Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential

This paper concerns solutions generated by impulses for a class of second order BVPs with convex potential. Necessary and sufficient conditions for the existence of non-constant solutions are derived via variational methods and critical point theory.

Theorem A ( [11,Theorem 1.8]).Assume that F satisfies condition (A) and F(t, •) is strictly convex for a.e.t ∈ [0, T].Then the following conditions are equivalent: where u(t) = (u 1 (t), u 2 (t)) T .When the second order BVPs (1.1) and (1.2) have no solution, the present paper concerns about generating solutions by impulses.More precisely, in this paper we will consider the necessary and sufficient conditions for the existence of solutions generated by impulses of the above two BVPs, that is, solutions of the problem (1.1) generated by and solutions of the problem (1.2) generated by where 0 < t 1 < T is the instant where the impulse occurs, ∆( ui Here, a solution of the problem (1.1) (resp.(1.2)) with the impulsive condition (1.4) (resp.(1.5)) is said to be generated by impulses if the problem (1.1) (resp.(1.2)) does not possess any solution.
Impulsive effects arise from the real world and are used to describe sudden, discontinuous jumps.Due to their significance, a number of papers [2,10,15] have provided the qualitative properties of such equations.Some efforts have been made in studying the existence of solutions of impulsive problems via variational methods, see, for instance, [1, 3, 4, 6-8, 12, 13, 16, 17, 21].In these results, the nonlinear term plays a more important role than the impulsive terms do in guaranteeing the existence of solutions.While, by strengthening the role of impulses, some sufficient conditions for the existence of solutions generated by impulses are established.In 2011, Zhang and Li [24] established sufficient conditions for the following system to possess at least one non-zero periodic solution and at least one non-zero homoclinic solution and these solutions are generated by impulses when f ≡ 0.
After that, Han and Zhang [5] considered the following asymptotically linear or sublinear Hamiltonian systems with impulsive conditions.q(t) = f (t, q(t)), for t ∈ (s k−1 , s k ), ∆ q(s k ) = g k (q(s k )). (1.6) And sufficient conditions for the existence of periodic and homoclinic solutions generated by impulses are derived.In 2013, Sun, Chu and Chen [19] established sufficient conditions for the existence of a positive periodic solution generated by impulses for the following second-order singular differential equations with impulsive conditions.
In 2014, Zhang, Wu and Dai [23] obtained sufficient conditions to guarantee the system (1.6) has infinitely many non-zero periodic solutions generated by impulses.In 2015, by using Ricceri's Variational Principle, Heidarkhani, Ferrara and Salari [8] investigated sufficient conditions for the existence of infinitely many periodic solutions generated by impulses for the following perturbed second-order impulsive differential equations.
On the other hand, some attempts have been made on the necessary and sufficient conditions for the existence of solutions (not generated by impulses) for impulsive boundary value problems.By the method of upper and lower solutions, Hou and Yan [9] established some necessary and sufficient conditions for the existence of solutions for singular impulsive boundary value problems on the half-line; Using the variational method, Sun and Chu [18] recently established a necessary and sufficient condition for the existence of periodic solutions for a impulsive singular differential equation.
However, to the best of our knowledge, relatively little attention is paid to the necessary and sufficient conditions for the existence of solutions generated by impulses.As a result, the goal of this paper is to fill the gap in this area.Result of this paper for the problem (1.1) is presented as follows.
Theorem 1.1.Assume that F satisfies the assumption (A), F(t, •) is strictly convex for a.e.t ∈ [0, T] and the equations T 0 ∇F(t, x)dt = 0 have no solution in R N .If I i > 0 for each i = 1, 2, . . ., N, then the following properties are equivalent: (α 1 ) The problem (1.1) has at least one non-constant solution generated by impulses (1.4) in H 1 T .
(β 1 ) There exists x ∈ R N such that When N = 1, Theorem 1.1 is also valid for the problem (1.2).However, for the scalar problem, the convexity of F(t, •) implies that F x (t, •) is nondecreasing in R, so the strictness of convexity of F(t, •) may be dropped, and a better result is obtained. (1.7) In the following, an example is given to illustrate Theorem 1.1.
This proves the assertion by Theorem 1.1.

Preliminaries
Let C ∞ T be the space of indefinitely differentiable T-periodic functions from R to R N .
is a Hilbert space with the inner product where (•, •) denotes the inner product in R N , and the corresponding norm is Let u(t) ≡ u(t) − u, where u = (1/T) The assumption (A) and all .
Since the behavior of a function on a set of measure zero does not affect its integral and So ü exists and (1.1a) holds.Moreover, the existence of weak derivative of u and u implies that (1.1b) holds.It follows from (1.1b) that which combining with (1.1a) and (2.1) yields which implies (1.4) holds.
For the reader's convenience, we now recall some facts.).If ϕ is weakly lower semi-continuous on a reflexive Banach space X and has a bounded minimizing sequence, then ϕ has a minimum on X.

Main result
In this section, the main results of this paper are proved.Proof.Let {u n } be a weakly convergence sequence to u 0 in H 1 T , then {u n } converges uniformly to u 0 on [0, T].Then there exists a constant C 1 > 0 such that u n ∞ ≤ C 1 for n = 0, 1, 2, . . ., so the continuity of I i implies that where T .Thus Φ is weakly lower semi-continuous on H 1 T .Lemma 2.3 shows that it remains to prove that Φ is coercive.In view of (3.1), H(x) has a minimum at some point x ∈ R N for which where 0 I i (s)ds and On the other hand, it follows from the assumption (A) and the convexity of F(t, •) and By Sobolev's inequality, we have → ∞, the above inequality, (3.3), (3.4) and (3.1) imply that Φ is coercive.

Impulsive differential systems
Theorem 3.2.Assume that F satisfies the assumption (A).If F(t, •) is strictly convex for a.e.t ∈ [0, T] and I i > 0 for each i = 1, 2, . . ., N. Then the following properties are equivalent: Proof.If u 0 is a solution of the problem (1.1)-(1.4),integrating both sides of (1.1a) over [0, T] and using the boundary condition (1.1b) and the impulsive condition (1.4), we have Define the strictly convex function H : R N → R by Since ∇ H(u 0 ) = 0 by (3.5), Lemma 2.2 implies that H(x) → +∞ as |x| → ∞.It follows from the convexity of F(t, •) and I i > 0 that where Thus it follows from Lemma 2.2 that there exists x ∈ R N such that ∇H(x) = 0 and (α 2 ) implies (β 1 ).
Proof of Theorem 1.1.Since the equations T 0 ∇F(t, x)dt = 0 have no solution in R N , it follows from Theorem A that the problem (1.1) has no solution.So (α 2 ) implies that the problem (1.1) has at least one solution generated by impulses (1.4).What is more, the solution is not a constant.In fact, suppose that the solution u(t) = C, a.e.t ∈ [0, T], then by (1.1a), this implies ∇F(t, C) = 0, a.e.t ∈ [0, T], then T 0 ∇F(t, C)dt = 0, which is a contradiction, thus (α 1 ) holds.This proves the assertion by Theorem 3.2.

Impulsive differential equations
We begin with the following lemma on impulsive linear boundary value problem.Proof.If u 0 (t) is a solution of (3.6), then integrating (3.6a) over [0, T] and using the boundary conditions and the impulsive condition, we have which implies (3.7) holds.For the sufficiency, if (3.7) holds, it could be verified that (3.6) has the following solution. where Theorem 3.4.Assume that F satisfies the assumption (A) where N = And (1.7) is derived follows from the intermediate value theorem.
For the sufficiency, consider first the following problem By (1.7), Lemma 3.3 implies that the problem (3.8) has a solution w * (t).
The subsequent discussions on the problem (1.2)-(1.5)will be divided into three cases.
Case I.
T 0 F x (t, x)dt + I(x) = 0 for all x ≥ x.In this case, condition (1.7) implies T 0 F x (t, x) − F x (t, x)dt + I(x) − I(x) = 0 for all x ≥ x.
What is more, F x (t, •) and I are nondecreasing functions, so we have Case II.
T 0 F x (t, x)dt + I(x) = 0 for all x ≤ x.The proof of Case II is similar to that of Case I and will be omitted.
Case III.There exist x 1 < x < x 2 such that
Theorem 1.2.Assume that F satisfies the assumption (A) where N = 1, F(t, •) is convex for a.e.t ∈ [0, T] and the equation T 0 F x (t, x)dt = 0 has no solution in R. If I ≥ 0, then problem (1.2) has at least one non-constant solution generated by impulse (1.5) in H 1 T if and only if there exists x