Positive Solutions Using Bifurcation Techniques for Boundary Value Problems of Fractional Differential Equations

and Applied Analysis 3 Lemma 6. Let α > 0; then, the differential equation


Introduction
During the last few decades, fractional calculus and fractional differential equations have been studied extensively since fractional-order models are found to be more adequate than integer-order models in some real-world problems.In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth involves derivatives of fractional order.For details and examples, see [1][2][3][4][5][6][7] and references therein.Recently, there have been a few papers which deal with the boundary value problem for fractional differential equation.For example, in [8], Tian and Liu investigated the following singular fractional boundary value problem (BVP, for short) of the form    0 +  () +  (,  ()) = 0, 0 <  < 1, where    0 + is Caputo's fractional derivatives,  − 1 <  ≤ ,  ≥ 4, and  : (0, 1) × (0, +∞) → [0, +∞) is continuous; that is, (, ) may be singular at  = 0, 1 and  = 0.By constructing a special cone, they obtained that there exist positive numbers  * and  * * with  * <  * * such that the above system has at least two positive solutions for  ∈ (0,  * ) and no solution for  >  * * under some suitable assumptions such as the following.
In [9], Bai and Lü consider the following nonlinear fractional differential equation Dirichlet-type boundary value problem: where 1 <  ≤ 2 is a real number and   0 + is the standard Riemann-Liouville differentiation.The corresponding Green function is derived.By means of some fixed point theorems on cone, the existence and multiplicity of positive solutions for BVP (2) were investigated.
In [10], Jiang and Yuan further investigated BVP (2).Comparing with [9], they deduced some new properties of the Green function, which extended the results of integerorder Dirichlet boundary value problems.Based on these new properties and Krasnoselskii fixed point theorem, the existence and multiplicity of positive solutions for BVP (2) were considered.
In this paper, by using bifurcation techniques, we consider the following boundary value problem of fractional differential equation: where 1 <  ≤ 2,   0 + is the standard Riemann-Liouville differentiation,  > 0 is a given constant, and  : [0, 1] × R + → R + is a given continuous function satisfying some assumptions that will be specified later.
It is remarkable that the method used in references mentioned above was fixed point theorems and the same kind of conditions was used such that the nonlinearity (, ) satisfies superlinear or sublinear condition at 0 and ∞, which is similar to (A1).To the best of our knowledge, there is no paper studying such fractional differential equations using bifurcation ideas.As we know, the bifurcation technique is widely used in solving BVP of integer-order differential equations (see, e.g., [11][12][13] and references therein).In [14], by virtue of bifurcation ideas, the authors studied a kind of BVP of differential inclusions.The purpose of present paper is to fill this gap.By using bifurcation techniques and topological degree theory, the existence of positive solutions of BVP (3) is investigated.The main features of present paper are as follows.First, the nonlinearity (, ) is asymptotically linear at 0 and ∞, not super-linear or sub-linear (see the condition (H1) in Section 2 and example in Section 4).Next, the main method used here is bifurcation techniques and topological degree, not fixed point theorem on cone, which is different from the references.
The paper is organized as described below.At the end of this section, for completeness, we list some results on bifurcation theory from interval and topological degree of completely continuous operators.Section 2 contains background materials and preliminaries.In Section 3, by using bifurcation techniques and topological degree theory, bifurcation results from infinity and trivial solution are established.Then the main results of present paper are given and proved.Finally in Section 4, an example is worked out to demonstrate the main results.
Lemma 1 (Schmitt and Thompson [15]).Let  be a real reflexive Banach space.Let  : R ×  to  be completely continuous such that (, 0) = 0, for all  ∈ R. Let ,  ∈ R ( < ) be such that  = 0 is an isolated solution of the equation for  =  and  = , where (, 0), (, 0) are not bifurcation points of (4).Furthermore, assume that where   (0) is an isolating neighborhood of the trivial solution.Let Then there exists a connected component C of T containing [, ] × 0 in R × , and either Lemma 2 (Schmitt [16]).Let  be a real reflexive Banach space.Let  : R ×  to  be completely continuous, and let ,  ∈ R ( < ) be such that the solution of (4) is, a priori, bounded in  for  =  and  = ; that is, there exists an  > 0 such that for all  with ‖‖ ≥ .Furthermore, assume that for  > 0 sufficiently large.Then there exists a closed connected set C of solutions of (4) that is unbounded in [, ] × , and either Lemma 3 (Guo [17]).Let Ω be a bounded open set of real Banach space , and let  : Ω →  be completely continuous.
If there exists then

Background Materials and Preliminaries
For convenience, we present some necessary definitions and results on fractional calculus theory (see [6]).
Definition 5.For a function ℎ given on the interval [, ], the th Riemann-Liouville fractional-order derivative of ℎ is defined by where  = [] + 1.
For more detailed results of fractional calculus, we refer the reader to [6].Now let us list the following assumption satisfied throughout the paper.
To solve BVP (3), we first consider the following linear boundary problem of fractional differential equation: where  ∈ [0, 1].We cite the following two lemmas from references.
From Lemmas 8 and 9 and the well-known Krein-Rutman Theorem, one can obtain the following lemma.
Lemma 11.The operator   defined by (28) is completely continuous and has a unique characteristic value  1 (), which is positive, real, and simple and the corresponding eigenfunction () is of one sign in (0, 1); that is, () =  1 ()  () for all  ∈ .
Notice that the operator   can be regarded as

Main Results
The main results of present paper are the following two theorems.
Theorem 12. Suppose that either Then BVP (3) has at least one positive solution.
Theorem 13.Suppose the following.
To prove Theorems 12 and 13, we first prove the following lemmas.
From the definition of (, ), it is easy to see that Taking a subsequence and relabeling if necessary, suppose   →  in [0, 1].Then ‖‖ = 1 and  ∈ .