Positive solutions of fractional differential equations at resonance on the half-line

This article deals with the differential equations of fractional order on the half-line. By the recent Leggett-Williams norm-type theorem due to O’Regan and Zima, we present some new results on the existence of positive solutions for the fractional boundary value problems at resonance on unbounded domains. MSC:26A33, 34A08, 34A34.

The problem (.) happens to be at resonance in the sense that the kernel of the linear operator D α + is not less than one-dimensional under the boundary value conditions. Fractional calculus is a generalization of the ordinary differentiation and integration. It has played a significant role in science, engineering, economy, and other fields. Some books on fractional calculus and fractional differential equations have appeared recently (see [-]); furthermore, today there is a large number of articles dealing with the fractional differential equations (see [-]) due to their various applications.
In [], the researchers dealt with the existence of solutions for boundary value problems of fractional order of the form C D α + y(t) = f t, y(t) , t ∈ [, +∞), y() = y  , y is bounded in [, +∞), http://www.boundaryvalueproblems.com/content/2012/1/64 where  < α ≤  and f : [, +∞) × R → R is continuous. The results are based on the fixed point theorem of Schauder combined with the diagonalization method. In [], Su and Zhang studied the following fractional differential equations on the halfline using Schauder's fixed point theorem Employing the Leray-Schauder alternative theorem, in [], Zhao and Ge considered the fractional boundary value problem However, the articles on the existence of solutions of fractional differential equations on the half-line are still few, and most of them deal with the problems under nonresonance conditions. And as far as we know, recent articles, such as [, , ], investigating resonant problems are on the finite interval.
Motivated by the articles [-], in this article we study the differential equations (.) under resonance conditions on the unbounded domains. Moreover, we have successfully established the existence theorem by the recent Leggett-Williams norm-type theorem due to O'Regan and Zima. To our best knowledge, there is no article dealing with the resonant problems of fractional order on unbounded domains by the theorem.
The rest of the article is organized as follows. In Section , we give the definitions of the fractional integral and fractional derivative, some results about fractional differential equations, and the abstract existence theorem. In Section , we obtain the existence result of the solution for the problem (.) by the recent Leggett-Williams norm-type theorem. Then, an example is given in Section  to demonstrate the application of our result.

Preliminaries
First of all, we present some fundamental facts on the fractional calculus theory which we will use in the next section.

Definition . ([-])
The Riemann-Liouville fractional integral of order ν >  of a function h : (, ∞) → R is given by provided that the right-hand side is pointwise defined on (, ∞).
where N is the smallest integer greater than or equal to ν.
Now, let us recall some standard facts and the fixed point theorem due to O'Regan and Zima, and these can be found in [, , -].
Let X, Z be real Banach spaces. Consider an operation equation where L : dom L ⊂ X → Z is a linear operator, N : X → Z is a nonlinear operator. If dim Ker L = codim Im L < +∞ and Im L is closed in Z, then L is called a Fredholm mapping of index zero. And if L is a Fredholm mapping of index zero, there exist linear continuous projectors P : X → X and Q : Z → Z such that Ker L = Im P, Im L = Ker Q and X = Ker L⊕Ker P, Z = Im L⊕Im Q. Then it follows that L P = L| dom L∩Ker P : dom L∩Ker P → Im L is invertible. We denote the inverse of this map by K P . For Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. It is known that the coincidence equation Lu = Nu is equivalent to The following lemma is valid for every cone in a Banach space.

Lemma . ([, ]) Let C be a cone in the Banach space X. Then for every u
for all x ∈ C.
Let γ : X → C be a retraction, i.e., a continuous mapping such that γ (x) = x for all x ∈ C.
Then the equation Lx = Nx has a solution in the set C ∩ (  \  ). Let Remark . It is easy for us to prove that (X, · X ) and (Z, · Z ) are Banach spaces.
Define Next, similar to the compactness criterion in [, ], we establish the following criterion, and it can be proved in a similar way.

Lemma . U is a relatively compact set in X if and only if the following conditions are satisfied:
(a) U is uniformly bounded, that is, there exists a constant R >  such that for each u ∈ U , u X ≤ R.
(c) The functions from U are equiconvergent, that is, given ε > , there exists

Main results
In this section, we will present the existence theorem for the fractional differential equation on the half-line. In order to prove our main result, we need the following lemmas. Lemma . Let g ∈ Z. Then u ∈ X is the solution of the following fractional differential equation: if and only if Proof In view of Lemmas . and ., we can certify the conclusion easily, so we omit the details here.

Lemma . The operator L is a Fredholm mapping of index zero. Moreover,
and Proof It is obvious that Lemma . implies (.) and (.). Now, let us focus our minds on proving that L is a Fredholm mapping of index zero.
where g ∈ Z. Evidently, Ker Q = Im L, Im Q = {g|g = ce -t , t ≥ , c ∈ R}, and Q : Z → Z is a continuous linear projector. In fact, for an arbitrary g ∈ Z, we have that is to say, Q : Z → Z is idempotent. Let g = g -Qg + Qg = (I -Q)g + Qg, where g ∈ Z is an arbitrary element. Since Qg ∈ Im Q and (I -Q)g ∈ Ker Q, we obtain that Z = Im Q + Ker Q. Take z  ∈ Im Q ∩ Ker Q, then z  can be written as z  = ce -t , c ∈ R , for z  ∈ Im Q. Since z  ∈ Ker Q = Im L, by (.), we get that Q(z  ) = Q(ce -t ) = cQ(e -t ) = ce -t = , which implies that c = , and then z  = . Therefore, Now, dim Ker L =  = dim Im Q = codim Ker Q = codim Im L < +∞, and observing that Im L is closed in Z, so L is a Fredholm mapping of index zero.
Let P : X → X be defined by It is clear that P : X → X is a linear continuous projector and Also, proceeding with the proof of Lemma ., we can show that X = Im P ⊕ Ker P = Ker L ⊕ Ker P. http://www.boundaryvalueproblems.com/content/2012/1/64 Consider the mapping K P : Im L → dom L ∩ Ker P Note that (K P L)u = K P (Lu) = u, ∀u ∈ dom L ∩ Ker P, and (LK P )g = L(K P g) = g, ∀g ∈ Im L.
Thus, JQN + K P (I -Q)N : X → X is given by Then, it is easy to verify that Now, we state the main result on the existence of the positive solutions to the problem (.) in the following.
Theorem . Let f : [, +∞) × R → R satisfy the condition (H). Assume that there exist six nonnegative functions α i (t) (i = , , ), β j (t) (j = , ) and μ(t) such that and Then the problem (.) has at least one positive solution in dom L.
Proof For the simplicity of notation, we denote and Consider the cone where R  ∈ (R  , R), R  ∈ (, R  ), ε  ∈ (ε  , ). Clearly,  and  are an open bounded set of X.
Step : In view of Lemma ., the condition  • of Theorem . is fulfilled.
Step : By virtue of Lemma ., we can get that QN : X → Z is continuous and bounded and K P (I -Q)N : X → X is compact on every bounded subset of X, which ensures that the assumption  • of Theorem . holds.
Step : Let u ∈ Ker L ∩  , then u(t) = ct α- , t ≥ , c ∈ R. Inspired by Aijun and Wang which shows that  • is true.