EXISTENCE OF SOLUTIONS FOR A FRACTIONAL ADVECTION-DISPERSION EQUATION WITH IMPULSIVE EFFECTS VIA VARIATIONAL APPROACH∗

In this paper, based on the variational approach and iterative technique, the existence of nontrivial weak solutions is derived for a fractional advection-dispersion equation with impulsive effects, and the nonlinear term of fractional advection-dispersion equation contain the fractional order derivative. In addition, an example is presented as an application of the main result.


Introduction
In this paper, we investigate the existence of nontrivial weak solutions for following a fractional advection-dispersion equation (FADE for short) with impulsive effects   [20] investigate the FADE (1.2) with b(t) = c(t) = 0, T = 1, and the boundary conditions u(0) = u(1) = 0. For more background information on FADE, see [1,2,4,13,14,18,21,[25][26][27] and so on. Recently, by the critical point theory, Jiao and Zhou in [9] consider the symmetric FADE of the following form    d dt Li et al. in [12] study the existence of solutions to fractional boundary-value problems with a parameter by using critical point theory and variational methods    − d dt Especially, differential equations with impulsive effects are intensively investigated recently. It can be used to describe discontinuous jumps and sudden changes of their states in optimal control and so on. Therefore, it is worth to study. There are few works that the existence of solutions for fractional advection-dispersion equations with impulsive effects and impulsive fractional differential equations. Chai and Chen in [3] investigated the following impulsive fractional boundary problem Under the condition 0 < a 1 ≤ a(t) ≤ a 2 , the authors proved the existence of at least one nontrivial solution by using the variational method and iterative technique.
Nyamoradi and Tayyebi in [17] study the existence of weak solutions for following impulsive fractional differential equations by using critical point theory and variational methods are continuous functions. In early time, Wang et al. in [22] apply Minimax principle and saddle point theorem to study the existence of weak solutions of problem (1.6).
Obviously, if we choose α = 1, and I j = 0 (j = 1, 2, . . . , n), then the FADE (1.1) reduces to the second-order FADE of the following form There have been many methods to investigate the existence of solutions of problem (1.7) such as fixed point theory and monotone iterative method and so on. (see [7,8,24] and references therein). Inspired by the works described above, we aim to investigate the existence of nontrivial weak solutions for a fractional advection-dispersion equation with impulsive effects. Different from the previous paper, the main characteristics of the present paper are as follows. Firstly, the nonlinear term of fractional advectiondispersion equation contain the fractional order derivative. As far as we know, there are no works for the impulsive fractional advection-dispersion equation with nonlinearity involving fractional derivatives of unknown function, although many excellent results about impulsive fractional differential equation are obtained. Secondly, the approach is different from the [6,9,12,16,17,22,23]. The tool of this article is variational method and iterative technique, which has been adopted in [3,20]. Comparison with [20], the assumed conditions in this paper are different from the conditions in [20], and the result depends on the parameter. Finally, comparisons with [3] and [11], the hypothetical conditions are weaker than those in [3,11]. For example, functions φ, contain constants s 1 , s 2 , l, m, d. The parameter ϱ in [11] is a non-negative real, but, the parameter ϱ can be either positive or negative in this paper.
The paper consists of four sections. In sect. 2, we present some preliminaries and lemmas to be used later. In sect. 3, we discuss the existence of nontrivial weak solutions for FADE (1.1). In sect. 4, we take an example to illustrate our main results.

Preliminaries and lemmas
In this section, some definitions and lemmas are presented, which are to be used to prove our main results.
which is equivalent to (2.1).
Owing to the continuity of f and I j , the functional

Definition 2.5 ( [15]
). Suppose that X is a Banach space and ϕ ∈ C 1 (X, R). We say that ϕ satisfies the Palais-Smale (P.S.) condition if any sequence {u n } ⊂ X such that ϕ(u n ) is bounded and ϕ ′ (u n ) → 0 as n → ∞ possesses a convergent subsequence in X. Lemma 2.4 (Theorem 2.2, [19]). Let X be a real Banach space and ϕ ∈ C 1 (X, R) satisfying P.S. condition. Suppose ϕ(0) = 0, and Then ϕ possesses a critical value c ≥ β. Moreover c can be characterized as For convenience, put

Main result
We are now in a position to give some conditions that will be used in the proof of our main result.
Proof. We give the proof of this theorem by five steps.
Step 3. We show I w satisfies P.S. condition. Let {u n } ⊂ J α 0 be a P.S. sequence, that is {I w (u n )} is bounded and 3), (2.5)-(2.7), and Holder's inequality, we assert In view of 0 < τ j , τ, ξ < 2, ζ > 2, and {I w (u n )} is bounded and I ′ w (u n ) → 0 as n → ∞, we know that {u n } ⊂ J α 0 is bounded. Moreover, it has a weakly convergent subsequence u n k i ⇀ u ∈ J α 0 in view of the reflexivity of J α 0 . It follows from Proposition 2.4 we know that u n → u in C[0, T ]. We still denote {u n k i } by {u n }. Since f and I j (j = 1, 2, . . . , n) are continuous, and In view of the fact that I ′ w (u n ) → 0, u n ⇀ u as n → ∞, the boundedness of the sequence {u n − u}, we obtain as n → ∞. So, u n → u in J α 0 . Hence, functional I w satisfies the P.S. condition. From Lemma 2.4, we know that there exists a point u ∈ J α 0 satisfying I ′ w ( u) = 0 and I w ( u) ≥ ω 1 > 0.
Step 5. We certificate that the iterative sequence {u n } constructed in the previous step is convergent to a nontrivial weak solution u of FADE (1.1). Assume that the sequence {u n } is divergent on J α 0 , that is, there exists a number ε > 0, for any positive number N such that for each n, n + 1 > N, we have ∥u n+1 − u n ∥ α ≥ ε.
This implies that u is a weak solution of FADE (1.1). Similarly, we can certificate that lim n→∞ I un−1 (u n ) = I u ( u). Because I un−1 (u n ) ≥ ω 1 > 0, we conclude that I u ( u) ≥ ω 1 > 0, it indicates that u is a nontrivial weak solution of FADE (1.1). Hence, our claim is proved.