Eigenvalue criteria for existence of positive solutions of impulsive differential equations with non-separated boundary conditions

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R E S E A R C H

Open Access

Eigenvalue criteria for existence of positive

solutions of impulsive differential equations

with non-separated boundary conditions

Ruixi Liang

1*

_{and Jianhua Shen}

2

*_{Correspondence:}

[email protected] 1_{Department of Mathematics and}

Statistics, Central South University, Changsha, Hunan 410075, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.

MSC: 34B37; 34B18

Keywords: impulsive diﬀerential equation; positive solution; ﬁxed point theorem; non-separated periodic boundary value condition

1 Introduction

Letωbe a ﬁxed positive number. In this paper, we are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:

–p(t)u(t)+q(t)u(t) =λft,u(t), t=ti,t∈J, (.a) –u[](ti)

=Ii

u(ti)

, i= , , . . . ,m, (.b)

u() =u(ω), u[]() =u[](ω). (.c)

Here,u[]₍_t_{) =}_p₍_t₎_u₍_t_{) denotes the quasi-derivative of}_u₍_t_{). The condition (.c) is called}

a non-separated periodic boundary value condition for (.a).

We assume throughout, and with further mention, that the following conditions hold. (H) Let J= [,ω], and  <t<t <· · ·<tm <ω, f ∈C(J×R+_,_R+_), _Ii _∈_C₍_R+_,_R+_), R+_{= [, +}_∞_).₍_u[]₍_ti_{)) =}_u[]₍_t+

i) –u[](t–i), whereu[](ti+) (respectivelyu[](t–i)) denotes

the right limit (respectively, the left limit) ofu[]₍_t_{) at}_t₌_t i.

(H)

ω





p(t)dt<∞,

ω



q(t)dt<∞,

p> ,q≥ andq≡ a.e. on [,ω].

A functionu(t) deﬁned onJ–₌_J_\{_t_,_t_{, . . . ,}_tm_}_{is called a solution of BVP (.)}

((.a)-(.c)) if its ﬁrst derivativeu(t) exists for eacht∈J–_,_p₍_t₎_u₍_t_{) is absolutely continuous on}

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each close subinterval ofJ–_{, there exist ﬁnite values}_u[]₍_t±

i ), the impulse conditions (.b)

and the boundary conditions (.c) are satisﬁed, and the equation (.a) is satisﬁed almost everywhere onJ–_.

For the case ofIi=  (i= , , . . . ,m), the problem (.) is related to a non-separated

peri-odic boundary value problem of ODE. Atici and Guseinov [] have proved the existence of a positive and twin positive solutions to BVP (.) by applying a ﬁxed point theorem for the completely continuous operators in cones. In [], Graef and Kong studied the following periodic boundary value problem:

⎧ ⎨ ⎩

–(p(t)u)+q(t)u=h(t)f(t,u), t∈(,ω),

u() =u(ω), u[]_{() =}_u[]₍_ω_), (.)

whereh(t) > . Based upon the properties of Green’s function obtained in [], the authors extended and improved the work of [] by using topological degree theory. They derived new criteria for the existence of non-trivial solutions, positive solutions and negative solu-tions of the problem (.) whenf is a sign-changing function and not necessarily bounded from below even over [,ω]×R+_{. Very recently, He}_{et al.}_{[] studied BVP (.) without}

im-pulses and generalized the results of [, ] via the ﬁxed point index theory. The problem (.) in the case ofp≡, the usual periodic boundary value problem, has been extensively investigated; see [–] for some results.

On the other hand, impulsive differential equations are a basic tool to study processes that are subjected to abrupt changes in their state. There has been a significant develop-ment in the last two decades. Boundary problems of second-order differential equations with impulse have received considerable attention and much literature has been published; see, for instance, [–] and their references. However, there are fewer results about pos-itive solutions for second-order impulsive differential equations. To our best knowledge, there is no result about nonlinear impulsive differential equations with non-separated pe-riodic boundary conditions.

Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem (.). By using fixed point theorems in a cone, criteria are established under some conditions onf(t,u) concerning the first eigenvalue correspond-ing to the relevant linear operator. More important, the impulsive terms are different from those of papers [, ].

2 Preliminaries

In this section, we collect some preliminary results that will be used in the subsequent sec-tion. We denote byϕ(t) andψ(t) the unique solutions of the corresponding homogeneous equation

–p(t)u(t)+q(t)u(t) = , t∈J, (.)

under the initial boundary conditions

ϕ() = , ϕ[]() = ; ψ() = , ψ[]() = . (.)

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Definition . For two differential functionsyandz, we define their Wronskian by

Wt(y,z) =y(t)z[](t) –y[](t)z(t) =p(t)y(t)z(t) –y(t)z(t).

Theorem . The Wronskian of any two solutions for equations(.)is constant. Espe-cially,Wt(ϕ,ψ)≡.

Proof Suppose thatyandzare two solutions of (.), then

Wt(y,z)

=p(t)y(t)z(t) –y(t)z(t) =y(t)p(t)z(t) –p(t)y(t)z(t) = ;

therefore, the Wronskian is constant. Further, from the initial conditions (.), we have

Wt(ϕ,ψ)≡. The proof is complete.

Consider the following equation:

⎧ ⎨ ⎩

–(p(t)u(t))+q(t)u(t) = , t∈J,

u() =u(ω), u[]_{() =}_u[]₍_ω_). (.)

From Theorem . in [], equation (.) has a Green functionG(t,s) >  for all s,t∈J, which has the following properties:

(G) G(t,s)is continuous intandsfor allt,s∈J.

(G) IfA=min_≤t,s≤ωG(t,s)andB=max≤t,s≤ωG(t,s), thenB>A> . (G)

G(t,s) =ψ(ω)

D ϕ(t)ϕ(s) – ϕ[](ω)

D ψ(t)ψ(s)

+

⎧ ⎨ ⎩

ψ[](ω)–

D ϕ(t)ψ(s) –

ϕ(ω)–

D ϕ(s)ψ(t), ≤s≤t≤ω,

ψ[](ω)–

D ϕ(s)ψ(t) –

ϕ(ω)–

D ϕ(t)ψ(s), ≤t≤s≤ω.

Combining with Theorem ., we can also prove that

(G)

G(,s) =G(ω,s), p(t)∂G

∂t

(,s)

=p(t)∂G

∂t

(ω,s)

,

ω



q(t)G(t,s)ds= .

Remark  From paper [], we can getG(t,s) whenq(t) = c

p(t)(c> ) andp(t) > ,

G(t,s) = 

c(ecω(dx/p(x))_{– )}

⎧ ⎨ ⎩

ecst(dx/p(x))_{+ e}c[

ω (dx/p(x))+

s

t(dx/p(x))]_, _≤_s≤_t≤_ω_, ects(dx/p(x))_{+ e}c[

ω (dx/p(x))+

t

s(dx/p(x))]_, _≤_t≤_s≤_ω_,

A= e

c/_ω(dx/p(x))

c[ec_ω(dx/p(x))_{– ]}, B=

 + ec_ω(dx/p(x))

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Especially, in the case ofp(t)≡,q(t)≡c₍_c_{> ), Green’s function}_G₍_t_,_s_{) has the form}

G(t,s) =  c(ecω_{– )}

⎧ ⎨ ⎩

ec(t–s)+ ec(ω+s–t)_, __≤_s_≤_t_≤_ω_,

ec(s–t)_{+ e}c(ω+t–s)_, __≤_t_≤_s_≤_ω_,

A= e

(cω/)

c(ecω_{– )}, B=

 + ecω c(ecω_{– )}.

Deﬁne an operator

(Tu)(t) =

ω



G(t,s)u(s)ds,

then it is easy to check thatT:C(J)→C(J) is a completely continuous operator. By virtue of the Krein-Rutman theorem, the authors in [] got the following result.

Lemma . The spectral radius r(T) > and T has a positive eigenfunction corresponding to its ﬁrst eigenvalueλ= (r(T))–.

In what follows, we denote the positive eigenfunction corresponding to λ by φ and

max_t∈Jφ(t) = . Deﬁne a mapping and a coneKin a Banach spaceC(J) by

( u)(t) =λ

ω



G(t,s)fs,u(s)ds+

m

i=

G(t,ti)Ii

u(ti)

, t∈J,

K=u∈C(J),u(t)≥δu,

whereδ=A_B,u=max_t∈J|u(t)|.

Lemma . The ﬁxed point of the mapping is a solution of(.).

Proof Clearly, uis continuous int. Fort=tk,

( u)(t) =λ

ω

 ∂G

∂tf

s,u(s)ds+

m

i= ∂G

∂t(t,ti)Ii

u(ti).

Using (G) and (G), we have ( u)() = ( u)(ω), ( u)[]_{() = (}_u₎[]₍_ω_{) and}

( u)[](tk) =p

t+_k( u)t_k+–p(tk)( u)(tk)

=

ψ[](ω) – 

D +

ϕ(ω) – 

D

p(tk)ϕ(tk)ψ(tk) –p(tk)ϕ(tk)ψ(tk)Iku(tk)

= –ψ

[]₍_ω_{) +}_ϕ₍_ω_{) – }

D Ik

u(tk) = –Ik

u(tk)

,

p(t)( u)(t)=

λ

ω



p(t)∂G

∂tf

s,u(s)ds+

m

i= p(t)∂G

∂t(t,ti)Ii

u(ti)

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=q(t)λ

ω



G(t,s)fs,u(s)ds–λft,u(t)+q(t)

m

i=

G(t,ti)Iiu(ti)

=q(t)( u)(t) –λft,u(t),

which implies that the ﬁxed point of is the solution of (.). The proof is complete.

The proofs of the main theorems of this paper are based on ﬁxed point theory. The following two well-known lemmas in [] are needed in our argument.

Lemma .[] Let X be a Banach space and K be a cone in X.Supposeandare open subsets of X such that∈⊂ ¯⊂,and suppose that

:K∩(¯\)→K

is a completely continuous operator such that

• inf_u∈K∩∂ u> ,u=μ uforu∈K∩∂andμ≥,andu=μ ufor

u∈K∩∂and <μ≤,or

• inf_u∈K∩∂ u> ,u=μ uforu∈K∩∂andμ≥,andu=μ ufor

u∈K∩∂and <μ≤.

Then has a ﬁxed point in K∩(¯\).

Lemma .[] Let X be a Banach space and K be a cone in X.Supposeandare open subsets of X such that∈⊂ ¯⊂,and suppose that

:K∩(¯ \)→K

is a completely continuous operator such that

• There existsu∈K\{}such thatu= u+μuforu∈K∩∂andμ> ,

u ≤ uforu∈K∩∂,or

• There existsu∈K\{}such thatu= u+μuforu∈K∩∂andμ> , u ≤ uforu∈K∩∂.

Then has a ﬁxed point in¯\.

3 Main results

Recalling thatδwas deﬁned after Lemma ., for convenience, we introduce the following notations. Assume that the constantr>  andγ is some positive function onJ,

¯

f_γr=sup

f(t,u)

γ(t)u,t∈J,u∈[δr,r]

,

fγr=inf

f(t,u)

γ(t)u,t∈J,u∈[δr,r]

,

¯

fγ=_r→lim₊f¯γr, fγ=_r→lim₊fγr, ¯fγ∞=_r→+∞lim f¯γr, fγ∞=_r→+∞lim fγr,

¯

I_ir=sup

Ii(u)

u ,u∈[δr,r]

, I_ir=inf

Ii(u)

u ,u∈[δr,r]

,

¯

I_i= lim

r→+I¯

r

i, Ii=_r→lim+I

r

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Theorem . Assume that there exist positive constantsα,β such that fα

q ≥,f¯

β

q ≥, Iα

i ≥,I

β

i ≥and

 <λ∈

 –Am_i=Iα

i fα

q

, –B

m i=I¯

β

i

¯

fqβ

. (.)

Then(.)has at least one positive solution u such thatmin{α,β} ≤ u ≤max{α,β}.

Proof Clearly,α=β, letα=min{α,β},β=max{α,β}. Deﬁne the open sets

α=

u∈C(J) :u<α, β=

u∈C(J) :u<β.

Then :K∩(¯β\α) is completely continuous. By (.) and the deﬁnition offqα,Iiα,f¯

β

q,

¯

I_iβ, there existsε>  such that

 – ( –ε)Am_i=I_iα

( –ε)fα

q

≤λ≤ – ( +ε)B

m i=¯I

β

i

( +ε)f¯qβ

, (.)

f(t,u)≥( –ε)f_qαq(t)u, Ii(u)≥( –ε)I_iαu, i= , , . . . ,m,δα≤u≤α, (.)

and

f(t,u)≤( +ε)f¯_qβq(t)u, Ii(u)≤( +ε)I¯_iβu, i= , , . . . ,m,δβ≤u≤β. (.)

Letu≡. We show that

u= u+μ, ∀u∈K∩∂αandμ> . (.)

If not, there existu∈K∩∂αandμ>  such thatu= u+μ. Letu(ρ) =mint∈Ju(t). Noting thatδα≤u≤αfor anyt∈J, we obtain that fort∈J,

u(t) = ( u)(t) +μ

=λ

ω



G(t,s)fs,u(s)ds+

m

i=

G(t,ti)Iiu(ti)+μ

≥( –ε)λf_qα

ω



G(t,s)q(s)u(s)ds+A( –ε)

m

i=

I_iαu(ti) +μ

≥( –ε)λf_qαu(ρ)

ω



G(t,s)q(s)ds+Au(ρ)( –ε)

m

i= I_iα+μ

≥( –ε)

λf_qα+A m

i= I_iα

u(ρ) +μ

≥u(ρ) +μ,

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On the other hand, for∀u∈K∩∂β,δβ≤u(t)≤β, we have

( u)(t) =λ

ω



G(t,s)fs,u(s)ds+

m

i=

G(t,ti)Iiui(t)

≤( +ε)λf¯_qβ

ω



G(t,s)q(s)u(s)ds+B( +ε)

m

i=

¯

I_iβu(ti)

≤( +ε)λf¯_qβu

ω



G(t,s)q(s)ds+B( +ε)u m

i=

¯

I_iβ

≤( +ε)

λf¯_qβ+B m

i=

¯

I_iβ

u ≤ u.

From Lemma . it follows that has a ﬁxed pointu∈K∩(¯β\α). Furthermore,α≤ u ≤β andu(t)≥δα> , which means thatu(t) is a positive solution of Eq. (.). The

proof is complete.

In the next theorem, we make use of the eigenvalueλand the corresponding

eigenfunc-tionφintroduced in Lemma ..

Theorem . Assume that there exist positive constantsα,β such that fα

γ ≥,fγβ≥,

Iα

i ≥,I

β

i ≥and

 <λ∈

_λ



ω

 φ(s)ds–δ

m i=Iiαφ(ti) δfα

γ

ω

 φ(s)ds

,δλ

ω

 φ(s)ds–δ

m i=¯I

β

i φ(ti) δf¯γβ

ω

 φ(s)ds

, (.)

hereγ ≡on J.Then(.)has at least one positive solution u such thatmin{α,β} ≤ u ≤

max{α,β}.

Proof Obviously,α=β, putα=min{α,β},β=max{α,β}. Deﬁne the open sets

α=

u∈C(J) :u<α, β=

u∈C(J) :u<β.

At ﬁrst, we show that :K∩(¯β\α)→K. For anyu∈K∩(¯β\α), from (G), we have

 < ( u)(t)≤B

λ

ω



fs,u(s)ds+

m

i= Ii

u(ti)

<∞.

On the other hand,

( u)(t)≥A

λ

ω



fs,u(s)ds+

m

i=

Iiu(ti)

≥A

B u.

It is easy to check that :K∩(¯β\α)→Kis completely continuous. Next, we show that

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If not, there existμ∈(, ] andu∈K∩∂βsuch thatμ u=u. Hence,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–(p(t)u_(t))+q(t)u(t) =μλf(t,u(t)), t∈J–,

–(u[]_ (tk)) =μIk(u(tk)), k= , . . . ,m,

u() =u(ω), u[] () =u[] (ω).

(.)

Multiplying the ﬁrst equation of (.) byφand integrating from  toω, we obtain that

–

ω



p(t)u_(t)φ(t)dt+

ω



q(t)u(t)φ(t)dt=μλ

ω



fs,u(s)φ(s)ds. (.)

One can ﬁnd that

ω



p(t)u_(t)φ(t)dt=μ m

i=

Iiu(ti)φ(ti) +

ω



q(t) –λ

φ(t)u(t)dt. (.)

Substituting (.) into (.), we get

–μ m

i=

Iiu(ti)φ(ti) +λ

ω



φ(t)u(t)dt=μλ

ω



ft,u(t)φ(t)dt.

Noting thatδu ≤u≤ u, therefore,

–u m

i=

¯

I_iβφ(ti) +λδu

ω



φ(t)dt≤λ¯fγβ

ω



φ(t)dtu,

which implies that

λ≥δλ

ω

 φ(s)ds–

m i=¯I

β

iφ(ti)

¯

fγβ

ω

 φ(s)ds

,

a contradiction. Finally, we show that

inf

u∈K∩∂α

u> , μ u=u, ∀u∈K∩∂αandμ≥.

Sincef(t,u) andIi(u) are negative foru∈[δα,α] andt∈J, the condition (.) implies that

fα

γ > . Hence,

ω

 f(s,u(s))ds>  foru∈K∩∂αand for anyu∈K∩∂α,

( u)(t) =λ

ω



G(t,s)fs,u(s)ds+

m

i=

G(t,ti)Ii

u(ti)

≥Aλ

ω



fs,u(s)ds> .

Suppose that there existμ≥ andu∈K∩∂αsuch thatμ u=u, that is,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–(p(t)u_(t))+q(t)u(t) =μλf(t,u(t)), t∈J–,

–(u[]_ (tk)) =μIk(u(tk)), k= , . . . ,m,

u() =u(ω), u[]_ () =u[]_ (ω).

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Multiplying the ﬁrst equation of (.) byφand integrating from  toω, we obtain that

–

ω



p(t)u_(t)φ(t)dt+

ω



q(t)u(t)φ(t)dt=μλ

ω



fs,u(s)φ(s)ds. (.)

One can get that

ω



p(t)u_(t)φ(t)dt=μ m

i= Ii

u(ti)

φ(ti) +

ω



p(t)φ(t)u(t)dt

=μ m

i=

Iiu(ti)φ(ti) +

ω



q(t) –λ

φ(t)u(t)dt. (.)

Substituting (.) into (.), we get

–μ m

i= Ii

u(ti)

φ(ti) +λ

ω



φ(t)u(t)dt=μλ

ω



ft,u(t)

φ(t)dt.

Noting thatδu ≤u≤ u, therefore,

–δu m

i=

I_iαφ(ti) +λu

ω



φ(t)dt≥μλ

ω



fs,u(s)ds

≥λδfγα

ω



u(s)φ(s)ds

≥λδfα γ

ω



φ(s)dsu> .

It is impossible forλ

ω

 φ(s)ds–δ

m

i=Iiαφ(ti)≤. Whenλ

ω

 φ(s)ds–δ

m

i=Iiαφ(ti) > ,

λ<λ

ω

 φ(s)ds–δ

m i=Iiαφ(ti) δfα

γ

ω

 φ(s)ds

,

a contradiction.

From Lemma . it follows that has a ﬁxed point u∈K∩(¯β\α). Furthermore,

α≤ u ≤βandu≥δα> , which means thatu(t) is a positive solution of Eq. (.). The

proof is complete.

Corollary . Assume that f

γ > ,fγ∞> ,Ii> ,I∞i > and

 <λ∈

_λ



ω

 φ(s)ds–δ

m i=Iiφ(ti) δf

γ

ω

 φ(s)ds

,δλ

ω

 φ(s)ds–

m

i=I¯i∞φ(ti)

¯

fγ∞

ω

 φ(s)ds

or

 <λ∈

λ

ω

 φ(s)ds–δ

m

i=Ii∞φ(ti) δfγ∞

ω

 φ(s)ds

,δλ

ω

 φ(s)ds–

m i=¯Iiφ(ti)

¯

f

γ

ω

 φ(s)ds

,

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Corollary . Assume that there exists a constantαsuch that fρ γ > ,I

ρ

i >  (ρ= ,αand

∞)and

¯

fα γ

ω

 φ(s)ds+

m i=I¯iαφ(ti)

δ_ωφ(s)ds <λ<min

δfγ+

δm_i=I_iφ(ti)

ω

 φ(s)ds

,δfγ∞+

δm_i=I_i∞φ(ti)

ω

 φ(s)ds

,

hereγ ≡on J.Then there exists one open interval: ∈such that(.)has at least two positive solutions forλ∈.

Example  Consider the equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–u(t) + u(t) =λf(t,u), t∈J,t=ti,

–u(t) =Ii(u(ti)), i= , , . . . ,m,

u() =u(), u() =u(),

(.)

wherep(t) = ,q(t) =  and

f(t,u) =

⎧ ⎨ ⎩

uρ_, _u_≤_,

, u> , Ii(u) =

⎧ ⎨ ⎩

, u≤,

√u, u> ,

hereρ>  andi= , , . . . ,m. SinceI

i = +∞,I¯i∞=  andf¯q∞= , by Theorem ., (.) has

at least one positive solution for anyλ> .

Example  Consider the equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–u(t) + u(t) =λf(t,u), t∈J,t=ti,

–u(ti) =Ii(u(ti)), i= , , . . . ,m,

u() =u(), u() =u(),

(.)

wheref(t,u) =e–u ,Ii(u) =

u (m+i).

It is well known that, for the problem consisting of the equation –u=λu,t∈(, ), and the boundary condition

u() =u(), u() =u(), (.)

the ﬁrst eigenvalue is  (see, for example, [, p.]). It follows that the ﬁrst eigenvalue is

λ=  for the problem consisting of the equation

–u+ u=λu, t∈(, ),

and the boundary condition (.). Meanwhile, we can obtain the positive eigenfunction

φ(t)≡ corresponding toλ. It is also easy to check thatδ= √

e +e,f



γ = +∞,fγ∞=  and

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, P.R. China.}2_Department

of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, P.R. China.

Acknowledgements

The authors would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274, 11171085) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).

Received: 29 August 2012 Accepted: 28 December 2012 Published: 14 January 2013 References

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doi:10.1186/1687-2770-2013-3