Solvability of Some Boundary Value Problems for Fractional p-Laplacian Equation

and Applied Analysis 3 where Ah (t) = −I β 0 h (t) 󵄨󵄨󵄨󵄨󵄨t=1


Introduction
Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger.This subject, as old as the problem of ordinary differential calculus, can go back to the times when Leibniz and Newton invented differential calculus.As is known to all, the problem for fractional derivative was originally raised by Leibniz in a letter, dated September 30, 1695.A fractional derivative arises from many physical processes, such as a non-Markovian diffusion process with memory [1], charge transport in amorphous semiconductors [2], and propagations of mechanical waves in viscoelastic media [3], and so forth.Moreover, phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are also described by differential equations of fractional order [4][5][6][7][8].For instance, Pereira et al. [9] considered the following fractional Van der Pol equation:    () +  ( 2 () − 1)   () +  () = 0, 1 <  < 2, (1) where   is the fractional derivative of order  and  is a control parameter that reflects the degree of nonlinearity of the system.Equation ( 1) is obtained by substituting the capacitance by a fractance in the nonlinear RLC circuit model.
Recently, fractional differential equations have been of great interest due to the intensive development of the theory of fractional calculus itself and its applications.For example, for fractional initial value problems, the existence and multiplicity of solutions (or positive solutions) were discussed in [10][11][12][13].On the other hand, for fractional boundary value problems (FBVPs), Agarwal et al. [14] considered a twopoint boundary value problem at nonresonance, and Bai [15] considered a -point boundary value problem at resonance.For more papers on FBVPs, see [16][17][18][19][20][21] and the references therein.
The turbulent flow in a porous medium is a fundamental mechanics problem.For studying this type of problems, Leibenson [22] introduced the -Laplacian equation as follows: where Obviously,   is invertible and its inverse operator is   , where  > 1 is a constant such that 1/ + 1/ = 1.
Motivated by the works mentioned previously, in this paper, we investigate the existence of solutions for fractional -Laplacian equation of the form subject to either boundary value conditions or where 0 < ,  ≤ 1, 1 <  +  ≤ 2,   0 + is a Caputo fractional derivative, and  : Note that the nonlinear operator   0 +   (  0 + ) reduces to the linear operator   0 +   0 + when  = 2 and the additive index law holds under some reasonable constraints on the function () [30].The rest of this paper is organized as follows.Section 2 contains some necessary notations, definitions, and lemmas.In Section 3, based on Schaefer's fixed point theorem, we establish two theorems on existence of solutions for FBVP ( 5) and ( 6) (Theorem 7) and FBVP ( 5) and ( 7) (Theorem 8).Finally, in Section 4, an explicit example is given to illustrate the main results.Our results are different from those of bibliographies listed in the previous texts.

Preliminaries
For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory, which can be found, for instance, in [31,32].
Definition 1.The Riemann-Liouville fractional integral operator of order  > 0 of a function  : (0, +∞) → R is given by provided that the right side integral is pointwise defined on (0, +∞).
Definition 2. The Caputo fractional derivative of order  > 0 of a continuous function  : (0, +∞) → R is given by where  is the smallest integer greater than or equal to , provided that the right side integral is pointwise defined on (0, +∞).
In this paper, we take  = [0, 1] with the norm By means of the linear functional analysis theory, we can prove that  is a Banach space.

Existence Results
In this section, two theorems on existence of solutions for FBVP (5) and (6) and FBVP ( 5) and ( 7) will be given under nonlinear growth restriction of .
As a consequence of Lemma 3, we have the following results that are useful in what follows.

Lemma 5. Given ℎ ∈ 𝑌, the unique solution of
is Abstract and Applied Analysis Proof.Assume that () satisfies the equation of FBVP (13); then Lemma 3 implies that From the boundary value condition   0 + (1) = 0, one has Thus, we have By condition (0) = 0, we get  1 = 0.The proof is complete.
Define the operator  :  →  by where  :  →  is the Nemytskii operator defined by Clearly, the fixed points of the operator  are solutions of FBVP ( 5) and ( 6).
Our first result, based on Schaefer's fixed point theorem and Lemma 5, is stated as follows.
Then FBVP (5) and ( 6) has at least one solution, provided that Proof.We will use Schaefer's fixed point theorem to prove that  has a fixed point.The proof will be given in the following two steps.
Let Ω ⊂  be an open bounded subset.By the continuity of , we can get that  is continuous and (Ω) is bounded.Moreover, there exists a constant  > 0 such that |  0 +  + | ≤ , for all  ∈ Ω,  ∈ [0, 1].Thus, in view of the Arzelà-Ascoli theorem, we need only to prove that (Ω) ⊂  is equicontinuous.
Now it remains to show that the set Ω is bounded.
Our second result, based on Schaefer's fixed point theorem and Lemma 6, is stated as follows.
Proof.The proof work is similar to the proof of Theorem 7, so we omit the details.