Existence of Positive Solution for Fractional Differential Systems with Multipoint Boundary Value Conditions

Abstract

In this paper, we apply the fixed-point theorems of γ concave and convex operators to establish the existence of positive solutions for fractional differential systems with multipoint boundary conditions. Two examples are given to support our results.

1. Introduction

Consider the following system of nonlinear fractional differential equationswith the multipoint boundary conditions with the multipoint boundary conditionswhere is the standard Riemann–Liouville fractional derivative, , , , , , , and ,

Differential equations with fractional order have been applied in various areas of science and engineering. For their applications, there has been a sharp increase in studying fractional differential equations (see [1–18] and references therein). Meanwhile, the theory of boundary value problems with multipoint boundary conditions has various applications in applied fields, which have been studied by many authors (cf., e.g., [19–26]). Many authors have studied these problems by using different methods, such as monotone iterative technique, the method of upper and lower solutions, fixed-point theorems in cones, nonlinear alternatives of Leray–Schauder, and coincidence degree theory. However, concave (convex) operators are a class of important operators, which can be used in nonlinear differential and integral equations (cf., e.g., [27–31]). Moreover, few papers can be reported on the existence of solutions for coupled systems of fractional differential equations with multipoint boundary conditions by using fixed-point theorems of γ concave and convex operators.

In [25], we considered the following m-point boundary value problem for fractional differential equationwith the multipoint boundary conditions , where is the standard Riemann–Liouville fractional derivative, , is continuous, , , , , and

In [26], Henderson and Luca studied the following system of nonlinear second-order ordinary differential equationwith the multipoint boundary conditions and By using the Schauder fixed-point theorem, the existence of positive solutions was investigated.

Motivated by above papers, in this paper, we investigate the existence of positive solutions for systems (1)–(3).

In this paper, we need the following assumptions that we shall use in the sequel:  , and are increasing in x for ,   There exists a constant such that and ,   There exists a constant such that ,   , is increasing in x for ,   There exists a constant such that ,   , is nondecreasing in x, and is nonincreasing in y  and are bounded in   There exists such that , and there exists such that , where ,

Here are our main results.

Theorem 1. Suppose that hold. Then, equations (1)–(3) have a unique positive solution in , where . Moreover, for any initial value and , constructing successively the sequencewe have as .

Corollary 1. Suppose that holds. Then, systemwith the multipoint boundary conditionshas a unique positive solution in , where , is the standard Riemann–Liouville fractional derivative, is continuous, , , , , , , and , Moreover, for any initial value and , constructing successively the sequencewe have as .

Theorem 2. Suppose that hold. Then, equations (1)–(3) have exactly one positive solution , where with , and constructing successively the sequencewe have as .
The rest of this paper is organized as follows. In Section 2, we present some background materials and preliminaries. Section 3 deals with the existence results. In Section 4, two examples are given to illustrate the result.

2. Background Materials and Preliminaries

Definition 1 (see [6]). The fractional integral of order α with the lower limit for a function f is defined aswhere is the gamma function.

Definition 2 (see [6]). For a function , the Riemann–Liouville derivative of fractional order is defined as

Definition 3 (see [28]). Let E be a real Banach space and be a cone in E which defined a partial ordering in E by if and only if . is said to be normal if there exists a positive constant N such that implies . is called solid if its interior is nonempty.

Definition 4 (see [28]). An operator is said to be mixed monotone if is nondecreasing in x and nonincreasing in y, i.e., , and imply .

Definition 5 (see [28]). For all , the notation means that there exists and such that . Clearly, is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .

Definition 6 (see [30]). Let or and γ be a real number with . An operator is said to be if it satisfies

Definition 7 (see [31]). An operator is said to be homogeneous if it satisfiesAn operator is said to be subhomogeneous if it satisfies

Lemma 1 (see [25]). Let . Then, the fractional differential equationhas a unique solution which is given bywherein whichwhere

Lemma 2. Let , then in Lemma 1 has the following property:(i)(ii)

Proof. For , we haveThus,From [25], we haveThis means that (i) holds. From Lemma 1, we know that (ii) is obvious.

Theorem 3 (see [31]). Let P be a normal cone in a real Banach space E and be an increasing operator and be an increasing subhomogeneous operator. Assume that(i)There is such that and (ii)There exists a constant such that , Then, operator equation has a unique solution in . Moreover, constructing successively the sequence for any initial value , we have as .

Theorem 4 (see [30]). Let P be a normal cone of the real Banach space E and be a mixed monotone operator. Suppose that(i)For fixed y, is concave; for fixed x, is convex, where (ii)There exist elements with and a real number such thatThen, A has exactly one fixed point in , and constructing successively the sequencefor any initial value , we have .

3. Main Results

In this section, we shall investigate the existence of positive solutions for systems (1)–(3). We consider the space equipped with the norm . Let , then P is a cone in E. Let .

From Lemma 1, we know that (1)–(3) can be translated into the following equationand (2)-(3) can be translated into the following equation

Thus, is a solution of (1)–(3) if and only if is a solution of system (26), and is a solution of (2)-(3) if and only if is a solution of system (27).

For convenience, we denote

Now, we prove Theorem 1, Corollary 1, and Theorem 2.

Proof of Theorem 1. Define two operatorsThus,Owing to [25], we know that . By , we have and . It is obvious that is the solution of problem (26) if and only if is the solution of . Step 1: A and B are two increasing operators. Set . It follows from (26) that From , we have According to and (29), we obtain Thus, A is an increasing operator. Similarly, we can see that B is an increasing operator. Step 2: A is a operator, and B is a subhomogeneous operator. In fact, for , from (30) and , we have Similarly, we can get Thus, Therefore, we can see that A is a operator, and B is a subhomogeneous operator. Step 3: and . Combining (29), (30), , and Lemma 2, one has, for , Let Noting that and , we obtain Therefore, which implies that By a closely similar way, we have . Step 4: There exists a constant such that and , .For , , by , we haveThis means that Similarly, we have .
Therefore, by simple computation, the conditions in Theorem 3 are satisfied. This implies that the operator equation has a unique solution in , and the operator equation has a unique solution in . Thus, has a unique solution in . For any initial value and , we can construct the following sequence:This follows that as .

Proof of Corollary 1. In Theorem 3, we let B be a null operator, Theorem 3 also holds. By Theorem 1, we conclude that Corollary 1 holds.

Proof of Theorem 2. Define the following operator:From , we know that is a mixed monotone operator. Step 1: we will prove that the condition (i) of Theorem 4 holds From , we know that, for  Step 2: we will verify that the condition (ii) of Theorem 4 holdsIt follows from that there exists and such thatLetObviously,Therefore, the conditions of Theorem 4 are satisfied. This means that (2)-(3) has exactly one positive solution , where with . By constructing successively the sequencewe obtain as .

4. Examples

Example 1. Let Consider the following boundary value problemwith the multipoint boundary conditionsHere,Thus,Set . Obviously, and are increasing with respect to the second argument, . For , we can notice thatFor , we deduce that whereThus, the assumptions of are satisfied. By Theorem 1, system (49)-(50) has a unique positive solution in , where , .

Example 2. Letting we consider the following problemwith the multipoint boundary conditionsHere,We deduce that Set . It is clear that and is nondecreasing in x, and is nonincreasing in y, , , and Moreover, for , we can find thatThen, all the conditions of Theorem 2 are fulfilled. Consequently, there exist , and system (56)-(57) has exactly one positive solution in .

Data Availability

No data were used to support the study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11771017) and the Key Research of Henan Higher Education Institutions (17A110001).