Variational Methods to Nonlinear Differential Equations with Non-instantaneous Impulses

GUO Jia( )

School of Mathematics and Computational Science， Hunan First Normal University， Changsha 410075， China

Abstract: The author aimed to investigate the solvability for nonlinear differential equations with not instantaneous impulses. Variational approach was adopted to obtain the existence of weak solutions as critical points. The findings of this study may serve as a reference for multiplicity of impulsive problems.

Key words: variational method; critical points; impulsive ordinary differential equations; non-instantaneous impulses

Introduction

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time， and many works considered the existence and multiplicity of weak solutions for impulsive differential equations， for some general and recent works see Refs. [1-7]. Nevertheless， it is worth noting that the impulses in these problems are instantaneous.

-u″(t)=σi(t)

-u″(t)+λu(t)=f(t，u(t))，
t∈si，ti+1，
i=0， 1， 2， …，N,

(1)

u′(t)=αi，t∈ti，si，i=1， 2， …，N，

(2)

(3)

u(0)=u(T)=0，u′(0)=α0.

(4)

1 Preliminaries

inducing the norm

Using Poincare’s inequality in Ref. [15]， we have that

Definition1[16]LetXbe a normed space. A functionalφ:X→-∞， +∞ is weakly lower semi-continuous if

Lemma2(Ref.[16]， Theorem 1.1.) Ifφis weakly lower semi-continuous on a reflexive Bananch spaceXand has a bounded minimizing sequence， thenφhas a minimum onX.

2 Main Results

To obtain the variational structure for the impulsive problem (NP) defined by Eqs. (1)-(4)， we use the similar method of Ref.[14]， and have

(5)

Moreover， in view of formula (1)， we have

(6)

Then， from Eqs. (4)-(6)， we obtain

(7)

Now we define

(8)

It is easy to check that

f(t，u(t))][v(t)-v(ti+1)]dt.

(9)

Therefore， a critical point ofφ， defined by (8)， gives us a weak solution of the impulsive problem (NP).

Lemma3The functionalφdefined by (8) is continuous， differentiable， and weakly lower semi-continuous.

ProofUsing the continuity off， we have the continuity and differentiability ofφ.

‖u‖≤liminfk→+∞‖uk‖.

By Lemma 1， we obtain thatukconverges uniformly touon [0，T]. Then

Thus，

φ(u)≤liminfk→+∞φ(uk).

That is，φis continuous， differentiable， and weakly lower semi-continuous.

Theorem1Suppose thatfis bounded. Then there is a critical point ofφ， and the impulsive problem (NP) has at least one solution.

ProofTakeM1>0 such that

f(t，u(t))≤M1

M2‖u‖2-M3‖u‖

andφis coercive. Using Lemma 2， we have thatφhas a minimum， which is a critical point ofφ. Then the impulsive problem (NP) has at least one solution.

Analogously we have:

Theorem2Suppose thatfis sublinear. Then there is a critical point ofφ， and the impulsive problem (NP) has at least one solution.

ProofTakea，b>0 andγ∈0， 1 such that

f(t，u(t))≤a+buγ

for any (t，u(t))∈[0，T]×R. It is easy to show that there existM2，M3>0，γ*<2 such that

φ(u)≥M2‖u‖2-M3‖u‖γ*

andφis coercive. As a consequence，φhas a critical point ofφ. Then the impulsive problem (NP) has at least one solution.

3 Conclusions

In this paper， we introduce the concept of a weak solution for a nonlinear equation with not instantaneous impulses. We use the classical minimizing theorem to reveal the variational structure of the nonlinear problem and get the existence of weak solutions. This will allow us in the future to deal with the multiplicity of impulsive problems.