EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR A CLASS OF FRACTIONAL STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS WITH IMPULSIVE CONDITIONS∗

In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional differential equations with non-homogeneous Sturm-Liouville conditions and impulsive conditions by using the critical point theory. In addition, at the end of this paper, we also give the existence results of infinite weak solutions of fractional differential equations under homogeneous Sturm-Liouville boundary value conditions. Finally, several examples are given to illustrate our main results.


Introduction
Fractional calculus can be applied to many fields, such as fluid flow, chemical physics and signal processing, probability and statistics, control, electrochemistry and so on (see [1,5,15,17,23,29,34] and references therein). In the past long time, the research on boundary value problems of fractional differential equations has been in full swing (see [7,11,24,32] and references therein).
The main focus of this paper is to study the existence and multiplicity of solutions to the boundary value problems of fractional differential equations by using the critical point theory. Some common methods have been used to discuss fractional differential problem in the literature, whether using fixed point theory (see [4,6] and references therein), topological degree theory (see [8,30] and references therein), upper and lower solution method (see [27,28] and references therein), or variational method, critical point theory (see [14,19] and references therein).
In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional Sturm-Liouville boundary value problems with impulsive condi- Our motivation for studying boundary value problem (for convenience, abbreviated as BVP) (1.1) is largely derived from the fact that it can be used to simulate physical phenomena such as anomalous diffusion. In other words, the traditional second-order convection-diffusion equations cannot accurately simulate diffusion. Therefore, we use the extension of classical advection -dispersion equation, namely fractional advection -dispersion equation, to simulate anomalous diffusion under certain conditions, or to describe nonsymmetric or symmetric transition and solute transportation and so on, for example, in [10,18,19,25,31] and their references.
In [25], the authors applied the critical point theory of non-differentiable functions to establish the existence of infinite solutions for the following boundary value As is known to all, the most prominent feature of impulsive differential equation is that it can fully consider the effect of instantaneous catastrophe on state, and can reflect the changing law of things more profoundly and accurately. With the development of science and technology, many scholars pay more and more attention to the theoretical significance and practical application of impulsive differential equation (see [12,13,22,26] and references therein).
Inspired by the above research, we study the BVP (1.1) in this paper. Compared with the BVP (1.2), the BVP (1.1) studies fractional order, which is undoubtedly the progress and innovation of our research. Further attention is paid to the BVP (1.3), the BVP (1.1) will study non-homogeneous Sturm-Liouville and impulsive boundary value conditions. Thus, the study of the BVP (1.1) is necessary and meaningful.

Preliminaries
For convenience, we will recall the necessary definitions and lemmas of fractional calculus.
Definition 2.1 ( [2,20]). Let u be a function defined on [a, b]. The left and right Riemann-Liouville fractional integrals of order 0 < γ ≤ 1 for the function u denoted by a D −γ t u(t) and t D −γ b u(t), respectively, are defined by provided the right-hand sides are pointwise defined on [a, b], where Γ > 0 is the gamma function. 2,20]). Let u be a function defined on [a, b]. The left and right Riemann-Liouville fractional derivatives of order 0 < γ ≤ 1 for the function u denoted by a D γ t u(t) and t D γ b u(t), respectively, are defined by 2,20]). Let u(t) ∈ AC([0, T ], R), then the left and right Caputo fractional derivatives of order 0 < γ ≤ 1 for the function u denoted by c a D γ t u(t) and and C([0, T ], R) be the p−Lebesgue space and continuous function space, respectively, with the norms It is obvious that E α,p is the space of functions u(t) ∈ L p ([0, T ], R) with an α order Caputo fractional derivative c 0 D α t u(t) ∈ L p ([0, T ], R). When p = 2, we denote E α,2 as X.
Lemma 2.1 ( [9]). The space X is a reflexive and separable Banach space. Lemma 2.2 ( [9,25]). Let 0 < α ≤ 1 and 1 ≤ p < +∞. For any u ∈ L p ([0, T ], R), we have In this paper, we treat the BVP (1.1) in the Hilbert space X with the inner product and the corresponding norm defined by Lemma 2.4. Let 1 2 < α ≤ 1 and u ∈ X, the norm u α,2 is equivalent to Proof. First, we will show that there exists a constant K > 0 such that u α,2 ≤ K u . According to Property 2.3, we deduce that Obviously, we can find a constant K > 0 so that (2.5) On the other hand, we must find a constant H > 0 satisfying u ≤ H u α,2 . Based on Property 2.3 again, we get that . Then from (2.1) and Hölder's inequality, we get It follows from Property 2.
. Then according to (2.2) and Hölder's inequality, we can obtain After calculation, we can get By integrating the above two formulas, one has So In other words, we can find a constant H > 0 so that and K is defined (2.5).
Proof. Similar to the proof of Lemma 5.4 in [25], we can take and then we can get u ∞ ≤ Λ u .
According to Property 2.1, then using Definition 2.3, the BVP (1.1) can be transformed into the following boundary value problem Above all, we have We are now introduce the concept of a weak solution for the BVP (2.7).
Definition 2.5. The weak solution of the BVP (2.7) is u ∈ X satisfying the following equation For ∀u ∈ X, we consider the functional J : X → R, that is J ∈ C 1 (X, R) as follows where u is defined by (2.4) and F (t, u) = u 0 f (t, s)ds for all (t, u) ∈ [0, T ] × R. It is easy to get that the functional J is differentiable on X and Lemma 2.8. If u ∈ X is a critical point of J in X, then u is a weak solution of the BVP (2.7).
Proof. Suppose u ∈ X is a critical point of J in X, that is, (2.10) Without loss of generality, for any i ∈ {0, 1, 2, · · · , m}, take v ∈ C ∞ 0 (t i , t i+1 ), then (2.10) can be sorted into According to Lemma 2.6, we get Multiply both sides of (2.11) by v, then integrate from 0 to T , and we have In addition, It follows from (2.10), (2.12) and (2.13) that Therefore, one has Lemma 2.9 ( [16]). If ϕ is sequentially weakly lower semi-continuous on a reflexive Banach space E and has a bounded minimizing sequence, then ϕ has a minimum on E. (2) Θ is bounded and weak sequentially closed, i.e., by definition, for each sequence {u n } in Θ such that u n u as n → +∞, we always have u ∈ Θ.
Then, ϕ possesses a critical value ≥ σ. Moreover, can be characterized as

Lemma 2.12 ( [21]
). Let E be an infinite-dimensional Banach space and let ϕ ∈ C 1 (E, R) be even, satisfy the (P S)-condition, and have ϕ(0) = 0. Suppose that E = V W , where V is finite dimensional, and ϕ satisfies the following (1) there exist > 0 and ρ > 0 such that ϕ(u) ≥ for all u ∈ E with u = ρ, and Then ϕ possesses an unbounded sequence of critical values.

Main result
In this section, the existence and multiplicity of weak solutions for the BVP (1.1) are considered.
Theorem 3.1. Assume that (H 1 ) and one of the following conditions hold, then the BVP (2.7) admits at least one weak solution, and thus the BVP (1.1) has at least one weak solution.
(2) (H 3 ) holds and λ ∈ (0, Proof. Let u k converges weakly to u in X, then u k converges uniformly to u in [0, T ]. Based on the continuity of f and I i (i = 1, 2, · · · , m), it is logical to get Notice that lim inf so we conclude that J(u) is sequentially weakly lower semi-continuous. If (H 2 ) holds, one can get that there exists L > 0 such that for a.e. t ∈ [0, T ] and u ∈ R, therefore, combined with (2.6), we have According to (H 1 ) and (2.6), we can obtain Combining the above two formulas and (2.8), we infer that On account of 0 ≤ θ < 2 and 0 ≤ σ i < 1, we get that lim u →+∞ J(u) = +∞. This means that J(u) has a bounded minimizing sequence. So the BVP (2.7) exists at least one weak solution, that is, the BVP (1.1) admits at least one weak solution. If (H 3 ) holds. (H 3 ) implies that there exists Q > 0 such that for a.e. t ∈ [0, T ] and u ∈ R. Similar to the above discussion, we can get Notice that λ ∈ (0, 1 2Λ 2 T 0 η(t)dt ), we get that lim u →+∞ J(u) = +∞. Lemma 3.1. Assume that (H 1 ) and (H 4 ) hold, then J satisfies the (P S)-condition.
Proof. Assume that {u k } k∈N ⊂ X is a sequence such that {J(u k )} k∈N is bounded and lim k→+∞ J (u k ) = 0. It follows from µ > 2, (H 1 ) and (H 4 ) that Because {u k } is bounded in X, there exists a subsequence of {u k }, which converges weakly to a subsequence of u in X. For convenience, we still record this subsequence as {u k }. Then {u k } converges uniformly to u on [0, T ]. Thus as k → +∞. Since lim k→+∞ J (u k ) = 0 and {u k } converges weakly to some u, we get as k → +∞. On the other hand, notice that According to the previous discussion, u k − u → 0 as k → +∞ is proved. That is, u k → u in X. In summary, J satisfies the (P S)-condition.
Remark 3.1. From (H 4 ), we know that there exists P > 0 such that and ).
That is, the BVP (1.1) exists at least two different weak solutions.
Proof. Let B r denotes the open ball in X with radius r and centered at 0 and let ∂B r andB r represent the boundary and closure of B r , respectively. It is easy to getB 1 Λ is a bounded weak closed set.
Next, we have known that J(u) is sequentially weakly lower semi-continuous in X, based on Lemma 2.10, we get J(u) has a local minimum point u 0 inB 1 Λ , in other word, J(u 0 ) ≤ J(0) = 0.
Thus, u 0 and u 2 are two different weak solutions of the BVP (2.7), moreover, they are also two different weak solutions of the BVP (1.1).
Next, let's consider the case of A = B = 0, then the BVP (1.1) becomes the following form Proof. According to the definition of J, it is obvious that J(0) = 0. In addition, according to (H 5 ), J is even. Notice the fact that all norms are equivalent in finite dimensional space, so for any finite dimensional spaceX in X, and for each u ∈X, we know that there exists a constant G > 0, so that Based on (3.2), for any u ∈X, there is a constant G > 0 such that T 0 F (t, u(t))dt ≥F 0 u µ L µ − P T ≥F 0 G µ u µ − P T, which yields that And because µ > 2 and 1 ≤σ i + 1 < 2, we get J(u) → −∞ as u ∈Ẽ and u → +∞. Then there exists R = R(Ẽ) such that J(u) ≤ 0 for all u ∈Ẽ satisfies u ≥ R. In view of (H 5 ), we know Since µ > 2, the above inequality implies that we can choose δ > 0 small enough such that J(u) ≥ > 0 for u = δ. Base on Lemma 2.12, the BVP (3.4) has infinitely many weak solutions.