VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

In this paper, the existence of solutions for the fourth-order linear and nonlinear differential equations with non-instantaneous impulses is studied by applying variational methods. The interesting point lies in that the variational structures corresponding to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses are established for the first time.


Introduction
In recent years, impulsive differential equations have emerged as a valuable branch of mathematics which is widely used in physical, chemistry, biology and engineering. Many authors studied the existence of solution for the problems with instantaneous impulses in research papers, we refer the readers to [3,14,15,[18][19][20].
However, in many realistic applications, we need to describe the change in the process which is transient but continues influence for a limited time interval. Noninstantaneous impulsive, a new class of impulse was introduced by Hernádez and ORegan in [8]. Many different motivations for the study of this type of problems have been proposed. For example, some dynamic changes of blood in patients during drug therapy, which cause impulsive jump starting abruptly at any fixed point and keep continues process for a finite time interval.
By and large, the study of the existence and multiplicity of solutions for noninstantaneous impulsive differential equations has attracted more and more attention, see, for example [1,2,7,8,10,11,16,21,22,24]. They are extensively studied by using various tools [7,16,22], such as monotone iterative technique, the theory of analytic semigroup, fixed point theory. In [8], Hernádez and O'Regan first introduced the non-instantaneous impulses and studied the mild and classical solutions for the non-instantaneous impulsive differential equation which was of the form x (t) = Ax(t) + f (t, x(t)), t ∈ (s i , t i+1 ], i = 0, 1, 2, ..., m, x(t) = g i (t, x(t)), t ∈ (t i , s i ], i = 1, 2, ..., m, (1.1) In [10], Bai and Nieto first revealed the variational structure of the following linear equation with non-instantaneous impulses (1.2) They got the existence and uniqueness of weak solutions as critical points by the variational methods. It is the first time that the critical point theory has been applied to consider this kind of problems. In [21], by applying variational methods, Tian and Zhang studied the existence of solutions for the following second-order differential equations with instantaneous and non-instantaneous impulses In recent years, a great deal of papers have studied the fourth-order differential equations [4,12,18,23].
Motivated by the above work, we investigate the existence of solutions for the fourth-order linear differential equation with non-instantaneous impulsive as follows . Furthermore, we also consider the existence result of the corresponding nonlinear problems as follows To the best of our knowledge, the structure of the fourth-order differential equation with non-instantaneous impulses has not been yet developed. In order to fill this gap, we consider the linear and nonlinear fourth-order differential equation with non-instantaneous impulse simultaneously. With the non-instantaneous impulsive effect and fourth-order taken into consideration, some difficulties need to be overcome, such as how to find the corresponding variational functional J should be overcome. The variational structure of the fourth-order differential equation with the non-instantaneous impulse are established for the first time. Furthermore, we overcome the difficulties that the weak solution is the classical solution under the non-instantaneous impulsive effect.
This paper is organized as follows. In Section 2, we demonstrate a few preliminaries, definitions, lemmas and our main tools which are to be employed in exhibiting our essential outcomes of the rest of the article. In Section 3, we discuss the existence of solutions for the linear problem (1.4). In Section 4, we give the main results that the nonlinear problem (1.5) has infinitely many pairs of distinct classical solutions. Moreover, a concrete example of application is given.

Preliminaries
Let us consider the space be equipped with the inner product which induces the usual norm It is clear that (X, · ) is a Hilbert space. Then the following norm is equivalent to the usual one. In fact, for any u ∈ X, let us recall Poincáre type inequality Thus, (X, · X ) is also a Hilbert space.
Proof. For any u ∈ X, by Hölder's inequality and Poincáre type inequality, and For the linear problem (1.4), we define functional J : X → R by (2.4) Obviously, J is a Gâteaux differentiable functional and its Gâteaux derivation at the point u ∈ X is For the nonlinear problem (1.5), we define the following functional on X Obviously, I is a Gâteaux differentiable functional and its Gâteaux derivation at the point u ∈ X is for all v ∈ X. .
To show Lemma 2.4, we need the following Fundamental Lemmas.
By the Fubini theorem, one has From (2.9) and (2.10), one has From (2.11) and (2.12), one has Proof. Clearly, u ∈ X which means that the boundary conditions are satisfied. If u is a weak solution of problem (1.4), then J (u), v = 0 for all v ∈ X , i.e., (2.13) We will divide three steps to complete the proof.
First of all, we show that u satisfies the equation in (1.4). Without loss of generality, we assume that (2.14) By integration by parts, one has (2.15) Substituting (2.15) into (2.14), we get i.e., u satisfies the equation in (1.4). Secondly, we will show the non-instantaneous impulsive conditions holds. Substituting (2.16) into (2.13), one has (2.17) By integration by parts, we have (2.20) Without loss of generality, we assume that By Lemma 2.3, one has u (t) = −A t ti u (s)ds + C 1 t + C 2 . Since u ∈ C(t i , s i ], one has u (t) + Au (t) = constant, t ∈ (t i , s i ], i = 1, 2, ..., N , which means the non-instantaneous impulsive equation In virtue of (2.20), by integration by parts, we obtain Obviously, v 2 (t) ∈ X, v 2 (s i ) = 0, v 2 (s j ) = 0 for j = i and v 2 (t i ) = 0 for i = 1, 2, ..., N . Substituting v 2 (t) into (2.23), we obtain u (s , combining two equations and u ∈ X, one has u (s + i ) = u (s − i ), i = 1, 2, ..., N. From the above, according to the definition 2.1, it follows that u is a classical solution of the problem (1.4). The proof is complete.
Similarly we can prove the following Lemma.
Lemma 2.5. If u ∈ X is a weak solution of problem (1.5), then u is a classical solution of problem (1.5).
In this paper, the following theorems will be used in our main results. Moreover, if a is also symmetric, then the function ϕ: H → R defined by attains its minimum at u.

Linear non-instantaneous impulsive problem (1.4)
We define a : and l : We see that finding the weak solutions of (1.4) is equivalent to finding u ∈ H 2 0 (0, T ) such that a(u, v) = l(v), for every v ∈ H 2 0 (0, T ).
Moreover, u is a classical solution and u minimizes the functional J(u).

Proof.
We will apply Theorem 2.6 to show the existence of solutions of linear problems (1.4). It is evident that a is bilinear, symmetric, and l is linear. Firstly, we will show a is bounded, by Hölder's inequality, (2.1) and Lemma 2.1, we have Now, we will show a is coercive. By the definition of a, one has si Bu 2 (t)dt.

(3.2)
For B ≥ 0, we can easily get a(u, u) ≥ u 2 X for every u ∈ H 2 0 ([0, T ]). For B < 0, by Lemma 2.1, one has Lastly, it remains to check that l is bounded.
Hence, l ∈ X . By applying Theorem 2.6, the functional J has a unique weak solution. Moreover, u is a classical solution and u minimizes the functional J.
4. Nonlinear non-instantaneous impulsive problem (1.5) In this part, we need the following conditions about the nonlinearity Such the above conditions produce the existence of infinitely many pairs of distinct solutions for nonlinear non-instantaneous impulse problem (1.5). Precisely, we obtain the following results Proof. It is obvious that I ∈ C 1 (X, R). By (G 3 ), I is an even functional with I(0) = 0. We complete the proof by the following three steps.
Step 1. We shall prove that the functional I satisfies the (PS)-condition, i.e., every sequence {u m } ⊂ X for which {I(u m )} is bounded and I (u m ) → 0 as n → ∞ possesses a convergent subsequence in X. By (2.6) and (2.7), one has β = 1 α , α > 2 By Lemma 2.1 and the assumption in Theorem 4.1, one has 1 + BM 2 > 0. So {u m } is bounded in X for B < 0. Combining (4.2) with (4.3), one has {u m } is bounded in X. Since X is reflexive Banach space, the fact {u m } is bounded in X means that one has weakly convergent subsequence. Without loss of generality, we still denote {u m } is the subsequence of {u m }. So, we have u m u in X. Following we will show {u m } strongly converges to u in X. One has (4.4) Similar to the proof of Proposition 1.2 in [19], u m u implies {u m } uniformly converges to u in C([0, T ]). So as m → 0. By (4.4) and (4.5), we obtain that u m − u 2 X → 0 as m → ∞, which means that {u m } strongly converges to u in X. Therefore, I(u) satisfies (PS)condition.
Choosing E = X and F = ∅, then F ⊥ = X. In virtue of (2.6) and (G 1 ), we have Step 3. We shall prove that the functional I is bounded from below on X. For B ≥ 0, by (G 2 ) and Remark 4.1, one has For Thus, there exists r > 0 such that I(u) > 0 for |u| > r. By I ∈ C 1 (X, R), it follows that I is bounded from below on X. Moreover, from the all above, it means that there exists α < β such that inf F ⊥ I(u) > α. By applying Theorem 2.7, I has infinitely many pairs of distinct critical points on X. Consequently, the problem (1.5) has infinitely many pairs of distinct classical solutions.  With regard to the problem (1.5), f i (t, u) = tu 0.6 , where F i (t, u) = 5 8 tu 1.6 . Let q = 1.8, ϕ i (t) = 5 16 t, µ = 1.7, φ i (t) = 2t. Obviously, f i (t, u) = tu 0.6 is odd in u. Besides, F i (t, u) = 5 8 tu 1.6 ≥ 5 16 t |u| 1.8 for |u| → 0. Moreover 0 < tu 1.6 ≤ 2t |u| 1.7 for |u| → ∞. Therefore, (G 1 ) ,(G 2 ) and (G 3 ) are satisfied. Applying Theorem 4.1, we obtain the problem (4.6) has infinitely many pairs of distinct classical solutions.