Research Article | Open Access
Multiple Positive Solutions to Nonlinear Boundary Value Problems of a System for Fractional Differential Equations
By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by where is the standard Riemann-Liouville fractional derivative, for and , subject to the boundary conditions , for , and , for , or , for , and , , for , Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.
The purpose of this paper is to consider the existence of multiple positive solutions for the following system of nonlinear fractional differential equations: where , for and , and , subject to a couple of boundary conditions. In particular, we first consider (1) subject to where , and . We then consider the case in which the boundary conditions are changed to where .
Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, and engineering and biological sciences; see [1–11] for example. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. For a small sample of such work, we refer the reader to [12–20] and the references therein. The situation of at least one positive solution has been studied in many excellent monograph; see [12–19, 21] and other references therein. In , by means of Schauder fixed point theorem, Su investigated the existence of one positive solution to the following boundary value problem for a coupled system of nonlinear fractional differential equations: where .
In , Goodrich established the existence of one positive solution to problems (1)-(2) and (1), (3) by using Krasnoselskii's fixed point theorem. Different from the above works mentioned, in this paper we will present the existence of at least two positive solutions to problems (1)-(2) and (1), (3) by using the similar method presented in . Moreover, under different conditions, we also present the existence of at least one positive solution to problems (1)-(2) and (1), (3) with .
For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our theorems.
Definition 1 (see ). Let with . Suppose that . Then the th Riemann-Liouville fractional integral is defined to be whenever the right-hand side is defined. Similarly, with and , we define the th Riemann-Liouville fractional derivative to be where is the unique positive integer satisfying and .
Lemma 2 (see ). Let be given. Then the unique solution to problem together with the boundary conditions , where and , is where is the Green function for this problem.
Lemma 4 (see ). Let be as given in the statement of Lemma 2. Then there exists a constant such that To prove our results, we need the following Krasnoselskii's fixed point theorem which can be seen in Guo and Lakshmikantham .
Lemma 5 (see ). Let be a Banach space, and let be a cone. Assume that are open bounded subsets of with , , and let be a completely continuous operator such that(i), and ; or(ii), and .
Then has a fixed point in .
3. Main Results
In our considerations, let represent the Banach space of when equipped with the usual supremum norm, . Then put , where is equipped with the norm for . Observe that is also a Banach space (see ). In addition, we define two operators by where is the Green function of Lemma 2 with replaced by and, likewise, is the Green function of Lemma 2 with replaced by . Now, we define an operator by We claim that whenever is a fixed point of the operator defined in (11), it follows that and solve problems (1)-(2). That is, a pair of functions is a solution of problems (1)-(2) if and only if is a fixed point of the operator defined in (11) (see ).
In the following, we will look for fixed points of the operator , because these fixed points coincide with solutions of problems (1)-(2). For use in the sequel, let and be the constants given by Lemma 4 associated, respectively, with the Green functions and , and define by , and notice that .
For the sake of convenience, we set
Now we list some assumptions:();();()there are numbers , where such that , .
Next, we define the cone by
Lemma 7. is a completely continuous operator.
Proof. The operator is continuous in view of nonnegativeness and continuity of , and .
Let be bounded; that is, there exists a positive constant such that , for all . Let ; then, for , we have Hence, is bounded.
On the other hand, given , setting , then, for each , , , and , one has . That is to say, is equicontinuity. In fact,
In the following, we divide the proof into two cases.
Case 1. If , then we have where .
Case 2. If , then we have By the means of the Arzela-Ascoli theorem, we have that is completely continuous. Similarly, is completely continuous. Consequently, is a completely continuous operator. This completes the proof.
In , Goodrich established the following result.
From Theorem 8, the following problem is natural: whether we can obtain some conclusions or not, if or In the rest of this paper, we give some answers to this problem.
For the sake of convenience, we make some assumptions:there exist constants , such that there exist constants , such that there are numbers , where such that ;there are numbers , where such that .
Proof. From Lemma 7, is a completely continuous operator. At first, in view of , we have , for ; , for , where satisfies . Set . So we define by . Then for each , we find that
So for .
Similarly, we find that for . Consequently, whenever . Thus, is cone expansion on .
Next, since , we have for ; for , where satisfies . Set . Let and . Then implies So we obtain So for .
Similarly, we find that for .
Consequently, , whenever . Thus, is cone expansion on .
Finally, let . For , from , , we have Similarly, we find that for .
Consequently, , whenever . Thus, is cone compression on .
So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with The proof is complete.
Proof. At first, in view of , we have , , for , where satisfies . Let .
Then for each , we find that Like Theorem 9, we get for .
Next, in view of , we have , , for , where satisfies . We consider two cases.
Case 1. Suppose that is unbounded; there exists such that Since , one has for . Then, for and , we obtain
Case 2. Suppose that is bounded; there exists such that for all . Taking , for and , we obtain Hence, in either case, we always may set such that for . Like Theorem 9, we get , for .
Finally, set . Then implies Hence we have Consequently, for . Like Theorem 9, we get for .
So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with which complete the proof.
In the following, for the sake of convenience, set Assume that there exist two positive constants such that, for ;, for .
Proof. With loss of generality, we may assume that .
Let . For , one has