous function. In 2010, by the use of the Lipschitz condition and the compression mapping principle, the existence of solutions of the following problem is obtained [2]:where is a real number, , is the standard Riemann-Liouville derivative, and is a continuous function. We refer the readers to other contributions in this line ([1–30], etc.).

The technique of barrier strips has been used by Kelevedjiev and Tersian in [31, 32] to study the solvability of integer order BVPs. Recently, we use it to study the fractional differential equation with Dirichlet boundary value condition [20]. In this paper, by using the topological transversality theorem, we consider the following equation:with one of the following boundary value conditions:where is a real number, is the conformable fractional order derivative, and is a continuous function. The existence results of solutions to the problem are obtained under which satisfies some barrier strip conditions.

2. Definitions and Lemmas about Fractional Calculus

Let .

Definition 1 (see [14]). Suppose is -order differentiable for ; the -order fractional derivative of is defined asprovided the limits of the right side exist.

Lemma 2 (see [12]). Let . Function is -order differentiable if and only if is -order differentiable; moreover, the following relation holds:

Definition 3 (see [14]). The -order fractional integral is defined aswhere is the -order integral.

Lemma 4 (see [20]). For , there holds

Lemma 5 (see [14]). Let be continuous on and -order differentiable on . Then, there exists such that

3. Topological Preliminaries and a New Function Space

We begin with a brief review of the topological results to be used in this paper; see [33]. Let be a convex subset of a Banach space , a metric space, and a continuous map. We say that is compact if is contained in a compact subset of . is completely continuous if it maps bounded subsets in into compact subsets of . A homotopy is said to be compact provided that given by for in is compact.

Let be open in . A compact map is called admissible if it is fixed point free on the boundary, , of . The set of all such maps will be denoted by .

Definition 6. A map in is inessential if there is a fixed point free compact map such that . A map in which is not inessential is called essential.

Lemma 7 (see [33]). Let be an arbitrary value in and be in and be the constant map   for in . Then is essential.

Definition 8. Two maps and in are called homotopic if there is a compact homotopy such that and and is admissible for each in .

The following theorem called topological transversality theorem which is very important to our results.

Lemma 9 (see [33]). Let and be in and be homotopic maps, . Then one of these maps is essential if and only if the other is.

Now, we construct a function space. Given , let . Define

By the linearity of integral operator , is a linear space. For , according to Lemma 4, there are . Definewhere . Next, we prove is a norm in the linear space , and is complete with this norm.

Theorem 10. is a Banach space.

Proof. It is easy to verify that satisfies the norm axioms. The following proof is about the completeness of .
Let be a Cauchy sequence in :where and . ThenBecause is a Cauchy sequence of the Banach space ,Thus every term of the above formula converges to 0. Bywe know is a Cauchy sequence in . By the completeness of , there exists such that . The second term isTaking into account that is a Cauchy sequence in , there isThat is to say, is a Cauchy sequence in . Denote its limit as . Continue this process, and we can prove that sequences , all are Cauchy sequences. Denote their limits as , respectively.
Let , and then and . The completeness of is proved.

4. Main Results

Denote

Theorem 11. Suppose is continuous. If there exist four constants , and , such thatthen Problem (3)-(4) has a solution such that

Proof. Consider the family of boundary value problemsDenoteDefine an operator aswhereThen standard arguments yield that is completely continuous. Clearly, the fixed point of operator is the solution for Problem (22)-(23).
Given , letWe will prove that, for some positive number , the map is admissible.
We claim that all the possible solutions of Problem (22)-(23) on have a priori bound without dependence on . Assume that the setsare not empty.
Choose . Assume there exists such thatTaking into account the fact that is continuous, we can take . The assumption on yields that and for . Consequently, is increasing on . Thuswhich is a contradiction to (29). So, there holdsand, in particular,a contradiction to (23). This shows that . The similar arguments can show that .
By and , there holds for , and then . On the other hand, by Lemma 5, for each , there exists such thatThus, . Finally (22) together with the continuity of and a priori estimations of and show that .
Set , and one hasThe above results indicate that is completely continuous and has no fixed point on and is admissible on for each . By Lemma 7, the constant map for is essential. Combined with Lemma 9, the topological transversality theorem, , is essential. So Problem (3)-(4) has a solution . Moreover, the above arguments show that the solution satisfiesThe proof is completed.

The next theorem can be proved by similar arguments.

Theorem 12. Let be continuous. Suppose there exist four constants , , such that and :Then Problem (3)-(4) has a solution such that

Example 13. Consider the following fractional BVPs:Let , , , and , and then the simple computation show that , , , andBy the use of Theorem 11, Problem (38) has a solution such that

5. Conclusion

By the use of the topological transversality theorem, some existence results for a class of fractional differential equations with certain boundary value conditions are obtained. The main condition is sign condition which is easy to be satisfied and checked.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by NSFC (11571207), the Taishan Scholar Project, and SDUST Graduate Innovation Project SDKDYC170343.