Existence of nonnegative solutions for a nonlinear fractional boundary value problem

Abstract This paper deals with the existence of solutions for a class of boundary value problem (BVP) of fractional differential equation with three point conditions via Leray-Schauder nonlinear alternative. Moreover, the existence of nonnegative solutions is discussed.


Introduction
Fractional differential equations are a natural generalization of ordinary differential equations. They can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, etc. It has been established that, in many situations these models provide more suitable results than analogous models with integer derivatives. As a consequence, the subject of fractional differential equations is gaining much importance and attention. For details, see [1 − 6,8] and references therein.
It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear initial value fractional differential equation in terms of special functions [20,25]. Recently, there are some papers dealing with the existence and multiplicity of solutions or nonnegative solutions of nonlinear initial value fractional differential equation by the use of techniques of nonlinear analysis of such fixed point theorems.
Existence of nonnegative solutions for a nonlinear fractional BVP 57 Liang and Zhang [22] studied the existence and uniqueness of positive solutions by properties of the Green function, the lower and upper solution method and fixed point theorem for the fractional boundary value problem D q 0 + u (t) + f(t, u(t)) = 0, 0 < t < 1, where 2 < q ≤ 3 and D q 0 + is the Riemann-Liouville fractional derivative. In [27], Mujeeb studied existence and multiplicity results by means of the Guo-Krasnosel'skii fixed point theorem, for a coupled system of nonlinear nonlocal boundary value problems for higher order fractional differential equations of the type where λ, µ > 0, n − 1 < α, β ≤ n for n ∈ N; ξ i , η i ∈ (0, 1) for i = 1, 2 and D q 0 + is the Caputo's fractional derivative. Ahmad et al. [2] studied the nonlinear fractional differential equation with nonlocal boundary value in which q ∈ (m − 1, m] , m ∈ N, m ≥ 2. Existence results are based on the contraction mapping principle and Krasnoselskii's fixed-point theorem.
Motived by all of the works above, in this work, we consider the existence and uniqueness of nonnegative solutions of boundary value problem for nonlinear fractional differential equation where f : [0, 1] × R × R → R is a given function, 2 < q < 3, 0 < σ < 1, 0 < η < 1 and c D q a + denotes the Caputo's fractional derivative. We remark that the Caputo fractional derivative is more suitable than the usual Riemann-Liouville derivative for the applications in several engineering problems 58 A. Guezane-Lakoud and K. Belakroum due to the fact that it has better relations with the Laplace transform and because the differentiation appears inside instead of outside of the integral, so to alleviate the effects of noise and numerical differentiation (see [23,29,30]). The rest of paper is organized as follows. In Section 2, we cite some definitions and lemmas needed in our proofs. Section 3 treats the existence and uniqueness of solution by using Banach contraction principle, Leray Schauder nonlinear alternative. Section 4 is devoted to prove the existence of nonnegative solutions with the help of Guo-Krasnoselskii theorem. Last, we give some examples illustrating the previous results.

Preliminaries and Lemmas
In this section we present some lemmas and definitions from fractional calculus theory, see [22], which will be needed throughout the paper. Definition 2.1. If g ∈ C [0, 1] and 0 < α, then the Riemann-Liouville fractional integral is defined by Let α ≥ 0, n = [α] + 1. If g ∈ C n [a, b] then the Caputo fractional derivative of order α of g defined by exists almost every where on [a, b] ([α] is the entire part of α).
The following lemmas give some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative. Then, and Lemma 2.5. [23] Let β > α > 0. Then, the formula c D α Now, we give solution of an auxiliary problem. Lemma 2.6. Let 2 < q < 3, 0 < σ < 1 and y ∈ C [a, b] . The unique solution of the fractional boundary value problem is given by Proof. Applying Lemmas 2.3 and 2.4 to (9), we get

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A. Guezane-Lakoud and K. Belakroum Differentiating both sides of (12) and using Lemma 2.5 it yields The first condition in (9) implies c 1 = c 3 = 0 , the second one gives . Substituting c 2 by its value in (12), we obtain that can be written as i.e., where G is defined by (11). The proof is complete.

Existence and Uniqueness Results
In this section we prove the existence and uniqueness of solutions in the Banach space E of all functions u ∈ C [0, 1] into R with the norm Existence of nonnegative solutions for a nonlinear fractional BVP 61 Then, we have the following lemma.

Lemma 3.1. The function u ∈ E is solution of the fractional BVP (1) if and only
where and Then, the fractional BVP (1) has unique solution u ∈ E.
To prove Theorem 3.2, we use the following property of Riemann-Liouville fractional integrals.
Now we prove Theorem 3.2.

Proof of Theorem 3.2.
We transform the fractional boundary value problem to a fixed point problem. By Lemma 3.1, the fractional boundary value problem (1) has a solution if and only if the operator T has a fixed point in E. Now we will prove that T is a contraction. Let u, v ∈ E. Then, With the help of (18), we obtain On other hand we have where Therefore, Applying (18) we get Let us estimate the term

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A. Guezane-Lakoud and K. Belakroum We have and consequently (29) becomes With the help of (22) it yields Taking into account (25)-(32), we obtain From here, the contraction principle ensures the uniqueness of solution for the fractional boundary value problem (1), this finishes the proof. Now, we give an existence result for the fractional boundary value problem (1).

g are defined as in Theorem 3.2 and
Then, the fractional boundary value problem (1) has at least one nontrivial solution u * in E.
To prove this theorem, we apply Leray-Schauder nonlinear alternative.

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A. Guezane-Lakoud and K. Belakroum Using similar techniques to get (25) Hence, Moreover, we have Using (30) we obtain and from (39) and (43), we get . Therefore, On the other hand, we have Using (30) and (40) it yields when t 1 → t 2 , in (45) and (48), T u(t 1) − T u(t 2 ) and c D q 0 + T u (t 1 ) − c D q 0 + T u (t 2 ) tend to 0. Consequently T (B r ) is equicontinuous. From Arzela-Ascoli Theorem we deduce that T is completely continuous operator. Now we apply Leray-Schauder nonlinear alternative to prove that T has at least one nontrivial solution in E. Letting Ω = {u ∈ E : u < r}, for any u ∈ ∂Ω , such that u = λT u, 0 < λ < 1, we get, with the help of (39) Taking into account (43), we obtain From (49), (50) and (34), we deduce that which contradicts the fact that u ∈ ∂Ω. Lemma 3.5 allows us to conclude that the operator T has a fixed point u * ∈ Ω and then the fractional boundary value problem (1) has a nontrivial solution u * ∈ E. The proof is completed.

Existence of nonnegative solutions
In this section we investigate the positivity of solution for the fractional boundary value problem (1). To do this, we show the following hypotheses.
where a ∈ C ([0, 1] , (0, ∞)) and f 1 ∈ C (R + × R, R + ) ; Let us rewrite the function u as Then, , s ≤ t, s > η, , s > t, s > η. (53) Now we give the properties of the Green function H (t, s) . and . Hence H (t, s) is nonnegative for all t, s ∈ It is easy to see that Ψ (s) = 0. Then, we have Now we look for lower bounds of H (t, s): Since Ψ (s) is nonnegative, we obtain Similarly, we can prove that H σ (t, s) has the following properties. The proof is completed.
We recall the definition of positive solution.
Proof. First, let us remark that under the assumptions on u and f, the function c D σ 0 + u is nonnegative. Applying the right-hand side of inequality (54), we get

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A. Guezane-Lakoud and K. Belakroum Moreover, (54) gives Combining (63) and (64) yield Hence, In view of the left hand side of (54), we obtain for all t ∈ [τ, 1] On the other hand, we have and with the help of (66), we deduce to min t∈(τ,1) This completes the proof.
Define the quantities A 0 and A ∞ by The case A 0 = 0 and A ∞ = ∞ is called superlinear case and the case A 0 = ∞, A ∞ = 0 is called sublinear case. The main result of this section is as follows. To prove Theorem 4.4, we apply the well-known Guo-Krasnosel'skii fixed point theorem on a cone. Theorem 4.5. [15] Let E be a Banach space and let K ⊂ E be a cone. Assume that Ω 1 and Ω 2 are open subsets of E with 0 ∈ Ω 1 ,Ω 1 ⊂ Ω 2 and let be a completely continuous operator such that (i) Au ≤ u , u ∈ K ∩ ∂Ω 1 , and Au ≥ u , u ∈ K ∩ ∂Ω 2 , (ii) Au ≥ u , u ∈ K ∩ ∂Ω 1 , and Au ≤ u , u ∈ K ∩ ∂Ω 2 .
Then, A has a fixed point in K ∩ Ω 2 \ Ω 1 .
Proof. To prove Theorem 4.4, we define the cone K by It is easy to check that K is a nonempty closed and convex subset of E, hence it is a cone. Using Lemma 4.3, we see that T K ⊂ K. From the proof of Theorem 3.4, we know that T is completely continuous in E.
Let us prove the superlinear case. First, since A 0 = 0, for any ε > 0, there exists R 1 > 0, such that
In order to illustrate our results, we give the following examples.
has a unique solution in E.