R E S E A R C H
Open Access
Multiple solutions to impulsive differential
equations
Hongmei Bao
**Correspondence: [email protected]
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, P.R. China
Abstract
In this paper, we study the existence of a second-order impulsive differential equations depending on a parameter
λ. By employing a critical point theorem, the
existence of at least three solutions is obtained.MSC: 34A37; 34K10
Keywords: multiple solutions; critical point theorem; impulses
1 Introduction
In recent years, the study of the existence of solutions to impulsive differential equation has aroused extensive interest, we refer the reader to [–] and the references therein.
In [], by using some existing critical point theorems, Xie and Luo investigated the ex-istence of multiple solutions of the following Neumann boundary value problem:
–p(t)u(t)+q(t)u(t) =λft,u(t), t=tj,t∈[, ],
p(tj)u(tj) =Ij
u(tj)
, j= , , . . . ,m, (.)
u() =u() = .
In [], Liang and Zhang considered the following boundary value problems: –p(t)u(t)=ft,u(t), t= tj,t∈[,T],
p(tj)u(tj) =Ij
u(tj)
, j= , , . . . ,m, (.)
u() =u(T), p()u() =p(T)u(T).
The authors gave some criteria to guarantee that the problem has at least one solution under some different conditions.
In [], Li and Shen were concerned with the existence of three solutions for the following boundary value problems:
–u(t) =λfu(t), t=tj,t∈[, ],
u(tj) =Ij
u(tj)
, j= , , . . . ,m, (.)
u() =u() = .
Motivated by the previous mentioned paper, in this paper, we will study the existence of at least three solutions for the following boundary value problems:
–p(t)u(t)+u(t) =λft,u(t), t=tj,t∈[, ],
p(tj)u(tj) =Ij
u(tj)
, j= , , . . . ,m, (.)
u() =u() = ,
where =t<t <· · ·<tm<tm+ = ,p∈PC([, ]),f ∈C([, ]×R,R), Ij∈C(R,R),
j= , , . . . ,m,p(tj)u(tj) =p(t+j)u(tj+) –p(t–j)u(tj–),p(tj+)u(tj+) andp(tj–)u(t–j) denote the
right and the left limits, respectively,λ∈[, +∞) is a real parameter.
2 Preliminaries
Letp=mint∈[,]p(t) > ,M=max{p, },X=W,[, ] with the norm
u=
p(t)u(t)+u(t)dt
.
Define the norm inC([, ]) byu∞=maxt∈[,]|u(t)|.
Lemma . For any u∈X,we haveu∞≤√Mu.
Proof Foru∈Xby the mean-value theorem, there existsτ ∈(, ) such thatu(s)ds=
u(τ). Hence, fort∈[, ], we have
u(t)=u(τ) +
t
τ
u(s)ds≤u(τ)+
u(s)ds
≤
u(s)ds+
u(s)ds
≤
u(s)ds
+ /p
p(t)u(t)dt
≤ Mu.
For everyu∈X, we define the functionalϕ(u) :X→Rby
ϕ(u) =(u) –λ(u); here
(u) = u
+
m
j=
u(tj)
Ij(s)ds
and
(u) =
F(t,u)dt,
We easily show thatϕis differentiable at anyu∈Xand
ϕ(u)v=
p(t)u(t)v(t) +u(t)v(t)dt+
m
j=
Ij
u(tj)
v(tj) –λ
ft,u(t)v(t)dt.
Obviously,is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse onX∗, andis a continuously Gâteaux differentiable functional whose Gâteaux derivative
is compact.
Lemma .([]) If u∈X is a critical point of the functionalϕ,then u is a classical solution of problem(.).
Suppose thatE⊂X. We denoteEω as the weak closure ofE, that is,u∈Eω if there exists a sequence{un} ⊂Esuch thatg(un)→g(u) for everyg∈X∗. Our main tool is the
following three critical points theorem obtained in [].
Lemma .([], Theorem .) Let X be separable and reflexive real Banach space.:
X→R a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X∗.
J:X→R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Assume that there exists x∈X such that(x) =J(x) = and that
(i) limx→+∞((x) –λJ(x)) = +∞for allλ∈[, +∞).
Further,assume that there are r> ,x∈X such that
(ii) r<(x).
(iii) supx∈
–((–∞,r))ωJ(x) <
r r+(x)J(x).
Then,for each
λ∈ =
(x)
J(x) –supx∈–((–∞,r))ωJ(x)
, r
supx∈
–((–∞,r))ωJ(x)
,
the equation
(x) –λJ(x) = (.)
has at least three solutions in X and,moreover,for each h> ,there exist an open interval
⊆
, hr
r(J(x)/(x)) –supx∈–((–∞,r))ωJ(x)
and a positive real numberσsuch that,for eachλ∈ , (.)has at least three solutions
in X whose norms are less thanσ.
3 Main results
Theorem . The following conditions are given.
(H)
m j=
u(tj)
(H) Letai> (i= , ),M> ,and <μ< such that
Proof Now we show the conditions (i)-(iii) of Lemma . are satisfied. For anyu∈X,|u| ≥M, andλ≥, and the assumptions (H)-(H) we have
which shows the condition (i) of Lemma . is satisfied. Letu=c∈Xandc>√cM
so the condition (ii) of Lemma . is obtained. By Lemma ., if(u)≤r, then
which implies that
On the other hand, we obtain
r
From the assumption (H) we have
sup u∈–(–∞,r)ω
(u) < r
r+(u)
(u),
which shows the condition (iii) of Lemma . is satisfied. Note that
The proof is complete.
4 Examples
Consider the following problem:
ClearlyM= . Letc= ,c= , it follows that
M
max
|u|≤cF(t,u)dt
= e– <c
c
M
+c
+
m
j=
c
Ij(s)ds –
F(t,c)dt=
(e– )
,
λ
= (e– ),
λ
=
e–e,
which shows that all conditions of Theorem . are satisfied, so the problem (.) admits at least three solutions forλ∈(e–e,(e–)).
Competing interests
The author declares that she has no competing interests.
Received: 4 August 2016 Accepted: 6 January 2017 References
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3. Li, J, Shen, J: Existence of three solutions to impulsive differential equations. J. Integral Equ. Appl.24, 273-281 (2012) 4. Li, J, Nieto, JJ: Existence of positive solutions for multi-point boundary value problem on the half-line with impulses.
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