POSITIVE SOLUTIONS FOR A P-LAPLACIAN TYPE SYSTEM OF IMPULSIVE FRACTIONAL BOUNDARY VALUE PROBLEM∗

Abstract In this paper, the aim is to discuss a class of p-Laplacian type fractional Dirichlet’s boundary value problem involving impulsive impacts. Based on the approaches of variational method and the properties of fractional derivatives on the reflexive Banach spaces, the existence results of positive solutions for our equations are established. Two examples are given at the end of each main result.

Fractional differential boundary value problems (BVPs for short) can describe many interesting nonlinear phenomena due to the fact that they have wild application background in multifarious fields of science, for instance, mathematics, biological processes, chemical engineering, underground water flow, thermo-elasticity and plasma physics ( [8,11,16]), etc. For this reason, many researchers have been attracted to focus on this kind of problems, and a large number of meaningful results have been obtained in resent years, (we refer the reader to the papers [5,22,29]). The classical approaches, such as the method of mixed monotone iterative, topological degree theory, fixed-point theorems and upper and lower solutions method, etc, are always used to investigate the existence results of positive solutions for nonlinear BVPs, and those methods have been developed maturely. Since the formulation of ordinary p-Laplacian operator was put forward by Leibenson in 1983 [21], there are numerous applications in nonlinear elastic mechanics, non-Newtonian fluid theory, and so on. Based on some classical metheds, many relevant existence results for fractional differential equations with generalized p-Laplacian operator have been established ( [12,13,23,31]). In [13], the following eigenvalue problem of nonlinear fractional differential equation with p-Laplacian operator was given where λ > 0 is a parameter, 2 < α ≤ 3, 1 < β ≤ 2, Φ p is the generalized p-Laplacian operator, f : (0, +∞) → (0, +∞) is continuous. The authors discussed the existence of at least one or two positive solutions for (1.2) by the Guo-Krasnosel'skii fixed point theorem in cones. Moreover, relying on the generalization of Leggett-Williams fixed point theorem, the existence of at least three positive solutions was obtained for the p-Laplacian type fractional equation involving both the Riemann-Liouville fractional derivatives and Caputo fractional derivatives in [23]. By virtue of some methods from nonlinear functional analysis including the contraction mapping theorem and the Brouwer fixed point theorem, the authors presented the existence and uniqueness of solution for a discrete fractional BVP with p-Laplacian operator in [31]. One the other hand, since Ambrosetti and Rabinowitz proposed Mountain pass theorem in 1973 [1], variational methods together with critical point theory have become useful and practical tools in dealing with the existence results for fractional differential equations in recent years [2,10]. The problem of p-Laplacian type fractional differential equation is also studied based on variational approachs [6,[17][18][19].
For example, Li et al. in reference [17] considered a class of ordinary p-Laplacian type equation with the form of Through the generalized Mountain pass theorem of Rabinowitz, the existence of periodic solutions was studied for problem (1.3). In [18], the existence of at least one nontrivial solution was discussed for the following fractional BVP with p-Laplacian operator by using the Mountain pass theorem and iterative technique Lipschitz continuous function with the Lipschitz constant L > 0. Furthermore, by using critical point theory, the existence of at least one weak solution was studied for a fractional differential equation with generalized p-Laplacian operator in [6]. Additionally, it is well known that the normal characteristic of impulsive effects is the sudden changes at some certain moments owing to instantaneous disturbances. Because the impulses always appear in many actual systems, such as in multi-agent systems, signal processing systems, automatic control systems, etc., the research on the impulsive problem of fractional differential BVPs has received great progress. Recently, many existence results about impulsive fractional differential equations via variational methods have been studied (see [3,4,9,10,25]).
Inspired by the mentioned work above, this paper devotes to investigate a class of p-Laplacian type impulsive fractional system with Dirichlet's boundary value conditions. Based on the variational approaches and the properties of fractional derivatives defined on reflexive Banach spaces, the existence results for the problem (1.1) are established. The main features for our paper are stated as follows: Firstly, we discuss the existence of positive solutions for problem (1.1) in the general case 1 < p < ∞, which is the generalization for some related results based on the particular case of p = 2. It's worth noting that the differential operator t D α T Φ p ( 0 D α t ) (α > 0, p > 1) is nonlocal and nonlinear, and it can be reduced to the linear differential operator t D α T 0 D α t under p = 2. Further, if take α = 1, the operator t D α T 0 D α t can be recovered the usual definition, i.e., a second-order differential operator d 2 dt 2 . So that, BVPs (1.1) can be recovered a integer-order Dirichlet's BVPs with impulsive impacts in the particular case of p = 2, α i = 1, i = 1, 2, ..., n, i.e., for i = 1, 2, ..., n. Secondly, we discuss a n-dimensional fractional differential equations rather than a single equation and consider the impulsive effects, which cause some aspects of the paper more complicated, such as the Euler-Lagrange functional related to the system (1.1), the procedure of applying the Mountain pass theorem. Finally, the positive solution is also studied in this paper. To the best of authors' knowledge, little work has been developed on studying the positive solutions for generalized p-Laplacian type impulsive fractional system by using the variational method. Therefore, it is worthwhile to be investigated. So that, our main results are different from those relevant literatures mentioned above, and moreover, complement the results in previous ones. This paper's organization is stated as follows. In section 2, some basic definitions and preliminary facts for fractional calculus are introduced. In section 3, we establish appropriate function spaces and the variational framework for problem (1.1), which are necessary for the discussion of this paper. Then, applying critical point theorems, the main results are obtained, and meanwhile, tow examples are given to illustrate the applications of our results in section 4. Finally, a conclusion is presented in section 5.

Preliminaries
In this section, some basic definitions and properties are introduced for fractional calculus, and some important theorems are given, which shall be used throughout this paper.
For with order γ > 0 are defined by 16,26]). Let x be a function defined on [a, b]. Define the left and right Riemann-Liouville fractional derivatives denoted by a D γ t and t D γ b with order n − 1 ≤ γ < n and n ∈ N as follows x(t) can be recovered the usual definitions, i.e., , R) and n − 1 < γ < n. Then the left and right Caputo fractional derivatives of order γ for x denoted as c a D γ t x(t) and c t D γ b x(t) respectively, are given by , R) and γ = n − 1, we can obtain the usual definitions for the Caputo fractional derivatives, i.e., following inequality: holds. (The proof immediately follows from Theorem 5 of [7].) Nextly, we point out some fundamental definitions and theorems, which will be used to obtain our main results.
is bounded and φ ′ (u k ) → 0 as k → ∞ possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (P.S. condition for short).
Then φ possesses a critical value c ≥ σ . Moreover, c can be characterized as where

Fractional derivative spaces and variational setting
In what follows, the appropriate function spaces and the variational framework for problem (1.1) are established.
with the norm
For any Then, we obtain

Lemma 3.2. The fractional derivative space X is a reflexive and separable Banach space.
Proof. The proof is similar to Lemma 9 of [19], we omit it here.
Obviously, u is a weak solution of (1.1) if u is a classical one. Define the Euler-Lagrange functional I : X → R related to BVPs (1.1) by Lemma 3.5. We claim that I ∈ C 1 (X, R).

Proof.
Denote Note that f is continuously differentiable and I ij are continuous for j = 1, 2, ..., m, i = 1, 2, ..., n. Thus, I 1 (u) and I 2 (u) are clearly continuous and differentiable on X, and we present the derivative of I at the point u(t) ∈ X as Using a standard argument in Theorem 4.1 of [15], we have I 1 (u(t)) ∈ C 1 (X, R).
In order to simplify the description of further discussion, some notations are stated here. Denote
Then, BVPs (1.1) admits at least one positive solution that minimizes I on X.

Conclusion
In this paper, by the methods of a critical point theorem and the Mountain pass theorem, the existence of at least one positive solution has been addressed for a class of p-Laplacian type fractional Dirichlet's boundary value problem involving impulsive impacts. Two examples have been given to illustrate the applications of our results.