EXISTENCE OF SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATION WITH P-LAPLACIAN THROUGH VARIATIONAL METHOD ∗

In this paper, a class of fractional differential equation with pLaplacian operator is studied based on the variational approach. Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for our equation is obtained. Additionally, we demonstrate the application of our main result through an example.

Fractional calculus is a broader concept, since it is a generalization of arbitrary order derivatives and integrals.With the development of fractional differential equation (FDE for short), a growing number of researchers have been aroused to discuss the existence of solutions for nonlinear FDE owing to the vast application space in different areas of science and engineering, such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc.For details, see [7,8,14,17,18].In recent years, the existence of solutions for nonlinear FDE has been established with all kinds of classical tools, such as fixed-point theorems, the method of upper and lower solutions, the topological degree theory and the critical point theory, etc. (see [1-3, 9, 12] and references therein).In [1], by using the Schauder fixed point theorem, the existence results were obtained for the fractional differential equation with three-point boundary conditions.By means of the Leray-Schauder degree theory and upper and lower solutions method, the existence of multiple solutions was proved for the fractional BVP in [12].Especially, because of the practicability and effectiveness of variational methods and critical point theory, more and more scholars have paid attention to tackling the existence of solutions for fractional BVP by applying those tools, such as [4,10,11,13,23,24], although it is often difficult to develop appropriate function spaces and variational frameworks for FDE containing both left and right fractional derivatives.For example, in [13], under suitable assumptions, the existence of at least one solution for the following FDE was obtained by applying the mountain pass theorem where 0 D α t and t D α T are the left and right Riemann-Liouville derivatives with order 0 < α ≤ 1, respectively.F : [0, T ] × R N → R, ∇F (t, u(t)) is the gradient of F at u. Recently, in [10], Heidarkhani et al. investigated the existence results for FDE with the following form Based on variational methods, the existence of one weak solution for BVP (1.4) was established.
In addition, the existence of solutions for fractional BVP with generalized p-Laplacian operator has been discussed via using variational methods in recent years.Chen in [5] considered the existence of at least one weak solution for a class of p-Laplacian type FDE by using variational method as below where 0 < α ≤ 1, 0 D α t and t D α T are the left and right Riemann-Liouville derivatives, respectively.ϕ p (s) =| s | p−2 s, p > 1.
However, with the advent of the fractional derivative contained in the nonlinearity f , we are not able to deal with the existence of solutions of BVP just relying on variational method and critical point theory directly.Therefore, in this paper, combining the variational method with iterative technique, the existence results are obtained for a class of generalized p-Laplacian type fractional boundary value problem with nonlinear function f including the fractional derivative c 0 D α t .The main contributions of our work include three points.Firstly, the suitable function space and the variational framework are developed reasonably for BVP (1.1).Then, a new criteria on the existence of solutions is obtained for BVP (1.1).Secondly, the nonlocal and nonlinear differential operator t D α T ϕ p ( 0 D α t ) can be reduced to the linear differential operator t D α T 0 D α t under p = 2. Thus, the content of this article is discussed based on the space of L p ([0, T ], R) (2 ≤ p < ∞), which is a generalization for the existing results based on the inner product space of L 2 ([0, T ], R).Finally, comparing with the published relevant results, some looser assumptions are given to guarantee the existence of solutions for BVP (1.1) in this paper.For instance, the literature [6] discussed a class of fractional equation whose nonlinear function f includes the fractional derivative, and the complex parameter conditions P 0 < 1 and Q0 1−P0 < 1 were required to ensure the existence of solutions of the equation.In our assumptions, the analogous restricted conditions do not appear.Hence, the conclusion obtained in the paper is more convenience for application and differ from the results mentioned above.
The organization of this paper is as follows.Section 2 shows a brief review of fractional calculus and the construct of theoretical framework.In section 3, the main result is proposed to guarantee the existence of solutions of BVP (1.1).Then, we demonstrate the application of our result through an example in Section 4. Finally, a conclusion is given in Section 5.

Preliminaries and lemmas
In this section, some associated definitions and basic lemmas are introduced, which will be used throughout this paper.
Let L p ([0, T ], R) (1 ≤ p < ∞) be the space of functions for which the p-th power of the absolute value is Lebesgue integrable with the norm C([0, T ], R) be the space of continuous functions with the norm 14,19]).Let x be a function on [0, T ].Define the left and right Riemann-Liouville fractional integrals with order 0 < α ≤ 1 by 14,19]).The left and right Riemann-Liouville fractional derivatives with order α are represented as where 0 < α ≤ 1 and x is a function defined on [0, T ].
Literatures [14] and [21] show that the Riemann-Liouville fractional integrals satisfy the following property.
Nextly, the suitable function space and the variational framework are developed to apply variational method.
Property 2.2.From [14], the following properties hold where c a D α t and c t D α T are the left and right Caputo fractional derivatives with order α, respectively.(See [14] for a detailed introduction of Caputo fractional derivatives and integrals).
According to [15], the following relationship holds, for any v(t) ∈ E α p .Hence, the definition of weak solution for the BVP (1.1) can be given as below.
Definition 2.4.We say u(t) ∈ E α p is a weak solution of the BVP (1.1).If the following identity holds, for any v(t) ∈ E α p .In order to obtain our result, let us first consider the functional I ξ : E α p → R for any fixed ξ(t) ∈ E α p as follows where u(t) ∈ E α p , F (t, x, y) = x 0 f (t, s, y)ds and H(x) = x 0 h(s)ds, for x, y ∈ R.
Since E α p is compactly embedded in C([0, T ], R) and f is continuous, we can know that I ξ is a continuous and Fréchet differentiable functional on E α p .The Fréchet derivative of I ξ at the point u ∈ E α p is given as Combining Definition 2.4 with (2.10), yields Conversely, if u(t) ∈ E α p is a nontrivial classical solution of BVP (1.1), u(t) is also a weak solution of BVP (1.1) obviously.The proof is completed.
Then, functional I possesses a critical value z * ≥ σ.Moreover, z * can be characterized as where

Main results
In this section, the existence of solutions for BVP (1.1) is established by using Theorem 2.1 and iterative technique.Firstly, some necessary assumptions are stated, which will be used in the further discussion of the main result.(H 1 ) There exist constants τ > p, a ≥ 0, b ≥ 0, d ≥ 0 and 0 < β, β < p, such that where G * is introduced in the sequel.
In order to describe easily for the further analysis, some notations are given as below.Denote Suppose that the conditions (H 1 )−(H 3 ) hold, and λ − L ≥ 0.Then, BVP (1.1) has at least one nontrivial solution on E α p .Proof.The proof will be shown as four steps.
Step 1.We claim that functional I ξ satisfies the P.S. condition.Suppose that For any fixed ξ(t) ∈ E α p with ξ α,p ≤ G. Combining (1.2) with h(0) = 0, one has | h(u) |≤ L | u | for any u ∈ R.Then, based on (1.2), (2.6), (2.8), (2.9) and (H 1 ), we have where Recalling I ξ (u k (t)) is bounded and p is a reflexive space, there exists a weakly convergent subsequence such that u ki u 0 in E α p .For convenience, we still take {u ki } as {u k }.In view of the fact that u k u 0 and I ξ (u k (t)) → 0 as k → ∞ on E α p , we derive which implies that as k → ∞.Hence, according to (3.4), we obtain 5) It is well known that there exist nonnegative constants a 1 and a 2 , for each υ 1 , υ 2 ∈ R n , the following inequalities hold (see [22]) Recalling p ≥ 2, from (3.6), there exists Then, from (3.5) and (3.8), we assert Hence, functional I ξ satisfies the P.S. condition.
Step 2. We will verify that functional I ξ satisfies the geometry conditions of mountain pass theorem.
Step 3. We can establish a sequence {u k } ∞ k=1 ⊂ E α p to satisfy For a fixed point x 0 (t) ∈ E α p with x 0 α,p ≤ G, there exists x(t) ∈ E α p to ensure I x0 (x(t)) = 0 and I x0 (x(t)) ≥ σ under the conclusion obtained in Step 2. Now, we prove that u k α,p ≤ G, for all k ∈ N.
At this point, taking account of (3.15), (3.16) and I x0 (x) = 0, we have Applying the Young inequality, we deduce , one has x p α,p ≤ G p , i.e., x α,p ≤ G.
Suppose that u k−1 α,p ≤ G, similar to the proof procedure above, we obtain that u k α,p ≤ G. Hence, u k α,p ≤ G, for all k ∈ N. From (2.6), we confirm that u k ∞ ≤ ΛG := G * .
Step 4. We will point out that {u k } ∞ k=1 converges to u * ∈ E α p , and u * is a solution of BVP(1.1) on E α p .According to the conclusion obtained in Step 3, we have {u k } ∞ k=1 ⊂ E α p is bounded.Since E α p is a reflexive space, there exists a weakly convergent subsequence such that u ki u * on E α p as k i → ∞.Without loss of generality, take {u ki } as {u k }.Then, from Lemma 2.3, one has Suppose that the sequence {u k } ∞ k=1 is divergent on E α p .Then, there exists a number ε 0 > 0, for any positive number N such that for each k, k > N , we have On the other hand, based on (H 3 ), (2.5), (2.6) and the Hölder inequality, for every v(t) ∈ E α p , a.e.t ∈ [0, T ], one has Namely, I u * (u * (t))v(t) = 0, for any v(t) ∈ E α p , and we can also guarantee that lim k→∞ The proof is completed.Remark 3.1.It is well known that the nonlocal and nonlinear differential operator t D α T ϕ p ( 0 D α t ) can be reduced to the linear differential operator t D α T 0 D α t under p = 2. Thus, the contents of our paper based on the space of L p ([0, T ], R) (2 ≤ p < ∞) are more general comparing with the existing relevant results based on the inner product space of L 2 ([0, T ], R).Moreover, we present some looser assumptions to establish the existence of solutions for BVP (1.1), which guarantee the conclusion obtained in the paper more convenience for application.For example, in reference [6], the complex parameter conditions P 0 < 1 and Q0 1−P0 < 1 are required to ensure the existence of solutions for the equation.The analogous restricted conditions do not appear in our assumptions.So far, little work has been done for the existence of solutions of p-Laplacian fractional boundary value problem with nonlinear function f including the fractional derivative.Therefore, it is worth studying further.
We claim that the conditions of (H Hence, all the conditions of Theorem 3.1 are satisfied.Namely, BVP (4.1) exists one nontrivial solution on E α p , for p = 3.

Conclusion
In this paper, a class of fractional differential equation with p-Laplacian has been investigated.Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for BVP (1.1) has been obtained.The reasonably function space and variational framework for BVP (1.1) have been developed to apply the variational approach.And iterative method has been used to obtain the solution of our equation.Finally, we have illustrated the application of our main result through an example.