Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses*

Wei Yongfang weiyonfang@163.com

University of Science and Technology, China

Shang Suiming shangsuiming2015@163.com

University of Science and Technology, China

Bai Zhanbing¹ zhanbingbai@163.com

University of Science and Technology, China

Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.

In recent years, the research on impulsive differential equations have attracted widespread attention. This is because many phenomena in life are not a continuous process and will change suddenly due to the influence of external factors. Therefore, it is more appropriate to use impulsive differential equations to describe such situations instead of simply using differential equations or difference equations. Impulsive differential equations have been widely used in recent years, especially, in the field of biological mathematics. For example, in the study of pharmacokinetic model, since oral and injected drugs often enter the human body in the form of impulse, it is more reasonable to use impulse differential equations to describe the changes of drug concentration in the human body.

Moreover, the impulsive differential equations have been fully applied in the fields of pest control and comprehensive environmental management, and many gratifying results have been obtained.

Impulses can be divided into instantaneous and noninstantaneous impulses due to the duration. However, in many applications, instantaneous and noninstantaneous impulses occur at the same time in some dynamic processes such as intravenous injection. Since the drug enters the blood and the subsequent absorption of the body is a sudden and continuous process, this situation can be explained as an impulsive behavior. The impulse suddenly starts to jump at any fixed point in time (drug enters the blood) and continues to occur within a limited time interval (the body absorbs the drug). Due to its wide range of applications, some scholars began to study the existence of solutions of differential equations with instantaneous and noninstantaneous impulses, and through fixed point theorems, upper and lower solution theorems, variational methods, etc., they obtained some excellent results [1,3,4,6–8,10,11,13–19]. Especially, in [4], Bai and Nieto first gave the variational structure of a linear equation with noninstantaneous impulses as the following problem:

It is the first time that the critical point theory has been applied to consider the problems with noninstantaneous impulses. Then Tian and Zhang in [15] studied the existence of classical solutions for differential equations with instantaneous and noninstantaneous impulses, they considered the following problem:

In addition, Bai et al. in [9] studied a three-point boundary value problem, they firstly gave the variational structure of the nonlocal boundary value problem, and gave a different idea to deal with the functionals by imposing the boundary value conditions on admissible space rather than the functionals.

Inspired by the above literatures, we try to study the problem, which the impulse suddenly starts to jump at any fixed point and continues to occur within a limited time interval . This paper considers the multiple solutions for a class of three-point boundary value problems (BVPs) with instantaneous and noninstantaneous impulses as follows:

for

The remainder of the paper is organized as follows. Section 2 will prove that the critical point of functional is the classical solution of BVPs (1). Section 3 will present the main results with the specific proof. Section 4 will give two examples to verify the results.

In this paper, we assume the following condition:

then the following norm

is equivalent to the norm . In fact, for all , there is ds, then by HÖlder inequality

let , we can obtain

Define the following functional on :

then for all , there is the derivative

Lemma 1..

Lemma 2. .

Proof. For each , there holds and

thus . The proof is complete.

Definition 1. Let be a real Banach space, . If any sequence with

contains a convergent subsequence, then the functional . is called satisfying the Palais– Smale (PS). condition.

Lemma 3. (See [5].) Let be a real Banach space, and let be a lower bounded functional, which satisfies the (PS)_c condition. Then have the minimum value in , that is, there exists such that , then is a critical point of .

Lemma 4. (See [2].) Let be a real Banach space, and let satisfy the (PS)_c condition. Assume and

(P1) there are constants such that , and (P2) there is an e ∈ Z \ B.such that J(e) 6 0.

(P2)there is an such that .

Then there exists a critical point of such that

Lemma 5. (See [11].) Let be a real Banach space, be even, and let the functional satisfy the (PS)_c condition. Assume that satisfies the following:

(i) ;

(ii) There exist

(iii) where is a finite dimensional subspace, there exists such that

Then possesses an unbounded sequence of critical values.

Lemma 6. The weak solution is the classical solution of problem (1).

Proof. Let is a weak solution of (1), then , that is, for all ,

Without loss of generality, let , and letThen substitute into equation (4). For , there is

Thus

Then substitute (5) into (4), there is

Assume , and . Then substituting into equation (6), we get that is a constant, that is, for ,

Substitute (7) into (6), there is

that is,

Then for ,

so there is

and for , there is

Then we can get is a classical solution of problem (1), the proof is completed.

Theorem 1. Assume that . and the following conditions hold:

(A2) There are constants andsuch that for

(i)

(ii)

(A3), there is

Then BVP_s(1) has at least two classical solutions.

Proof. Firstly, let be a sequence in such that is bounded and Then there exists nonnegative constants such that . By (3) there is

then

by (A2), 1, and , thus is bounded in .

Then there exist a subsequence of sequence such that in, so

by Lemma 1 there in , then

and by in there is

By [12, Eq. (2.2)] there exist such that

and by [14, Lemma 2.7], for , there is . Therefore, in , that is, the sequence has a convergent subsequence, then . satisfies the (PS)_ccondition.

Secondly, by (2) and assumption (A2) we get . For any . By assumption (A3), given , where , there exists such that and . Then for all , there is

Then taking into Lemma 5, setting , one has that (P1) holds. Moreover, is a lower bounded functional, which satisfies the (PS)_ccondition in , and is a real Banach space by . So by Lemma 3 there exists such that , then is a critical point of . Moreover, as .

Thirdly, by assumption (A2), for all , that is, , one has

where are positive constants. Then for , one has

where , then , that is, when is sufficiently large (that is, away from the origin), one has , thus (P2) holds. Then by Lemma 4 there is a critical point such that . Therefore, and are two different critical points of , that is, BVPs (1) has at least two classical solutions.

Proposition 1.Under the assumptions of Theorem 1, and are odd functions with respect to y and p is an odd number, then BVPs (1) has infinitely many classical solutions.

Proof. is a even functional by assumptions, and let be a finite dimensional subspace. By Theorem 1 it is easy to prove (i) and (ii) in Lemma 5 and choose an such that By Lemma 5 BVPs (1) possesses infinitely many classical solutions.

Example 1. Consider the following boundary value problem:

Compare with for , then (A1) holds. When , we can get

then (A2) holds, and

Then (A3) holds. Thus, by Theorem 1, problem (8) has at least two classical solutions.

Example 2. Consider the following boundary value problem:

Compare with (1), , then (A1) holds. When , we can get

then (A2) holds, and

then (A3) holds. Thus, by Theorem 1, problem (9) has at least two classical solutions.

The interesting points of this paper are the following:

• By using the variational method we study a three-point boundary value problem instead of the local boundary value problem such as [15];

• We study the second-order Laplace differential equations with instantaneous and noninstantaneous impulses;

• The nonlinear term and the impulsive term in this paper do not have to meet the sublinear growth conditions in [15];

• We choose an appropriate space instead of functional to contain the boundary conditions.

The authors would like to express their deep thanks to the referee for valuable suggestions that allowed us to revise and improve the manuscript.