Solvability of a boundary value problem at resonance

This paper concerns the solvability of a nonlinear fractional boundary value problem at resonance. By using fixed point theorems we prove that the perturbed problem has a solution, then by some ideas from analysis we show that the original problem is solvable. An example is given to illustrate the obatined results.

where f ∈ C([0, 1] × R × R, R), 2 < q < 3, c D α 0 + denotes the Caputo's fractional derivative. The problem (P) is called at resonance in the sense that the associated linear homogeneous boundary value problem has u(t) = ct 2 , c ∈ R as nontrivial solutions. In this case since Leray-Schauder continuation theory cannot be used, we will apply some ideas from analysis. Although these techniques have already been considered in Mawhin (1972) for ordinary differential equation but the present problem (P) is different since the nonlinearity f depends also on the derivative and the differential Eq. (1) is of fractional type.
Fractional boundary value problems at resonance have been investigated in many works such in Bai (2011), Han (2007, Infante and Zima (2008), where the authors applied Mawhin coincidence degree theory. Further for the existence of unbounded positive solutions of a fractional boundary value problem on the half line, see Guezane-Lakoud and Kılıçman (2014).
The organization of this work is as follows. In Sect. 2, we introduce some notations, definitions and lemmas that will be used later. Section 3 treats the existence and uniqueness of solution for the perturbed problem by using respectively Schaefer fixed point theorem and Banach contraction principal. Then by some analysis ideas, we prove that the problem (P) is solvable. Finally, we illustrate the obtained results by an example.

Preliminaries
In this section, we present some Lemmas and Definitions from fractional calculus theory that can be found in Nieto (2013), Podlubny (1999). a, b]) and α > 0, then the Riemann-Liouville fractional integral is defined by Definition 2 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 everywhere on [a, b] ([α] is the integer part of α).
Lemma 3 For α > 0, g ∈ C([0, 1], R), the homogenous fractional differential equation has a solution where, c i ∈ R, i = 0, . . ., n − 1, here n is the smallest integer greater than or equal to α. (2) Now we start by solving an auxiliary problem. Differentiating both sides of (6), it yields The first condition in (3) gives c 0 = c 1 = 0, the second one implies that I q 0 + y(1) = 0, hence (3) has solution if and only if I q 0 + y(1) = 0, then the problem (3) has an infinity of solutions given by Now we try to rewrite the function u. We have then substituting c by its value in (9) we obtain Hence the linear problem can be written as if v c is a solution of (11) with c = u(1) then u is a solution of (1). We apply Schaefer fixed point theorem to prove Theorem 1.

Theorem 2 Let A be a completely continuous mapping of a Banach space X into it self,
such that the set {x ∈ X : x = Ax, 0 < < 1} is bounded, then A has a fixed point.
Proof of Theorem 1 By Arzela-Ascoli Theorem we can easly show that T c is a completely continuous mapping.
Now, let us prove that the set {v ∈ E : v = T c v, 0 < < 1} is bounded. Endeed for ∈ (0, 1) such that v = T c (v), we have remarking that H(t, s) is continuous according to both variables s, t on [0, 1] × [0, 1], nonnegative and 0 ≤ H (t, s) ≤ 2 then using assumptions (14) and (15), we get thus, where G(t) = H (t) = (0.1) 1 + t 2 , hence we get In view of Theorem 3, T c has a unique fixed point v * c in E. It is easy to see that From the above discussion and Theorem 4 we conclude that the problem (24) is solvable in E.

Conclusion
The goal of this paper was to provide sufficient conditions in order to ensure the existence of solutions for the following fractional boundary value problem where f ∈ C([0, 1] × R × R, R), 2 < q < 3, c D α 0 + denotes the Caputo's fractional derivative. By using fixed point theorems we proved that the perturbed problem has a solution, then we also show that the original problem is solvable. An example is provided n order to illustrate the results.