Initial value problem of fractional order

Fractional initial value problems have been studied recently by many authors. In the paper of Yoruk, Gnana Bhaskar, and Agarwal (2013), Krasnoselskii-Krein, Nagumo’s type uniqueness result and successive approximations have been extended to differential equations of fractional order 0 < q < 1. Some results in literature are given for boundary value problems for ordinary differential equation, by Webb (2009) and Graef, Kong, and Wang (2008) in the case where the Green function associated to the posed problem is vanishing on a set of zero measure. By means of Guo–Krasnosel’skii fixed point theorem the existence of nontrivial positive solution is proved. (P): {


Introduction
This work is devoted to the study of positive solutions for the following fractional differential equation with initial conditions Where f : 0, 1 × ℝ × ℝ → ℝ is a given function, 2 < q < 3, D q 0 + denotes the Riemann's fractional derivative. We note that few papers dealing with fractional differential equations, considered the nonlinearity f in (P) depending on the derivative of u, due to this fact we need more assumptions on f and the problem becomes more complicated.
Fractional initial value problems have been studied recently by many authors. In the paper of Yoruk, Gnana Bhaskar, and Agarwal (2013), Krasnoselskii-Krein, Nagumo's type uniqueness result and successive approximations have been extended to differential equations of fractional order 0 < q < 1. Some results in literature are given for boundary value problems for ordinary differential equation, by Webb (2009) and Graef, Kong, and Wang (2008) in the case where the Green function associated to the posed problem is vanishing on a set of zero measure. By means of Guo-Krasnosel'skii fixed point theorem the existence of nontrivial positive solution is proved.

PUBLIC INTEREST STATEMENT
Under suitable conditions on the nonlinearity term, we prove the existence of positive solutions for an initial fractional value problem. The proofs are based on a fixed point theorem.
In this work, we discuss the existence of positive solutions for the problem (P). To prove our results, we assume some conditions on the nonlinear term f , then we use a cone fixed point theorem due to Guo-Krasnoselskii.
We start by solving an auxiliary problem which allows us to get the expression of the solution, let us consider the following linear problem (P 0 ): Lemma 2.5 Assume that y ∈ C([0, 1], ℝ), then the problem (P 0 ) has a unique solution given by: Proof Using Lemmas 2.3 and 2.4, we get : The condition u 0 = 0 implies that c = 0. Differentiating both sides of (2.5) and using the initial condition u � (0) = 0, it yields b = 0. The condition u �� 0 = 0 implies a = 0. Substituting a, b and c by their values in (2.5), we obtain |u (t)|. Define the operator T: E → E as follows: Lemma 2.6 The function u ∈ E is solution of the initial value problem (P) if and only if Tu(t) = u(t), for all t ∈ 0, 1 .

Main results
First, we state the assumptions that will be used to prove the existence of positive solutions: (H 2 ) There exists two positive constants g 1 and g 2 such that 0 < g 1 ≤ g(t) ≤ g 2 for all t ∈ 0, 1 .

The operator T: E → E becomes
Let us introduce the following notations Let K be the classical cone Recall the definition of a positive solution: Definition 3.1 A function u is called positive solution of problem (P) if u(t) ≥ 0, ∀t ∈ 0, 1 and it satisfies the differential equation and the initial conditions in (P). Now, we give the main result of this paper Theorem 3.2 Under the assumptions (H 1 ) and (H 2 ) and if f 1 is convex and decreasing to each variables (i.e. for u fix, f 1 (u,.) is decreasing according to the second variable and for v fix the function f 1 (.,v) is decreasing according to the first variable), then the problem (P) has at least one nontrivial positive solution in the cone K, in the case A 0 = +∞ and A ∞ = 0.
Jensen's inequality for a convex function is given by: The inequalities hold in reversed order if f is concave on Δ.
Then  has a fixed point in K ∩ Ω 2 �Ω 1 .
Proof of Theorem 3.2. Using Ascoli Arzela Theorem, we prove that T is a completely continuous operator. From A 0 = +∞, we deduce that for M ≥ Γ(q+2) g 1 , there exists r 1 > 0, such that if , we should prove the first statement of Theorem 3.5. Assume that u 1 ∈ K ∩ Ω 1 , then the mean value theorem implies (1 − s) q g(s)f 1 (u 1 (s), u � 1 (s))ds.