fractional boundary value problems

In this paper, we give sufficient conditions for the existenceor the nonexistence of positive solutions of the nonlinear fractional boundary value problem

We show, under suitable conditions on the nonlinear term f , that the fractional boundary value problem (1.1)-(1.2) has at least one or has non positive solutions.By employing the fixed point theorems for operators acting on cones in a Banach space (see, for example [7,8,13,14,15]).The use of cone techniques in order to study boundary value problems has a rich and diverse history.That is, some authors have used fixed point theorems to show the existence of positive solutions to boundary value problems for ordinary differential equations, difference equations, and dynamic equations on time scales, (see for example [1,2,3]).Moreover, Delbosco and Rodino [7] considered the existence of a solution for the nonlinear fractional differential equation D α 0 + u = f (t, u), where 0 < α < 1 and f : [0, a] × R → R, 0 < a ≤ +∞ is a given function, continuous in (0, a) × R.They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle.Bai and Lü [5] studied the existence and multiplicity of positive solutions of nonlinear fractional differential equation boundary value problem: where D α 0 + is the standard Riemann-Liouville differential operator of order α.Recently Bai and Qiu [4].considered the existence of positive solutions to boundary value problems of the nonlinear fractional differential equation where D α 0 + is the Caputo's fractional differentiation, and f : (0, 1] × [0, +∞) → [0, +∞), with lim t−→0 + f (t, .)= +∞ .They obtained results for solutions by using the Krasnoselskii's fixed point theorem and the nonlinear alternative of Leray-Schauder type in a cone.
Lü Zhang [18] considered the existence of solutions of nonlinear fractional boundary value problem involving Caputo's derivative In another paper, by using fixed point theory on cones, Zhang [19] studied the existence and multiplicity of positive solution of nonlinear fractional boundary value problem where D α t is the Caputo's fractional derivative.By using the Krasnoselskii fixed point theory on cones, Benchohra, Henderson, Ntoyuas and Ouahab [6] used the Banach fixed point and the nonlinear alternative of Leray-Schauder to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay where D α is the standard Riemman-Liouville fractional derivative, f : J × B → R is a given function satisfying some suitable assumptions, φ ∈ B, φ(0) = 0 and B is called a phase space.By using the Krasnoselskii fixed point theory on cones, El-Shahed [8] Studied the existence and nonexistence of positive solutions to nonlinear fractional boundary value problem where D α 0 + is the standard Riemann-Liouville differential operator of order α.Some existence results were given for the problem (1.1)-(1.2) with α = 3 by Sun [17].The BVP (1.1)-(1.2) arises in many different areas of applied mathematics and physics, and only its positive solution is significant in some practice.
For existence theorems of fractional differential equation and application, the definitions of fractional integral and derivative and related proprieties we refer the reader to [7,11,12,16].
The rest of this paper is organized as follows: In section 2, we present some preliminaries and lemmas.Section 3 is devoted to prove the existence and nonexistence of positive solutions for BVP (1.1)-(1.2).

Elementary Background and Preliminary lemmas
In this section, we will give the necessary notations, definitions and basic lemmas that will be used in the proofs of our main results.We also present a fixed point theorem due to Guo and Krasnosel'skii.
Definition 2 [10,11,15].For a function h given on the interval [a, b], the αth Riemann-Liouville fractional-order derivative of h, is defined by Lemma 4 [4] Assume that u ∈ C (0, 1) ∩ L (0, 1) with a fractional derivative of order α > 0. Then for some ) has a unique solution where and Proof.By applying Lemmas 3 and 4 , the equation (2.4) is equivalent to the following integral equation (2.9) for some arbitrary constants c 1 , c 2 , c 3 ∈ R. Boundary conditions (2.5), permit us to deduce there exacts values then, the unique solution of (2.4)-(2.5) is given by the formula where, This ends the proof.In order to check the existence of positive solutions, we give some properties of the functions G(t, s) and G 1 (t, s).
, by (2.6) and Lemma 6, it follows that and thus More that, (2.6) and Lemma 6 imply that, for any t ∈ [τ, 1] , Hence min This completes the proof.
Definition 8 Let E be a real Banach space.A nonempty closed convex set Definition 9 An operator is called completely continuous if it continuous and maps bounded sets into precompact sets To establish the existence or nonexistence of positive solutions of BVP (1.1)-(1.2),we will employ the following Guo-Krasnosel'skii fixed point theorem: Theorem 10 [12] Let E be a Banach space and let K ⊂ E be a cone in E. Assume that Ω 1 and Ω 2 are open subsets of E with 0 ∈ Ω 1 and Ω 1 ⊂ Ω 2 .Let T : K ∩ Ω 2 \Ω 1 −→ K be completely continuous operator.In addition, suppose either

Existence of solutions
In this section, we will apply Krasnosel'skii's fixed point theorem to the problem (1.1)-(1.2).We note that u(t) is a solution of (1.1)-(1.2) if and only if Let us consider the Banach space of the form We define a cone K by and an integral operator It is not difficult see that, fixed points of T are solutions of (1.1)-(1.2).Our aim is to show that T : K−→K is completely continuous, in order to use Theorem 10.
Proof.Since G(t, s), G 1 (η, s) ≥ 0, then T u(t) ≥ 0 for all u ∈ K.We first prove that T (K) ⊂ K.In fact, on the other hand, Lemma 6 imply that, for any t and, for u ∈ K min T u(t) ≥ γ T u .
Consequently, we have T (K) ⊂ K. Next, we prove that T is continuous.In fact, let it is uniformly continuous.Therefore, for any ǫ > 0, there exists δ > 0 such that That is, T : K−→K is continuous.Finally, let B ⊂ K be bounded, we claim that T (B) ⊂ K is uniformly bounded.Indeed, since B is bounded, there exists some m > 0 such that u ≤ m, for all u ∈ B. Let , this allows us to show that, One has where )ds . In order to estimate t 2 α − t 1 α and t 2 α−1 − t 1 α−1 , we can apply a method used in [4,18]; by means value theorem of differentiation, we have Thus, we obtain where By means of the Arzela-Ascoli theorem, T : K−→K is completely continuous.The proof is achieved.
In all what follow, we assume that the next conditions are satisfied.
Proof.By a similar as in the proof of Lemma 11 it is obvious that and by the absolute continuity of the integral, we have (3.3),Lemma 6(P3), and the absolute continuity of the integral, we have Then by Lemma 12, T : K−→K is completely continuous.Throughout this section, we shall use the following notations: Theorem 14 Suppose that f is superlinear, i.e.
Then BVP (1.1)-(1.2) has at least one positive solution for λ small enough and has no positive solution for λ large enough.
Proof.We divide the proof into two steps.
Thus, T (K(u 0 )) ⊆ K(u 0 ).By Shaulder's fixed point theorem we know that there exists a fixed point u ∈ K(u 0 ), which is a positive solution of BVP (1.1)-(1.2).The proof is complete.Now we consider the case f is sublinear.
Then BVP (1.1)-(1.2) has at least one positive solution for any λ ∈ (0, ∞).Next we construct the set Ω 2 .We consider two cases: f is bounded or f is unbounded.
Case (1): Suppose that f is bounded, say f (r) ≤ M for all r ∈ [0, ∞).In this case we choose So, T u ≤ u .