A class of BVPs for nonlinear fractional differential equations with p-Laplacian operator

In this paper, we study a class of integral boundary value problems for nonlinear differential equations of fractional order with p-Laplacian operator. Under some suitable assumptions, a new result on the existence of solutions is obtained by using a standard fixed point theorem. An example is included to show the applicability of our result.


Introduction
In this paper, we consider the following boundary value problems (BVPs for short) for nonlinear fractional differential equations with p-Laplacian operator: denotes the Caputo fractional derivative of order α and f : [0, 1] × R 2 → R is a continuous function.
Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be non-integer.More and more researchers have found that fractional differential equations play important roles in many research areas, such as physics, chemical technology, population dynamics, biotechnology and economics (see [10,13,15,17,19]).A fractional derivative arises from many physical processes, such as propagations of mechanical waves in viscoelastic media (see [14]), a non-Markovian diffusion process with memory (see [16]), charge transport in amorphous semiconductors (see [23]), etc.Moreover, phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by fractional differential equations (see [7][8][9]).For instance, Pereira et al.
(see [18]) considered the following fractional Van der Pol equation where D α is the fractional derivative of order α and λ is a control parameter which reflects the degree of nonlinearity of the system.Eq.( 2) is obtained by substituting a fractance for the capacitance in the nonlinear RLC circuit model.
In the past few decades, many important results with certain boundary value conditions related to Eq.( 3) had been obtained (see [3,11,29] and the references therein).However, EJQTDE, 2012 No. 70, p. 2 the research of BVPs for fractional p-Laplacian equations has just begun in recent years.
For example, T. Chen et al. [5] investigated the existence of solutions of the boundary value problem for fractional p-Laplacian equation with the following form Later, in [6], T. Chen and W. Liu studied an anti-periodic boundary value problem for the fractional p-Laplacian equation: , Under certain nonlinear growth conditions of the nonlinearity, the existence result was obtained by using Schaefer's fixed point theorem.
In [28], J. Wang and H. Xiang have considered the following p-Laplacian fractional boundary value problem: where 1 < γ, α ≤ 2, 0 ≤ a, b ≤ 1, 0 < ξ, η < 1, and D α 0 + is the standard Riemann-Liouville fractional differential operator of order α.Using upper and lower solutions method, they get some existence results on the existence of positive solution.
Since the p-Laplacian operator and fractional calculus arises from many applied fields, such as turbulent filtration in porous media, blood flow problems, rheology, modeling of viscoplasticity, material science, it is worth studying the fractional p-Laplacian equations.
As far as we know, there are relatively few results on BVPs for fractional p-Laplacian equations, and no paper is concerned with the existence results for fractional p-Laplacian BVPs (1).In this context, we study the problem (1) with integral boundary condition.
Integral boundary conditions have various applications in applied fields such as underground water flow, blood flow problems, chemical engineering, thermo-elasticity, population EJQTDE, 2012 No. 70, p. 3 dynamics, and so on.For a detailed description of the integral boundary conditions, we refer the reader to some recent papers [1,2,4,21,[24][25][26] and the references therein.
The rest of this paper is organized as follows: In section 2, we present some material to prove our main results.In section 3, by applying a standard fixed point principle, we prove the existence of solutions for nonlinear fractional differential equations with p-Laplacian operator.Finally, an example is given to illustrate the main result in section 4.

Preliminaries and lemmas
Firstly, we recall the following known definitions, which can be found, for instance, in [10,19].
Definition 2.1.The Riemann-Liouville fractional integral operator of order α > 0 of a function f : (0, ∞) → R is given by provided that the right side integral is pointwise defined on (0, +∞).
Definition 2.2.The Riemann-Liouville derivative of order α > 0 for a function f : [0, ∞) → R can be written as where n is the smallest integer greater than α.
Definition 2.3.The Caputo fractional derivative of order α > 0 for a function f : [0, ∞) → R can be written as where n is the smallest integer greater than α.
where n is the smallest integer greater than α.Furthermore, the Caputo derivative of a constant is equal to zero.EJQTDE, 2012 No. 70, p. 4 where AC[a, b] be the space of functions f which are absolutely continuous on [a, b].
. Then the following equality holds: where here n is the smallest integer greater than or equal to α.
Given a function f satisfying the Carathéodory conditions and a function u : Ω → R m , we can define another function by composition The composition operator N is called a Nemytskii operator.
Lemma 2.7 ([22, 27], Schaefer's fixed point theorem).Let X be a Banach space and T : is bounded, then T has at least a fixed point in X.

Main results
In this section, we deal with the existence of solutions of the problem (1).Define By means of the linear functional analysis theory, we can prove that X is a Banach space.
if and only if u ∈ C[0, 1] is a solution of the fractional integral equation h(s)ds Proof.Assume that u ∈ AC[0, 1] satisfies (4).For any t ∈ [0, 1], by Lemma 2.5, the first equality of (4) can be written as substituting it into the second equality of (4), we have That is On the other hand, assume that u satisfies (5).Using the fact that D α 0 + is the left inverse of I α 0 + , we get (4), which completes our proof.
} is a solution of the following fractional differential equation: if and only if u ∈ C[0, 1] is a solution of the fractional integral equation where, for any t ∈ [0, 1] Proof.At first, assume that u is a solution of (6).For any t ∈ [0, 1], by Lemma 2.5, the first equality of (6) can be written as Hence, Then the equation ( 6) can be written as follows When we set h(t) = φ q t 0 (t − s) β−1 Γ(β) ϕ(s)ds + F 1 ϕ(t) , then by Lemma 3.1, we have (7).
Conversely, we can obtain that the solution of ( 7) is the solution of the BVP (6) by calculation.This completes the proof.Now, we consider the fractional differential equation (1).Let us define an operator T : X → X as following by where N is a Nemytskii operator defined by Clearly, a fixed point of the operator T is a solution of the problem (1).
Lemma 3.3.The operator T : X → X is completely continuous.
Proof.At first, we show that T : X → X is continuous.
Let {u n } ⊆ X be a sequence with u n → u in X.We will show that T u n − T u → 0. Since uniformly for t ∈ [0, 1].By the continuity of φ q , we have And as a consequence, we have T u n − T u → 0 in X.This shows that T : X → X is continuous.
EJQTDE, 2012 No. 70, p. 10 Assume that {u n } ⊂ X is a bounded sequence.By using the Arzelá-Ascoli theorem, we can select a subsequence {T u n k } of {T u n } which is convergent with respect to the norm Then, by using the Arzelá-Ascoli theorem again, we can select a subsequence . Thus, we have lim As a consequence of above discussion, the operator T : X → X is completely continuous.
The proof is completed.
Proof.Set Ω = {u ∈ X | u = ρT u, ρ ∈ (0, 1]}.Now we are going to show that the set Ω is bounded.From (H 1 ), we have EJQTDE, 2012 No. 70, p. 11 which, together with the monotonicity of s q−1 , yields that For u ∈ Ω, we get u(t) = ρT u(t).Thus, we obtain that where for α ∈ (0, 1], 0 In view of condition (H 2 ), and from the above inequality, we can see that there exists a constant M > 0 such that EJQTDE, 2012 No. 70, p. 12 This shows that the set Ω is bounded.By Lemma 3.3, the operator T : X → X is completely continuous.As a consequence of Schaefer's fixed point theorem, T has at least a fixed point which is a solution of the Eq.( 1).The proof is completed.
is a EJQTDE, 2012 No. 70, p. 5 solution of the following fractional differential equation: