Periodic boundary value problems for second order functional differential equations with impulse

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R E S E A R C H

Open Access

Periodic boundary value problems for

second-order functional differential

equations with impulse

Yulin Zhao

1*

, Haibo Chen

2

and Bin Qin

3

*Correspondence:

zhaoylch@sina.com

1School of Science, Hunan

University of Technology, Zhuzhou, Hunan 412007, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we study the existence of multiple positive solutions of the

second-order periodic boundary value problems for functional differential equations with impulse. The proof of our main results is based upon the fixed point index theorem in cones.

Keywords: boundary value problem; impulsive functional differential equations; multiple solutions; cones

1 Introduction

In this paper, we will consider the existence of positive solutions for boundary value prob-lems of second order impulsive functional differential equations of the form

⎧ ⎪ ⎨ ⎪ ⎩

(ρ(t)u(t))+f(t,ut) = , tJ,t=tk,k= , . . . ,m,

u(tk) =Ik(utk), k= , , . . . ,m, u=ϕ, u(T) =A,

(.)

whereJ= [,T],f :J×Ris a continuous function,ϕ (Cτ be given in Section ),

τ≥,ρ(t)∈C(J, (,∞)),ut,ut(θ) =u(t+θ),θ∈[–τ, ].IkC(Cτ,R),  =t<t<

t<· · ·<tm<tm+=T,J= (,T)\{t, . . . ,tm}.u(tk) =u(tk+) –u(t–k),u(tk+) (u(tk–)) denote

the right limit (left limit) ofu(t) att=tk, andAR= (–∞, +∞).

Impulsive differential equations describe processes which experience a sudden change of their state at certain moments. The theory of impulse differential equations has been a significant development in recent years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, population dynamics, chemical technology and biotechnology; see [–].

Many papers have been published about the existence analysis of periodic boundary value problems of first and second order for ordinary or functional or integro-differential equations with impulsive. We refer the readers to the papers [–]. For instance, in [], He and Yu investigated the following problem:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u(t) =f(t,u(t),ut), t=tk,  <t<T, u(tk) =Ik(u(tk)), k= , , . . . ,m,

u(t) =u(), t∈[–τ, ), u() =u(T).

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By using the coincidence degree, Dong [] studied the following periodic boundary value problems (PBVP) for first-order functional differential equations with impulse:

⎧ ⎪ ⎨ ⎪ ⎩

u(t) =f(t,ut), t=tk,  <t<T, u(tk) =Ik(u(tk)), k= , , . . . ,m,

u() =u(T).

(.)

It is remarkable that the author requiredu(t) =u(t+ ) fort∈[–τ, ]. The author also obtained the existence of one solution of PBVP (.).

To study periodic boundary value problems for first and second order functional dif-ferential equations with impulse, the approaches used in [, –, –, –] are the monotone iterative technique and the method of upper and lower solutions. What they obtained is the existence of at least one solution if there is a pair of upper and lower solu-tions. However, in some cases it is difficult to find upper and lower solutions for general differential equations.

As we know, the fixed point theorem of cone expression and compression is extensively used to study the existence of multiple solutions of boundary value problems for second-order differential equations. In paper [], Ma considers the following periodic boundary value problem:

u(t) +f(t,ut) = ,  <t<T,

u=ϕ, u(T) =A,

(.)

and he obtained some sufficient conditions for the existence of at least one positive solu-tion of the PBVP (.).

In [], by applying the fixed point theorem of cone expression and compression, Liu investigates the existence of multiple positive solutions of the following problem:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u(t) =f(t,ut), t=tk,t∈(,T), u(tk) =Ik(u(tk)), k= , , . . . ,m,

u(t) =ϕ(t), t∈[–τ, ], u() =u(T).

Motivated by the results in [, , ], the aim of this paper is to consider the existence of multiple positive solutions for the PBVP (.) by using some properties of the Green function and the fixed point index theorem in cones.

This paper will be divided into three sections. In Section , we provide some preliminar-ies and establish several lemmas which will be used throughout Section . In Section , we shall give the existence theorems of multiplicity positive solutions of PBVP (.).

2 Preliminary and lemmas

Let :={ϕ: [–τ, ]→R;ϕ(t) is continuous everywhere except for a finite number of points¯tat whichϕt+) andϕt–) exist andϕ(t¯–) =ϕ(t)¯}, then is a normed space with the norm

ϕ[–τ,]= sup

θ∈[–τ,]

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LetJ∗= [–τ,T],C(J∗) andC(J) represent the set of a continuous and continuously

differentiable onJ∗, respectively. Moreover, foruC(J∗) we defineuJ∗=suptJ∗|u(t)|.

Furthermore, we denote:

PC(J∗) ={u:u(t)is a map fromJ∗intoRsuch thatu(t)is continuous att=tk, and

u(t+

k),u(tk–)exist andu(tk) =u(tk)fork= , , . . . ,m;u(t) =ϕ(t)fort∈[–τ, ]}.

PC(J) ={uPC(J) :u|

(tk,tk+)∈C

(t

k,tk+),u(t–k)andu(t+k)exist; andu(t–k) =u(tk) fork= , , . . . ,m}.PC+(J) ={uPC(J) :u(t),t[–τ,T]}.

Clearly,PC(J∗) is a Banach space with the normuJ∗=suptJ∗|u(t)|foru(t)PC(J∗).

PC(J) is also a Banach space with the normu

=max{uJ∗,uJ∗}.

We need to assume the following conditions:

(A)  –TG(s,s)ds> ;ϕ(θ)≥forθ∈[–τ, ], andAφ(); (A) f(t,u)≥fort∈[,T]anduPC+(J);

(A) Ik(k= , , . . . ,m) are continuous andIk(u)≥foruPC+(J∗).

Lemma .(see []) Let E be a Banach spaces and KE be a cone in E.Let r> and

r={xK:x<r}.Assume that S:rK is a completely continuous operator such

that Sx=x for x∂r.

(i) IfSxxforx∂r,theni(S,r,K) = . (ii) IfSxxforx∂r,theni(S,r,K) = .

Lemma . For any y,akPC(J,R),andη,AR.Then the problem

⎪ ⎨ ⎪ ⎩

(ρ(t)u(t))+y(t) = , tJ,t=tk,k= , , . . . ,m,

u(tk) =ak, k= , , . . . ,m,

u() =η, u(T) =A

(.)

has a unique solution

u(t) =η+(A–η)

φ(T) φ(t) +

T

G(t,s)y(s)ds+

<tk<T

G(t,tk)ρ(tk)ak, (.)

where

G(t,s) =

φ(T)

(φ(T) –φ(t))φ(s), ≤stT,

φ(t)(φ(T) –φ(s)), ≤tsT, φ(t) =

t

ds

ρ(s)< +∞. (.) Proof Integrating the first equation of (.) over the interval [,t] fort∈[,T), we get

u(t) =u() +ρ()u()φ(t) –

t

φ(t) –φ(s)y(s)ds

<tk<t

φ(t) –φ(tk)

ρ(tk)ak. (.)

It follows from the boundary conditionsu() =η,u(T) =Aand (.) that

u() = 

ρ()φ(T)

(A–η) +

T

φ(T) –φ(s)y(s)ds+

<tk<T

φ(T) –φ(tk)

ρ(tk)ak

.

Together with (.), we obtain (.).

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Lemma . The Green function G(t,s)defined in(.)has the following properties:

(i) G(t,s)≥,∀t,s∈[,T];

(ii) G(t,s)G(s,s) < +∞,∀t,s∈[,T];

(iii)  <σG(s,s)G(t,s),∀t∈[α,β],s∈[,T],whereα∈(,t],β∈[tm,T),and

 <σ:=min

φ(T) –φ(β)

φ(T) –φ(α),

φ(α)

φ(β)

< .

Form Lemma ., the problem (.) is equivalent to the integral equation:

(Su)(t) :=u(t) =

⎧ ⎪ ⎨ ⎪ ⎩

ϕ() +(Aφ(ϕT())) φ(t) +TG(t,s)f(s,us)ds

+<t

k<TG(t,tk)ρ(tk)Ik(utk), t∈[,T],

ϕ(t), t∈[–τ, ].

(.)

Definition . A functionuPC(J)C(J) is called a positive solution of PBVP (.),

if it satisfies the PBVP (.) andu(t)≥ onJ∗, andu(t)≡ onJ. Define a coneKinPC(J∗) as follows:

K=xPCJ∗:x≥ onJ∗andx(t)σx[,T],∀t∈[α,β]

. (.)

Lemma . The operator S:KK is completely continuous.

Proof It is easy to show thatS(K)K holds. LetBbe a bounded subset inPC(J∗). By virtue of the Ascoli-Arzela theorem, we only show that S(B) is bounded inPC(J∗) and S(B) is equicontinuous. For anyu andθ∈[–τ, ],t∈[,T], we haveut(θ) =u(t+θ).

Therefore, the set{ut:uB,t∈[,T]}is uniformly bounded with respect tot∈[,T] on

. Then there exist two constantsL> ,L>  such that

max uB,s∈[,T]

f(s,us)<L, max uB,<tk<T

Ik(utk)<L, k= , , . . . ,m. (.)

Taking

L= max t,s∈[,T]

G(t,s), L= max

t,tk∈[,T]

G(t,tk), k= , , . . . ,m. (.)

It follows from (.), (.), and (.) thatS(B) is bounded inPC(J∗). LetuBandt,t∈[–τ,T] witht<t. There are three possibilities: Case I. If ≤t<tT, then

(Su)(t) – (Su)t

T

G(t,s) –Gt,sf(s,us)ds

+Aφ()

φ(T) φ(t) –φ

t

+

<tk<T

G(t,tk) –G

t,tkρ(tk)Ik(utk).

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Case III. If –τt<  <tT, then

(Su)(t) – (Su)t≤(Su)t– (Su)()+(Su)() – (Su)(t)

T

Gt,sG(,s)f(s,us)ds

+Aϕ()

φ(T) φ(t) – +ϕ() –ϕ(t)

+

<tk<T

Gt,tk

G(,tk)ρ(tk)Ik(utk).

Clearly, in either case, it follows from the continuity ofG(t,s) and the uniform continu-ity ofϕin [–τ, ] that for anyε> , there exists a positive constantδ, independent oft, tandu, whenever|tt|<δ, such that|(Su)(t) –S(u)(t)| ≤εholds. Therefore,S(B) is

equicontinuous.

3 The main results

In this section, we shall consider the existence of multiple positive solutions for the peri-odic boundary value problems (.).

For convenience sake, we set

f∞:= lim vPC+,v

[–τ,]→∞

f(t,v)

v[–τ,]

, Ik∞:= lim

vPC+,v [–τ,]→∞

Ik(v)

v[–τ,]

, k= , , . . . ,m.

For the first theorem we need the following hypotheses:

(C) There exists a constanta>  such that forvPC+(J∗) :v[–τ,]≤a,

f(t,v)Mmax

v[–τ,],ϕ[–τ,]

, ∀t∈[,T], Ik(v)≥Mkmax

v[–τ,],ϕ[–τ,]

, k= , , . . . ,m,

whereMandMkare two positive constants satisfying:

σ

M

β

α G

 ,s

ds+ α<tk<β

G

 ,tk

ρ(tk)Mk

≥. (.)

(C) There exists a constantb>  satisfying:

b>max

ϕ[–τ,],a, ( –D)–A Dbe given in (.)

such that

f(t,v)Mv[–τ,], ∀t∈[,T] and Ik(v)≤Mkv[–τ,], k= , , . . . ,m

forvPC+(J) :v

[–τ,]≤b, whereM∗> ,Mk>  are constants satisfying:

D:= max t∈[,T]

M

T

G(t,s)ds+

m

k=

G(t,tk)ρ(tk)M∗k

< . (.)

Theorem . Assume that(C), (C),f∞=∞,and Ik∞=∞hold.Then PBVP(.)has at

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Hence, Lemma . implies that

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Thus,Shas a fixed pointu∗inb\a, and a fixed pointu∗∗inb∗\b. They are positive

solutions of the PBVP (.) and

 <u∗[–τ,T]<b<u∗∗[–τ,T].

The proof is complete.

For the second theorem we need the following hypotheses: (C) There exists a constanta>  satisfying

a>max

ϕ[–τ,],

 –D∗–A D∗is given in (.),

such that forvPC+(J) :v

[–τ,]≤a,

f(t,v)λv[–τ,], ∀t∈[,T] and Ik(v)≤λkv[–τ,], k= , , . . . ,m,

whereλ,λkare positive constants satisfying:

D∗:= max t∈[,T]

λ

T

G(t,s)ds+

m

k=

G(t,tk)ρ(tk)λk

< . (.)

(C)Hk:R+→R+ (k= , , . . . ,m) are continuous nonincreasing functions such that

|Ik(v)| ≤Hk(v[–τ,]) forvPC+(J∗).

(C) There exists a constant b with b >a such that for any vPC+(J∗) :σb ≤

v[–τ,]≤b,

f(t,v)λmax

v[–τ,],ϕ[–τ,]

, ∀t∈[,T] and

Ik(v)≥λkmax

v[–τ,],ϕ[–τ,]

, k= , , . . . ,m,

whereλ> ,λk> , and

σ

λ

β

α G

 ,s

ds+ α<tk<β

G

 ,tk

ρ(tk)λk

≥.

Theorem . Assume that(C)-(C),and f∞= hold.Then PBVP(.)has at least two positive solutions u∗,u∗∗with <u∗[–τ,T]<b<u∗∗[–τ,T].

Proof For anyuKwithu[–τ,T]=a. According to (.) and assumption (C), we have

Su(t)=

⎧ ⎪ ⎨ ⎪ ⎩

|ϕ() +(Aφ(ϕT())) φ(t) +TG(t,s)f(s,us)ds

+<t

k<TG(t,tk)ρ(tk)Ik(utk)|, t∈[,T],

|ϕ(t)|, t∈[–τ, ],

⎧ ⎪ ⎨ ⎪ ⎩

maxt∈[,T]{λ∗ T

G(t,s)us[–τ,]ds

+<t

k<TG(t,tk)ρ(tk)λ

kutk[–τ,]}+A,

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On the other hand, it follows from (.) and assumption (C) that

(Su) ∂b. Thus an application of Lemma . again shows that

i(S,b,K) = . (.)

In view of assumption (C) andf∞= , there are two possibilities:

Case . Suppose thatf is unbounded, then there exists a constantdsatisfying:

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whereεis a positive constant satisfying

max t∈[,T]

εd

T

G(t,s)ds+

m

k=

G(t,tk)ρ(tk)Hk(d)

+Ad. (.)

IfuKwithu[–τ,T]=d, then it follows from (.), (.), and (.) that Su(t)

maxt∈[,T]{εdT

G(t,s)ds+

<tk<TG(t,tk)ρ(tk)Hk(d)}+A,

ϕ[–τ,]

d. (.)

Case . Suppose thatf is bounded. Then there exists a constantNsuch that

f(t,v)N, t∈[,T],∀vPC+J∗.

Taking

d≥max

ϕ[–τ,], max t∈[,T]

N

T

G(t,s)ds+

<tk<T

G(t,tk)ρ(tk)Hk()

+A,b

.

ForuKandu[–τ,T]=d, from (.), we have Su(t)

maxt∈[,T]{N T

G(t,s)ds+

<tk<TG(t,tk)ρ(tk)Hk()}+A,

ϕ[–τ,]

d.

Choosed=max{d,d}. Hence, in either case, we always may set

d=uK:u[–τ,T]<d

,

such thatSu[–τ,T]≤ u[–τ,T]foru∂d. Thus, Lemma . yields

i(S,d,K) = . (.)

Sincea<b<d, it follows from (.), (.), (.), and the additivity of the fixed point

index that

i(S,b\a,K) = –, i(S,d\b,K) = .

Thus,Shas a fixed pointu∗inb\a, and a fixed pointu∗∗ind\b. They are positive

solutions of the PBVP (.) and

 <u∗[–τ,T]<b<u∗∗[–τ,T].

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Example . Consider the following PBVP:

⎧ ⎪ ⎨ ⎪ ⎩

(etu(t))= ( +t)F(sup

θ∈[–,]|ut(θ)|),

u() =I(supθ[–,]|u(+θ)|),

u(t) =( +sint), t∈[–, ], u() = ,

(.)

where

F(x) =

⎧ ⎪ ⎨ ⎪ ⎩

 –√x, x∈[, .), x, x∈[., ),

 x

, x[, +),

I(x) =

⎧ ⎪ ⎨ ⎪ ⎩

x+ ., x∈[, .),

x, x∈[., ), 

x

– , x[, +).

(.)

Then PBVP (.) has at least two positive solutions u∗,u∗∗ with  <u∗[–,]<  <

u∗∗[–,].

Proof PBVP (.) can be regarded as a PBVP of the form (.), where

f(t,x) = +tFx, I(x) =I

x, ϕ(t) = 

( +sint), (.)

andρ(t) =et,A= ,τ= –,T= .

First we have  –G(s,s)ds≈.. By choosing α = .,β = ., we get σ = . < . On the other hand, it follows from (.) and (.) thatf∞=∞andI∞=∞ are satisfied. Finally, we show that (C) and (C) hold. ChooseM= ,M= ,a=,

b= ,M∗= ,M∗=. By calculation, we get

σ

M .

.

G

 ,s

ds+G

 ,

 

ρ

 

M

≈. > , D≈.

and

maxϕ[–τ,],a, ( –D)–A

≈. <b= .

Then it is not difficult to see that the conditions (C) and (C) hold.

By Theorem ., PBVP (.) has at least two positive solutions u∗, u∗∗ with  <

u∗[–,]<  <u∗∗[–,].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Author details

1School of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, P.R. China.2Department of Mathematics,

Central South University, Changsha, Hunan 410075, P.R. China.3School of Electrical and Information Engineering, Hunan

University of Technology, Zhuzhou, Hunan 412007, P.R. China.

Acknowledgements

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Hunan Provincial Natural Science Foundation of China (Nos. 13JJ3106 and 12JJ2004); it is also supported by the National Natural Science Foundation of China (Nos. 61074067 and 11271372).

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