Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

R E S E A R C H

Open Access

Multiple positive solutions for first-order

impulsive singular integro-differential

equations on the half line in a Banach space

Yanlai Chen

_{and Baoxia Qin}

*_{Correspondence:}

[email protected]

1_{School of Economics, Shandong}

University, Jinan, Shandong 250100, People’s Republic of China Full list of author information is available at the end of the article

Abstract

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type.

MSC: 45J05; 34G20; 47H10

Keywords: impulsive singular integro-differential equation in a Banach space; infinite three-point boundary value problem; fixed-point theorem of cone expansion and compression with norm type

1 Introduction

In recent years, multiple solutions of boundary value problems for impulsive differential

equations in scalar spaces had been extensively studied (see, for example, [–]). In recent papers [] and [], Professor D. Guo discussed two infinite boundary value problems for

nth-order impulsive nonlinear singular integro-differential equations of mixed type on the

half line in a Banach space. By constructing a bounded closed convex set, apart from the

singularities, and using the Schauder fixed-point theorem, he obtained the existence of

positive solutions for the infinite boundary value problems. But such equations are

sub-linear, and there are no results on existence of two positive solutions. Now, in this paper,

we shall discuss the existence of two positive solutions for first-order superlinear singular

equations by means of a different method,i.e., by using the fixed-point theorem of cone

expansion and compression with norm type (see [, ]), and the key point is to introduce

a new coneQ.

LetEbe a real Banach space andPbe a cone inE, which defines a partial ordering in

Ebyx≤yif and only ify–x∈P.Pis said to be normal if there exists a positive constant

Nsuch thatθ ≤x≤yimpliesx ≤Ny, whereθ denotes the zero element ofE, and the smallestN is called the normal constant ofP. Ifx≤yandx=y, we writex<y. Let

P+=P\{θ},i.e.,P+={x∈P:x>θ}. For details on cone theory, see [].

Consider the infinite three-point boundary value problem for a first-order impulsive nonlinear singular integro-differential equation of mixed type on the half line inE:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u(t) =f(t,u(t), (Tu)(t), (Su)(t)), ∀t∈J₊,

u|t=tk=Ik(u(tk)) (k= , , , . . .),

u(∞) =γu(η) +βu(),

()

where J= [,∞),J+= (,∞),  <t<· · ·<tk<· · ·,tk→ ∞,J+ =J+\{t, . . . ,tk, . . .}, f ∈ C[J+×P+×P×P,P],Ik∈C[P+,P] (k= , , , . . .), ≤γ < ,β+γ > ,tm–<η<tm(for somem),u(∞) =lim_t→∞u(t) and

(Tu)(t) = t



K(t,s)u(s)ds, (Su)(t) =

∞



H(t,s)u(s)ds, ()

K∈C[D,R+],D={(t,s)∈J×J:t≥s},H∈C[J×J,R+],R+denotes the set of all

nonneg-ative numbers.u|t=tk denotes the jump ofu(t) att=tk,i.e.,

u|t=tk=u

t_k+–ut–_k,

whereu(t_k+) andu(t–

k) represent the right and left limits ofu(t) att=tk, respectively. In the following, we always assume that

lim

t→+ f(t,u,v,w) =∞, ∀u∈P+,v,w∈P ()

and

lim

u→θ+ f(t,u,v,w) =∞, ∀t∈J+,v,w∈P, ()

(whereu→θ+_{meansu>θ,u →),i.e.,f(t,u,v,w) is singular att=  andu=θ. We}

also assume that

lim

u→θ+ Ik(u) =∞ (k= , , , . . .), ()

i.e.,Ik(u) (k= , , , . . .) are singular atu=θ.

Let PC[J,E] = {u:u is a map fromJ into E such that u(t) is continuous at t=tk, left continuous att=tk, andu(t+_k) exists,k= , , , . . .} andBPC[J,E] ={u∈PC[J,E] :

sup_t∈Ju(t)<∞}. It is clear thatBPC[J,E] is a Banach space with norm

uB=sup

t∈J

_{u(t) .}

LetBPC[J,P] ={u∈BPC[J,E] :u(t)≥θ,∀t∈J}andQ={u∈BPC[J,P] :u(t)≥β–( –

γ)u(s),∀t,s∈J}. Obviously,BPC[J,P] andQare two cones in spaceBPC[J,E] andQ⊂ BPC[J,P].u∈BPC[J,P]∩C_[J

+,E] is called a positive solution of the infinite three-point

boundary value problem () if u(t) >θ fort∈J andu(t) satisfies (). LetQ+={u∈Q:

2 Several lemmas

Let us list some conditions. (H)supt∈J

t

K(t,s)ds<∞,supt∈J

∞

 H(t,s)ds<∞and

lim

t→t

_∞



Ht,s–H(t,s)ds= , ∀t∈J.

In this case, let

k*=sup

t∈J

t



K(t,s)ds, h*=sup

t∈J

∞



H(t,s)ds.

(H) There exista∈C[J+,R+] andg∈C[R++×R+×R+,R+] such that

f(t,u,v,w) ≤a(t)gu,v,w, ∀t∈J+,u∈P+,v,w∈P,

and

a*=

∞



a(t)dt<∞,

whereR++={x∈R+:x> }.

(H) There existγk≥ (k= , , , . . .) andF∈C[R++,R+] such that

Ik(u) ≤γkF

u, ∀u∈P+(k= , , , . . .),

and

γ*= ∞

k=

γk<∞.

(H) For any t∈J+ andr>p> ,f(t,Ppr,Pr,Pr) ={f(t,u,v,w) :u∈Ppr,v,w∈Pr}and Ik(Ppr) ={Ik(u) :u∈Ppr} (k= , , , . . .) are relatively compact inE, wherePr={u∈P: u ≤r}andPpr={u∈P:p≤ u ≤r}.

Remark Obviously, condition (H) is satisfied automatically whenEis finite dimensional.

Remark It is clear: If condition (H) is satisfied, then the operatorsT andS defined by

() are bounded linear operators fromBPC[J,E] intoBPC[J,E] andT ≤k*_{,S ≤h}*_;

moreover, we haveT(BPC[J,P])⊂BPC[J,P] andS(BPC[J,P])⊂BPC[J,P].

We shall reduce the infinite three-point boundary value problem () to an impulsive integral equation. To this end, we consider the operatorAdefined by

(Au)(t) = 

β+γ– 

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

+ t



fs,u(s), (Tu)(s), (Su)(s)ds+

<tk<t

Ik

u(tk)

, ∀t∈J. ()

In what follows, we writeJ= [,t],Jk= (tk–,tk] (k= , , , . . .).

Lemma  Let cone P be normal and conditions(H)-(H)be satisfied.Then operator A

defined by()is a continuous operator from Q+into Q;moreover,for any q>p> ,A(Qpq) is relatively compact.

Proof Letu∈Q+anduB=r. Thenr>  and

u(t)≥β–( –γ)u(s)≥θ, ∀t,s∈J, so,

u(t) ≥N–β–( –γ)uB, ∀t∈J, ()

whereNdenotes the normal constant of coneP, and consequently,

N–β–( –γ)r≤ u(t) ≤r, ∀t∈J. () By condition (H) and (), we have

ft,u(t), (Tu)(t), (Su)(t) ≤Ma(t), ∀t∈J, ()

where

M=maxg(x,y,z) :N–β–( –γ)r≤x≤r, ≤y≤k*r, ≤z≤h*r, which implies the convergence of the infinite integral

∞



ft,u(t), (Tu)(t), (Su)(t)dt ()

and ∞



ft,u(t), (Tu)(t), (Su)(t)dt ≤

∞



ft,u(t), (Tu)(t), (Su)(t) dt≤Ma*. ()

On the other hand, by condition (H) and (), we have

Ik

u(tk) ≤Dγk (k= , , , . . .), ()

where

which implies the convergence of the infinite series ∞ k= Ik u(tk)

() and ∞ k= Ik u(tk)

≤ ∞ k= Ik

u(tk) ≤Dγ*. ()

It follows from (), (), and () that

(Au)(t) ≤ 

β+γ– 

∞

η

fs,u(s), (Tu)(s), (Su)(s) ds

+ ( –γ) η



_f

s,u(s), (Tu)(s), (Su)(s) ds

+ ∞

k=m

Ik

u(tk) + ( –γ) m–

k=

Ik

u(tk)

+ t



fs,u(s), (Tu)(s), (Su)(s) ds+

<tk<t

Ik

u(tk)

≤ 

β+γ– 

∞



fs,u(s), (Tu)(s), (Su)(s) ds+ ∞

k=

Ik

u(tk)

+

∞



fs,u(s), (Tu)(s), (Su)(s) ds+ ∞

k=

Ik

u(tk)

= β+γ

β+γ– 

_∞



fs,u(s), (Tu)(s), (Su)(s) ds+ ∞

k=

Ik

u(tk)

≤ β+γ

β+γ– 

Ma*+Dγ*, ∀t∈J,

which implies thatAu∈BPC[J,P] and

AuB≤ β+γ

β+γ – 

Ma*+Dγ*. ()

Moreover, by (), we have

(Au)(t)≥ 

β+γ – 

_∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

and

(Au)(t)≤ 

β+γ – 

_∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

+

∞



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

Ik

u(tk)

, ∀t∈J. ()

It is clear,

∞



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

Ik

u(tk)

≤ 

 –γ

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

, ()

so, () and () imply

(Au)(t)≤



β+γ – +   –γ

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

, ∀t∈J. ()

It follows from () and () that

(Au)(t)≥ 

β+γ – 



β+γ– +   –γ

–

(Au)(s)

=β–( –γ)(Au)(s), ∀t,s∈J. ()

Hence,Au∈Q,i.e.,AmapsQ+intoQ.

Now, we are going to show thatAis continuous. Letun,u¯∈Q+,un–u¯B→ (n→ ∞). Write¯uB= r¯(¯r> ) and we may assume that

¯

r≤ unB≤r¯ (n= , , , . . .).

So, by (),

and

N–β–( –γ)r¯< N–β–( –γ)r¯≤ u(t)¯ ≤r¯< ¯r, ∀t∈J. ()

Similar to (), it is easy to get

Aun–Au¯B

≤ β+γ

β+γ – 

_∞



fs,un(s), (Tun)(s), (Sun)(s)

–fs,u(s), (T¯ u)(s), (S¯ u)(s)¯ ds

+ ∞

k=

Ik

un(tk)

–Ik

¯ u(tk)

(n= , , , . . .). ()

It is clear that

ft,un(t), (Tun)(t), (Sun)(t)

→ft,u(t), (T¯ u)(t), (S¯ u)(t)¯ asn→ ∞,∀t∈J, ()

and, similar to () and observing () and (), we have

ft,un(t), (Tun)(t), (Sun)(t)

–ft,u(t), (T¯ u)(t), (S¯ u)(t)¯ ≤Ma(t) =d(t),

∀t∈J(n= , , , . . .);d∈L[J,R+], ()

where

M=maxg(x,y,z) :N–β–( –γ)¯r≤x≤r, ¯ ≤y≤k*r, ¯ ≤z≤h*r¯.

It follows from (), (), and the dominated convergence theorem that

lim

n→∞

∞



ft,un(t), (Tun)(t), (Sun)(t)

–ft,u(t), (T¯ u)(t), (S¯ u)(t)¯ dt= . ()

On the other hand, for any> , we can choose a positive integerjsuch that

D ∞

k=j+

γk<, ()

where

D=maxF(x) :N–β–( –γ)¯r≤x≤r¯.

And then, choose an positive integernsuch that

j

k=

Ik

un(tk)

–Ik

¯

From (), (), and observing condition (H) and (), (), we get

∞

k=

Ik

un(tk)

–Ik

¯

u(tk) <+ D ∞

k=j+

γk< , ∀n>n,

hence,

lim

n→∞

∞

k=

Ik

un(tk)

–Ik

¯

u(tk) = . ()

It follows from (), (), and () thatAun–Au¯B→ asn→ ∞, and the continuity ofAis proved.

Finally, we prove thatA(Qpq) is relatively compact, whereq>p>  are arbitrarily given. Letvn∈A(Qpq) (n= , , , . . .). Then, by (),

N–β–( –γ)p≤ vn(t) ≤q, ∀t∈J(n= , , , . . .). ()

Similar to (), (), (), and observing (), we have

_f

t,vn(t), (Tvn)(t), (Svn)(t) ≤Ma(t), ∀t∈J+(n= , , , . . .), ()

Ik

vn(tk) ≤Dγk (k,n= , , , . . .) ()

and

AvnB≤ β+γ

β+γ – 

Ma*+Dγ*

(n= , , , . . .), ()

where

M=max

g(x,y,z) :N–β–( –γ)p≤x≤q, ≤y≤k*q, ≤z≤h*q

and

D=max

F(x) :N–β–( –γ)p≤x≤q.

ConsiderJi= (ti–,ti] for any fixedi. By () and (), we have

(Avn)

t– (Avn)(t) ≤

t

t

fs,vn(s), (Tvn)(s), (Svn)(s) ds

≤M

t

t

a(s)ds, ∀t,t∈Ji,t>t(n= , , , . . .), ()

which implies that the functions{wn(t)}(n= , , , . . .) defined by

wn(t) =

⎧ ⎨ ⎩

(Avn)(t), ∀t∈Ji= (ti–,ti],

(Avn)(t+i–), ∀t=ti–

((Aun)(t+i–) denotes the right limit of (Aun)(t) att=ti–) are equicontinuous on¯Ji= [ti–,ti]. On the other hand, for any> , choose a sufficiently largeτ>ηand a sufficiently large positive integerj>msuch that

M

∞

τ

a(s)ds<, D

∞

k=j+

γk<. ()

We have, by (), (), (), (), and (),

wn(t) = 

β+γ– 

τ

η

fs,vn(s), (Tvn)(s), (Svn)(s)

ds

+

∞

τ

fs,vn(s), (Tvn)(s), (Svn)(s)

ds

+ ( –γ) η



fs,vn(s), (Tvn)(s), (Svn)(s)

ds+

j

k=m Ik

vn(tk)

+ ∞

k=j+

Ik

vn(tk)

+ ( –γ) m–

k=

Ik

vn(tk)

+ t



fs,vn(s), (Tvn)(s), (Svn)(s)

ds + i– k= Ik vn(tk)

, ∀t∈ ¯Ji(n= , , , . . .) ()

and

_τ∞fs,vn(s), (Tvn)(s), (Svn)(s)

ds < (n= , , , . . .), ()

∞

k=j+

Ik

vn(tk)

< (n= , , , . . .). ()

It follows from (), (), (), and ([], Theorem ..) that

αW(t)≤ 

β+γ – 

 τ

η

αfs,V(s), (TV)(s), (SV)(s)ds+ 

+ ( –γ) η



αfs,V(s), (TV)(s), (SV)(s)ds+ j

k=m

αIk

V(tk)

+ 

+ ( –γ) m–

k=

αIk

V(tk)

+  t



αfs,V(s), (TV)(s), (SV)(s)ds

+ i–

k=

αIk

V(tk)

, ∀t∈ ¯Ji, ()

(TV)(s), (SV)(s)⊂Pq*fors∈J, wherep*=N–β–( –γ)pandq*=max{q,k*q,h*q}, we see

that, by condition (H),

αfs,V(s), (TV)(s), (SV)(s)= , ∀s∈J ()

and

αIk

V(tk)

=  (k= , , , . . .). ()

It follows from () to () that

αW(t)≤ 

β+γ – , ∀t∈ ¯Ji,

which implies by virtue of the arbitrariness ofthatα(W(t)) =  fort∈ ¯Ji.

By the Ascoli-Arzela theorem (see [, Theorem ..]), we conclude thatW={wn:n= , , , . . .}is relatively compact inC[¯Ji,E], hence,{wn(t)}has a subsequence which is con-vergent uniformly onJ¯i, so,{(Avn)(t)}has a subsequence which is convergent uniformly onJi. Sinceimay be any positive integer, so, by diagonal method, we can choose a sub-sequence{(Avni)(t)}of{(Avn)(t)}such that{(Avni)(t)}is convergent uniformly on eachJk

(k= , , , . . .). Let

lim

i→∞(Avni)(t) =w(t), ∀t∈J.

It is clear thatw∈PC[J,P]. By (), we have

AvniB≤

β+γ β+γ– 

Ma*+Dγ*

(i= , , , . . .),

which implies thatw∈BPC[J,P] and

wB≤ β+γ

β+γ – 

Ma*+Dγ*

.

Let>  be arbitrarily given and choose a sufficiently large positive numberτ such that

M

∞

τ

a(s)ds+D

tk≥τ

γk<. ()

For anyτ<t<∞, we have, by (),

(Avni)(t) – (Avni)(τ) =

t

τ

fs,vni(s), (Tvni)(s), (Svni)(s)

ds

+

τ≤tk<t

Ik

vni(t)

(i= , , , . . .),

which implies by virtue of (), (), and () that

(Avni)(t) – (Avni)(τ) ≤M

t

τ

a(s)ds+D

τ≤tk<t

Lettingi→ ∞in (), we get

w(t) –w(τ) ≤, ∀t>τ. ()

On the other hand, since{(Avni)(t)}converges uniformly tow(t) on [,τ] asi→ ∞, there

exists a positive integerisuch that

(Avni)(t) –w(t) <, ∀t∈[,τ],i>i. ()

It follows from () to () that

(Avni)(t) –w(t) ≤ (Avni)(t) – (Avni)(τ) + (Avni)(τ) –w(τ)

+ w(τ) –w(t) < , ∀t>τ,i>i. ()

By () and (), we have

Avni–wB≤, ∀i>i,

hence,Avni–wB→ asi→ ∞, and the relative compactness ofA(Qpq) is proved.

Lemma  Let cone P be normal and conditions (H)-(H)be satisfied. Then u∈Q+∩

C_[J

+,E]is a positive solution of the infinite three-point boundary value problem()if and

only if u∈Q+is a solution of the following impulsive integral equation:

u(t) = 

β+γ– 

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

+ t



fs,u(s), (Tu)(s), (Su)(s)ds+

<tk<t

Ik

u(tk)

, ∀t∈J, ()

i.e.,u is a fixed point of operator A defined by()in Q+.

Proof Foru∈PC[J,E]∩C[J₊,E], it is easy to get the following formula:

u(t) =u() + t



u(s)ds+

<tk<t

ut_k+–u(tk)

, ∀t∈J. ()

Letu∈Q+∩C[J+,E] be a positive solution of the infinite three-point boundary value

problem (). By () and (), we have

u(t) =u() + t



fs,u(s), (Tu)(s), (Su)(s)ds+

<tk<t

Ik

u(tk)

We have shown in the proof of Lemma  that the infinite integral () and the infinite series () are convergent, so, by taking limits ast→ ∞in both sides of (), we get

u(∞) =u() +

∞



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

Ik

u(tk)

. ()

On the other hand, by () and (), we have

u(∞) =γu(η) +βu() ()

and

u(η) =u() + η



fs,u(s), (Tu)(s), (Su)(s)ds+ m–

k=

Ik

u(tk)

. ()

It follows from () to () that

u() = 

β+γ – 

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

,

and, substituting it into (), we see thatu(t) satisfies equation (),i.e.,u=Au. Conversely, assume thatu∈Q+is a solution of Equation (). We have, by (),

u() = 

β+γ – 

_∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

()

and

u(η) = 

β+γ – 

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

+ η



fs,u(s), (Tu)(s), (Su)(s)ds+ m–

k=

Ik

u(tk)

Moreover, by taking limits ast→ ∞in (), we see thatu(∞) exists and

u(∞) = 

β+γ – 

∞

η

fs,u(s), (Tu)(s), (Su)(s)ds

+ ( –γ) η



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=m Ik

u(tk)

+ ( –γ) m–

k=

Ik

u(tk)

+

∞



fs,u(s), (Tu)(s), (Su)(s)ds

+ ∞

k=

Ik

u(tk)

. ()

It follows from () to () that

γu(η) +βu() =u(∞).

On the other hand, direct differentiation of () gives

u(t) =ft,u(t), (Tu)(t), (Su)(t), ∀t∈J₊,

and, it is clear, by (),

u|t=tk=Ik

u(tk)

(k= , , , . . .).

Hence,u∈C_[J

+,E] andu(t) satisfies (). Sinceu∈Q+, so () holds anduB> , hence

u(t) >θfort∈J. Consequently,u(t) is a positive solution of the infinite three-point

bound-ary value problem ().

Lemma  (The fixed-point theorem of cone expansion and compression with norm

type; see [, Theorem ] or [, Theorem ..]) Let P be a cone in real Banach space E and , be two bounded open sets in E such thatθ ∈, ¯⊂, and operator

A:P∩(¯\)→P be completely continuous,whereθ denotes the zero element of E and

¯

idenotes the closure ofi(i= , ).Suppose that one of the following two conditions is satisfied:

(a) Ax ≤ x, ∀x∈P∩∂; Ax ≥ x, ∀x∈P∩∂,

where∂idenotes the boundary ofi(i= , ).

(b) Ax ≥ x, ∀x∈P∩∂; Ax ≤ x, ∀x∈P∩∂.

Then A has at least one fixed point in P∩(¯\).

Remark  Lemma  is different from the Krasnoselskii fixed-point theorem of cone

ex-pansion and compression (see [, Theorem .]). In Krasnoselskii’s theorem, the condi-tion corresponding to (a) is

It is clear, conditions (a) and (a) are independent each other. On the other hand, in Kras-noselskii’s theorem,andare balls with centerθ.

3 Main theorems

Let us list more conditions.

(H) There existu∈P+,b∈C[J+,R++] andτ∈C[P+,R+] such that

f(t,u,v,w)≥b(t)τ(u)u, ∀t∈J+,u∈P+,v,w∈P,

and

τ(u)

u → ∞ asu∈P+,u → ∞,

and

b*=

_∞



b(t)dt<∞.

Remark  Condition (H) means thatf(t,u,v,w) is superlinear with respect tou.

(H) There existu∈P+,c∈C[J+,R++] andσ∈C[P+,R+] such that

f(t,u,v,w)≥c(t)σ(u)u, ∀t∈J+,u∈P+,v,w∈P,

and

σ(u)→ ∞ asu∈P+,u →,

and

c*=

∞



c(t)dt<∞.

Theorem  Let cone P be normal and conditions(H)-(H)be satisfied.Assume that there

exists aξ> such that N(β+γ)

β+γ – 

Mξa*+Dξγ*<ξ, ()

where N denotes the normal constant of P,and

Mξ=maxg(x,y,z) :N–β–( –γ)ξ≤x≤ξ, ≤y≤k*ξ, ≤z≤h*ξ, () Dξ=maxF(x) :N–β–( –γ)ξ≤x≤ξ ()

(for g(x,y,z),F(x),a*_andγ*_{;see conditions(H}

)and(H)).Then the infinite three-point

boundary value problem()has at least two positive solutions u*_,u**_∈Q

+∩C[J+,E]such

Proof By Lemma  and Lemma , operatorAdefined by () is continuous fromQ+into

Qand we need to prove thatAhas two fixed pointsu*_andu**_inQ

+such that  <u*B<

ξ<u**B.

By condition (H), there exists ar>  such that

τ(u)≥β(β+γ– )N



( –γ)_b*_u 

u, ∀u∈P+,u ≥r, ()

so,

f(t,u,v,w)≥β(β+γ – )N

_u

( –γ)_b*_u 

b(t)u, ∀t∈J+,u∈P+,v,w∈P,u ≥r. ()

Choose

r>max

Nβ( –γ)–r,ξ

. ()

Foru∈Q,uB=r, we have by () and (),

u(t) ≥N–β–( –γ)uB=N–β–( –γ)r>r, ∀t∈J, ()

so, (), (), (), and () imply

(Au)(t)≥  –γ

β+γ – 

∞



fs,u(s), (Tu)(s), (Su)(s)ds

≥ βN

( –γ)b*_u 

∞



u(s) b(s)ds

u

≥ NuB

b*_u 

∞



b(s)ds

u=

NuB u

u, ∀t∈J, ()

and consequently,

AuB≥ uB, ∀u∈Q,uB=r. ()

By condition (H), there existsr>  such that

σ(u)≥(β+γ – )Nξ ( –γ)c*_u



, ∀u∈P+,  <u<r, ()

so,

f(t,u,v,w)≥(β+γ – )Nξ ( –γ)c*_u



c(t)u, ∀t∈J+,u∈P+,v,w∈P,  <u<r. ()

Choose

 <r<min{r,ξ}. ()

Foru∈Q,uB=r, we have by () and (),

so, we get by (), (), and (),

(Au)(t)≥  –γ

β+γ – 

∞



fs,u(s), (Tu)(s), (Su)(s)ds

≥ Nξ

c*_u 

∞



c(s)ds

u=

Nξ

u

u, ∀t∈J,

which implies

(Au)(t) ≥ξ>r, ∀t∈J,

and consequently,

AuB>uB, ∀u∈Q,uB=r. ()

On the other hand, foru∈Q,uB=ξ, by condition (H), condition (H), (), and (),

we have

ft,u(t), (Tu)(t), (Su)(t) ≤Mξa(t), ∀t∈J+ ()

and

Ik

u(tk) ≤Dξγk (k= , , , . . .). ()

It is clear, by (),

(Au)(t)≤ β+γ

β+γ – 

∞



fs,u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

Ik

u(tk)

, ∀t∈J. ()

It follows from () to () that

AuB≤N(β+γ)

β+γ– 

Mξa*+Dξγ*. ()

Thus, () and () imply

AuB<uB, ∀u∈Q,uB=ξ. ()

From () and (), we know  <r<ξ<r, and by Lemma ,A:Qrr→Qis completely

continuous, whereQrr={u∈Q:r≤ uB≤r}, hence, (), (), (), and Lemma 

imply thatAhas two fixed pointsu*_,u**_∈Q

+such thatr<u*B<ξ<u**B≤r. The

proof is complete.

Theorem  Let cone P be normal and conditions(H)-(H)and(H)be satisfied.Assume

that

g(x,y,z)

uniformly for y,z∈R+,and

F(x)

x → as x→ ∞ ()

(for g(x,y,z)and F(x),see conditions(H)and(H)).Then the infinite three-point boundary

value problem()has at least one positive solution u*∈Q+∩C[J+,E].

Proof As in the proof of Theorem , we can chooser>  such that () holds (in this

case, we putξ=  in () and ()). On the other hand, by () and (), there existsr> 

such that

g(x,y,z)≤(x+y+z), ∀x>r,y≥,z≥ ()

and

F(x)≤x, ∀x>r, ()

where

=

β+γ – 

N(β+γ)[( +k*_+h*_)a*_+γ*_]. ()

Choose

r>max

Nβ( –γ)–r,r

. ()

Foru∈Q,uB=r, we have by () and (),

u(t) ≥N–β–( –γ)r>r, ∀t∈J

so, () and () imply

g u(t) , (Tu)(t) , (Su)(t) ≤ u(t) + (Tu)(t) + (Su)(t)

≤

 +k*+h*r, ∀t∈J ()

and

F u(tk) ≤ u(tk) ≤r (k= , , , . . .). ()

It follows from (), conditions (H), condition (H), (), (), and () that

(Au)(t) ≤N(β+γ)

β+γ – 



 +k*+h*r

∞



a(s)ds+r

∞

k=

γk

= N(β+γ)r

β+γ– 

and consequently,

AuB≤ uB, ∀u∈Q,uB=r. ()

Sincer>rby virtue of (), we conclude from (), (), and Lemma  thatAhas a fixed

pointu*_∈Q

+such thatr<u*B≤r. The theorem is proved.

Example  Consider the infinite system of scalar first-order impulsive singular

integro-differential equations of mixed type on the half line:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u_n(t) = e–t

n√_t{_(un+(t) +

∞

m=um(t))+_(

∞

m=um(t))–}

+ e–t

n√_t{(

t

e–(t+)sun(s)ds)+_(

∞



un+(s)ds (+t+s) )},

∀ <t<∞,t=k(k= , , , . . . ;n= , , , . . .),

un|t=k= 

–k

n{un(k) +_(

∞

m=um(k))–



} _{(k= , , , . . . ;n= , , , . . .),}

un(∞) =_un(_) + un() (n= , , , . . .).

()

Conclusion Infinite system () has at least two positive solutions{u*_n(t)}(n= , , , . . .)

and{u**

n(t)}(n= , , , . . .) such that

 < inf

≤t<∞ ∞

n=

u*_n(t)≤ sup

≤t<∞ ∞

n=

u*_n(t) <  < sup

≤t<∞ ∞

n=

u**_n(t),

inf

≤t<∞ ∞

n=

u**_n(t) > .

Proof LetE=l _{={u= (u}

, . . . ,un, . . .) :

_∞

n=|un|<∞}with norm u=

_∞

n=|un|and

P={(u, . . . ,un, . . .) :un≥,n= , , , . . .}. ThenPis a normal cone inEwith normal con-stantN= , and infinite system () can be regarded as an infinite three-point bound-ary value problem of form (). In this situation, u= (u, . . . ,un, . . .), v= (v, . . . ,vn, . . .), w= (w, . . . ,wn, . . .),tk=k(k= , , , . . .),K(t,s) =e–(t+)s,H(t,s) = ( +t+s)–,η=_,γ =_,

β= ,f = (f, . . . ,fn, . . .) andIk= (Ik, . . . ,Ikn, . . .), in which

fn(t,u,v,w)

= e

–t

n√_t

 

un++

∞ m= um  +  _∞ m= um – + e –t

n√_t

v_n+

w



n+

,

∀t∈J+= (,∞),u∈P+={u∈P:u> },v,w∈P(n= , , , . . .) ()

and

Ikn(u) = –k n

un+

  _∞ m= um –

It is easy to see thatf ∈C[J+×P+×P×P,P],Ik∈C[P+,P] (k= , , , . . .) and condition

(H) is satisfied andk*≤,h*≤. We have, by (),

≤fn(t,u,v,w)≤ e–t n√_t

 

u+ u

–

+ e

–t

n√_t

v₊

w



≤ e–t n√_t

 u

₊ 

u

–₊ 

v

₊ 

w



,

∀t∈J+,u∈P+,v,w∈P(n= , , , . . .), ()

so, observing the inequality∞_n= 

n < , we get

f(t,u,v,w) = ∞

n=

fn(t,u,v,w)≤ e–t

√ t

 u

₊ 

u

–₊

v

₊ 

w



,

∀t∈J+,u∈P+,v,w∈P,

which implies that condition (H) is satisfied for

a(t) =e

–t √

t

and

g(x,y,z) =  x

₊ 

x+

 y

₊ 

z



with

a*=

_∞



e–t √

t dt< 



dt √

t+

_∞



e–tdt=  + e

–_<

.

By (), we have

≤Ikn(u)≤ –k n

u+ u

–_

, ∀u∈P+(k= , , , . . . ;n= , , , . . .), ()

so,

Ik(u) ≤  

–k

u+

u

–_

, ∀u∈P+(k= , , , . . .),

which implies that condition (H) is satisfied forγk=_–k(γ*=_) and

F(x) =x+  √x.

On the other hand, () implies

fn(t,u,v,w)≥ e–t n√_tu

_{, ∀t∈J}

and

fn(t,u,v,w)≥ e–t n√_tu

–_{, ∀t∈J}

+,u∈P+,v,w∈P(n= , , , . . .), ()

so, we see that condition (H) is satisfied forb(t) = e –t √t (b

*_< 

,),τ(u) =u _andu

=

(, . . . ,_n, . . .) and condition (H) is satisfied forc(t) = e –t √t (c

*_< 

,),σ(u) =u–and

u= (, . . . ,_n, . . .). In addition, from (), we have

f(t,u,v,w) ≥

_∞

n=

 n

e–t √tu

–_> e–t

√tu

–_{, ∀t∈J}

+,u∈P+,v,w∈P,

which implies that () and () hold,i.e.,f(t,u,v,w) is singular att=  andu=θ. Moreover, from (), we get

Ikn(u)≥ –k nu

–_{, ∀u∈P}

+(k= , , , . . . ;n= , , , . . .),

and so,

_{Ik(u) ≥}∞ n=

 n

–k

u

–_> –k

u

–_, ∀_u∈_P

+(k= , , , . . .),

which implies that () holds,i.e.,Ik(u) (k= , , , . . .) are singular atu=θ. Now, we check that condition (H) is satisfied. Lett∈J+andr>p>  be fixed, and{z(m)}be any sequence

inf(t,Ppr,Pr,Pr), wherez(m)= (z(m), . . . ,z (m)

n , . . .). Then, we have, by () and (),

≤z(_nm)≤ e

–t

n√_t

 r

₊ 

p+  r



(n,m= , , , . . .). ()

So,{z(nm)}is bounded, and, by diagonal method, we can choose a subsequence{mi} ⊂ {m} such that

z(mi)

n → ¯zn asi→ ∞(n= , , , . . .), ()

which implies by virtue of () that

≤ ¯zn≤ e–t n√_t

 r

₊ 

p+  r



(n= , , , . . .). ()

Consequently,z¯= (z¯, . . . ,z¯n, . . .)∈l=E. Let>  be given. Choose a positive integern

such that

e–t √

t

_∞

n=n+

 n

 r

₊ 

p+  r



<

. ()

By (), we see that there exists a positive integerisuch that

_z(mi)

n –z¯n<

n

It follows from () to () that

z(mi)_–z¯ ₌

∞

n=

z(mi)

n –¯zn≤

n

n=

z(mi)

n –z¯n+ ∞

n=n+

z(mi)

n

+ ∞

n=n+

|¯zn|< +

+

 =, ∀i>i,

hence,z(mi)→ ¯_zinEasi→ ∞_{. Thus, we have proved thatf(t,P}

pr,Pr,Pr) is relatively com-pact in E. Similarly, by using (), we can prove thatIk(Ppr) is relatively compact inE. Hence, condition (H) is satisfied. Finally, we check that inequality () is satisfied for

ξ= . In this case,

M≤max

g(x,y,z) : 

≤x≤, ≤y≤, ≤z≤ ≤  +  +  +  =   and

D=max

F(x) : 

≤x≤

≤ + √

  < .,

so,

N(β+γ)

β+γ – 

Ma*+Dγ*

<   ×  + .×   = ,

,< ,

i.e., inequality () is satisfied forξ= . Hence, our conclusion follows from Theorem .

Example  Consider the infinite system of scalar first order impulsive singular

integro-differential equations of mixed type on the half line:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u_n(t) =  n_t(+t){

√

un(t) + un+(t) + (

∞

m=um(t))–}

+ 

n_t(+t){(

t



un(s)ds +ts+s)

  + (∞

 e–ssin(t–s)un(s)ds)

 },

∀≤t<∞,t= k(k= , , , . . . ;n= , , , . . .);

un|t=k=e

–k

n(un+(k))   +–k

n(

∞

m=um(k))–

(k= , , , . . . ;n= , , , . . .),

un(∞) = un() + un() (n= , , , . . .).

()

Conclusion Infinite system () has at least one positive solution{u*

n(t)}(n= , , , . . .) such that

inf

≤t<∞

∞

n=

u*_n(t) > .

Proof LetE=l={u= (u, . . . ,un, . . .) :

∞

n=|un|<∞}with normu=

∞

n=|un|andP=

w= (w, . . . ,wn, . . .),tk= k(k= , , , . . .),K(t,s) = ( +ts+s)–,H(t,s) =e–ssin(t–s),

η= ,γ=_,β=_,f = (f, . . . ,fn, . . .) andIk= (Ik, . . . ,Ikn, . . .), in which

fn(t,u,v,w) = 

n_t_{( +t)}

un+ un++

_∞

m=

um

–

+ 

n_t_{( +t)}

v

  n+w

  n

,

∀t∈J+,u∈P+,v,w∈P(n= , , , . . .) ()

and

Ikn(u) = e–k

nu   n++

–k n

_∞

m=

um

–

(k= , , , . . . ;n= , , , . . .). ()

It is clear thatf ∈C[J+×P+×P×P,P],Ik∈C[P+,P] (k= , , , . . .) and condition (H) is

satisfied andk*_≤π

,h

*_{≤. We have, by () and (),}

≤fn(t,u,v,w)≤ 

n_t_{( +t)}

u+u–_+v _+w_,

∀t∈J+,u∈P+,v,w∈P(n= , , , . . .)

and

≤Ikn(u)≤ –k

n

u_+u–_, ∀_u∈_P

+(k= , , , . . . ;n= , , , . . .),

so, observing

∞

n=

 n <

∞

n=

 n < ,

we get

f(t,u,v,w) ≤  t( +t)

√u+ u–+ v_{+ w}_,

∀t∈J+,u∈P+,v,w∈P

and

Ik(u) ≤–k+

u_+u–_, ∀_u∈_{P+(k= , , , . . .),}

which imply that conditions (H) is satisfied for

a(t) =  t_{( +t)}

and

with

a*=

∞



dt

t_{( +t)}

< 



dt

t

+

∞



dt ( +t) =

 

and (H) is satisfied forγk= –k+(γ*= ) and

F(x) =x_+x–_.

By (), we have

fn(t,u,v,w)≥ 

n_t( +t)

u–, ∀t∈J+,u∈P+,v,w∈P(n= , , , . . .) ()

so, condition (H) is satisfied for

c(t) =  t( +t)

c*=a* < 

,

σ(u) =u–_andu

= (, . . . ,_n, . . .). Moreover, () implies

f(t,u,v,w) ≥

_∞

n=

 n



t_{( +t)}

u–>  t_{( +t)}

u–,

∀t∈J+,u∈P+,v,w∈P,

so, () and () are satisfied,i.e.,f(t,u,v,w) is singular att=  andu=θ. Similarly, () implies

Ik(u) ≥

_∞

n=

 n

–ku–> –ku–, ∀u∈P+(k= , , , . . .),

so, () is satisfied,i.e.,Ik(u) (k= , , , . . .) are singular atu=θ. Similar to the discussion in Example , we can prove thatf(t,Ppr,Pr,Pr) andIk(Ppr) (for fixedt∈J+andr>p> ;

k= , , , . . .) are relatively compact inE=l_{, so, condition (H}

) is satisfied. On the other

hand, we have

 < g(x,y,z) x+y+z= 

√ 

x x+y+z

 

(x+y+z)– _+x–_(x+y+z)–

+ 

y x+y+z

 

(x+y+z)–_{+ }

z x+y+z

 

(x+y+z)–

≤√x– _+x–_{+ x}–_{+ x}–_, ∀_{x> ,y}≥_,z≥_,

so, () is satisfied. Moreover, it is clear that () is satisfied. Hence, our conclusion follows

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read, and approved the final manuscript.

Author details

1_{School of Economics, Shandong University, Jinan, Shandong 250100, People’s Republic of China.}2_{Department of}

Mathematics, Qilu Normal University, Jinan, Shandong 250100, People’s Republic of China.

Acknowledgements

The author would like to thank Professor D. Guo for his valuable suggestions.

Received: 22 December 2012 Accepted: 18 March 2013 Published: 2 April 2013

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doi:10.1186/1687-2770-2013-69