Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems

This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem DC0+α u(t) = f(t, u(t), u′(t)), 0 < t < 1, u(1) = u′(1) = u′′(0) = 0, where 2 < α ≤ 3 is a real number, DC0+α is the Caputo fractional derivative, and f : [0,1]×[0, +∞) × R → [0, +∞) is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.

where 1 < ≤ 2 is a real number, 0 + is the Caputo fractional derivative, and : By means of a new fixed point theorem and Schauder fixed theorem, some results on the existence of positive solutions are obtained. Though the fractional boundary value problems have been studied by lots of authors, there are few pieces of work considering the case that the nonlinear term depends on the first order derivative ( ). In addition, to the best of our knowledge, there is no paper discussing the existence of multiple positive solutions for BVP (1). By constructing a special cone, using Guo-Krasnoselskii and Leggett-Williams fixed point theorems, two sufficient conditions are established for the existence of multiple positive solutions to BVP (1). In addition, by virtue of a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions of BVP (1) from its auxiliary problem. Finally, some illustrative examples are worked out to demonstrate the main results.
The organization of this paper is as follows. Section 2 contains some definitions and lemmas of fractional calculus theory which will be used in the next two sections. In

Preliminary Results
In this section, we introduce some necessary definitions and preliminary facts which will be used throughout this paper.
Definition 1 (see [15]). The Caputo fractional derivative of order > 0 of a continuous function : (0, ∞) → is given by where = [ ] + 1, [ ] denotes the integer part of the real number and provided that the right side integral is pointwise defined on [0, ∞).
Definition 2 (see [15]). The Riemann-Liouville standard fractional derivative of order > 0 of a continuous function ] : (0, ∞) → is given by where = [ ] + 1, [ ] denotes the integer part of the real number , and provided that the right side integral is pointwise defined on [0, ∞).
Definition 3 (see [15]). The Riemann-Liouville standard fractional integral of order > 0 of a continuous function : (0, ∞) → is given by provided that the right side integral is pointwise defined on [0, ∞).
Proof. In view of Lemma 4, (11) is equivalent to the integral equation The Scientific World Journal 3 for some ∈ , = 0, 1, 2. So, we have From the boundary condition (1) = (1) = (0) = 0, one has Therefore, by Definition 3, we conclude that the unique solution of BVP (11) is The proof is completed.
The following properties of the Green function play an important role in this paper.  (ii) There exist a positive number and a positive function Similarly, we can obtain that Hence, / < 0 for all , ∈ [0, 1). In addition, it is clear that (1, ) = 0 for 0 ≤ < 1. Therefore, we get that ( , ) > 0 for any , ∈ [0, 1).
(ii) In the following, we consider the existence of and ( ).
Firstly, if 0 ≤ ≤ < 1, then by the definition of ( , ), we have If 0 ≤ ≤ < 1, then by an argument similar to the case Secondly, for given 0 ≤ ≤ < 1, it is obvious that As 0 ≤ ≤ < 1, we also have that Thus, setting we immediately obtain that The proof is completed. Now, we list the following fixed point theorems which will be used in the next section.
For the sake of stating Leggett-Williams fixed point theorem, we first give the definition of concave functions.
Definition 11 (see [11]). The map is said to be a nonnegative concave functional on a cone of a real Banach space provided that : → [0, ∞) is continuous and for all , ∈ and 0 ≤ ≤ 1.

Let be a cone in a real
Then, has at least three fixed points 1 , 2 , and 3 with Remark 13. If = holds, then condition ( 1 ) of Lemma 12 implies condition ( 3 ).
Finally, in this section, we give a new extension of Krasnoselskii's fixed point theorem, which is developed in [24].

Main Results
In this section, we assume that It is easy to verify that is a cone in the space . Let the nonnegative continuous concave function on the cone be defined by By means of the Arzela-Ascoli theorem, one can obtain that the operator is completely continuous. The proof is completed. Now, we are in a position to state the main results. For convenience, denote Theorem 16. Assume that there exist two positive constants 2 > 1 > 0 such that Then, BVP (1) has at least one solution such that 1 ≤ ‖ ‖ ≤ 2 .
Proof. By Lemma 15, we know that the operator : → defined by (31) is completely continuous.
In view of Lemma 10, has a fixed point in ∩ (Ω 2 \ Ω 1 ) which is a solution of BVP (1). The proof is completed.
Theorem 17. Suppose that there exist constants 0 < < < such that the following assumptions hold: Proof. We will show that all the conditions of Lemma 12 are satisfied. First, if ∈ , then ‖ ‖ ≤ . By condition ( 3 ) and Lemma 9, we have which implies that ‖ ‖ ≤ for ∈ . Hence, : → . Next, by using the analogous argument, it follows from condition ( 1 ) that ‖ ‖ < for ∈ .

Examples
Example 1. Consider the following fractional BVP:

Conclusion
In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem (1) in the Caputo sense. Using Guo-Krasnoselskii and Leggett-Williams fixed point theorems, we establish the existence of multiple positive solutions to BVP (1). By virtue of a new extension of Krasnoselskii's fixed point theorem, we obtain a sufficient condition for the existence of multiple positive solutions of BVP (1) from its auxiliary problem. As applications, examples are presented to demonstrate the main results.