Existence of two positive solutions for a class of second order impulsive singular integro-differential equations on the half line

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R E S E A R C H

Open Access

Existence of two positive solutions for a

class of second order impulsive singular

integro-differential equations on the half line

Dajun Guo

*

*_{Correspondence:} [email protected] Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China

Abstract

In this paper, the author discusses the existence of two positive solutions for an inﬁnite boundary value problem of second order impulsive singular

integro-diﬀerential equations on the half line by means of the ﬁxed point theorem of cone expansion and compression with norm type.

MSC: 45J05; 47H10

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

1 Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years (see [–]). Many problems have been investigated for im-pulsive differential equations, imim-pulsive functional differential equations and imim-pulsive differential inclusions. These problems include existence of solutions, stability theory, ge-ometric properties, applications,etc.There is a vast literature on existence of solutions: by using upper and lower solutions together with the monotone iterative technique to obtain the extremal solutions [–]; by using fixed point theorems to obtain the existence of so-lution and multiple soso-lutions [–]; by using the Leray-Schauder degree theory or fixed point index theory to obtain multiple solutions [–]; by using the variational method to obtain the existence of solution and existence of infinite many solutions [–]. In re-cent article [], the author discussed the existence of two positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [] (see also [–]). Now, in this article, we shall discuss such problem for a class of second order equations. The discus-sion for second order equations is more complicated than the first order case. We must introduce a new Banach space and a new cone in it to control both the unknown function and its derivative so that we can still use the fixed point theorem of cone expansion and compression with norm type.

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Consider the inﬁnite boundary value problem (IBVP) for second order impulsive singu-lar integro-diﬀerential equation of mixed type on the half line:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u(t) =f(t,u(t),u(t), (Tu)(t), (Su)(t)), ∀t∈R₊₊, u|t=tk=Ik(u(t–k)) (k= , , , . . .),

u|t=tk=Ik(u¯ _(t–

k)) (k= , , , . . .), u() = , u(∞) =βu(),

()

whereRdenotes the set of all real numbers,R+={x∈R:x≥},R++={x∈R:x> },  <

t<· · ·<tk<· · ·,tk→ ∞,R++=R++\{t, . . . ,tk, . . .},f∈C[R++×R++×R++×R+×R+,R+],

Ik,¯Ik∈C[R++,R+] (k= , , , . . .),β> ,u(∞) =limt→∞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)∈R+×R+:t≥s},H∈C[R+×R+,R+].u|t=tkandu|t=tkdenote the jumps ofu(t) andu(t) att=tk, respectively,i.e.

u|t=tk=u

t_k+–ut–_k, 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, and u(t+

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

lim

t→+f(t,u,v,w,z) =∞, ∀u,v∈R++,w,z∈R+, ()

lim

u→+f(t,u,v,w,z) =∞, ∀t,v∈R++,w,z∈R+ ()

and

lim

v→+f(t,u,v,w,z) =∞, ∀t,u∈R++,w,z∈R+, ()

i.e. f(t,u,v,w,z) is singular att= ,u=  andv= . We also assume that

lim

v→+Ik(v) =∞ (k= , , , . . .) ()

and

lim

v→+¯Ik(v) =∞ (k= , , , . . .), ()

i.e. Ik(v) and¯Ik(v) (k= , , , . . .) are singular atv= . LetPC[R+,R] ={u:uis a real function

onR+such thatu(t) is continuous att=tk, left continuous att=tk, andu(tk+) exists,k= , , , . . .}andPC[R+,R] ={u∈PC[R+,R] :u(t) is continuous att=tk, andu(tk+) and u(t–

k) exist fork= , , , . . .}. Letu∈PC[R+,R]. For  <h<tk–tk–, by the mean value

theorem, there existstk–h<ξk<tksuch that

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hence the left derivative ofu(t) att=tk, which is denoted byu_–(tk), exists, and

u_–(tk) = lim h→+

u(tk) –u(tk–h) h =u

_t–

k

.

In what follows, it is understood thatu(tk) =u_–(tk). So, foru∈PC_[R

+,R], we haveu∈

PC[R+,R].

A functionu∈PC[R+,R]∩C[R++,R] is called a positive solution of IBVP () ifu(t) > 

fort∈R++andu(t) satisﬁes (). Now, we need to introduce a new spaceDPC[R+,R] and

a new coneQin it. Let

DPC[R+,R] = u∈PC[R+,R] : sup

t∈R++

|u(t)|

t <∞,t∈Rsup+

u(t)<∞

.

It is easy to see thatDPC[R+,R] is a Banach space with the norm

uD=max

uS,u_B,

where

uS= sup

t∈R++

|u(t)|

t , u

B=sup t∈R+

u(t).

LetW={u∈DPC_[R

+,R] :u(t)≥,u(t)≥,∀t∈R+}and

Q= u∈DPC[R+,R] : inf

t∈R++

u(t) t ≥β

–_u_S,_inf

t∈R+

u(t)≥β–u_B

.

Obviously,W andQare two cones in the spaceDPC_[R

+,R] andQ⊂W (for details on

cone theory, see []). LetQ+={u∈Q:uD> }andQpq={u∈Q:p≤ uD≤q}for q>p> .

2 Several lemmas

Remark  (a) Foru∈DPC[R+,R], we haveu() = . This is clear sinceu()=  implies

sup t∈R++

|u(t)| t =∞.

(b) Foru∈Q+, we haveu(t) >  fort∈R++andu(t) >  fort∈R+.

Lemma  For u∈Q,we have

uS≥β–u_B, u_B≥β–uS, ()

β–uD≤ uS≤ uD, β–uD≤u_B≤ uD () and

β–uD≤ u(t)

t ≤ uD, ∀t∈R++; β

–_u

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Proof Since () implies () and () and () imply (), we need only to show ().

For ﬁxed  <t<t, observingu() =  and by the mean value theorem, there exists

 <ξ<tsuch that

u(t) t =

u(t) –u() t =u

₍_ξ_).

So,

uS= sup

s∈R++

u(s) s ≥

u(t) t =u

₍_ξ₎_≥ _inf s∈R+

u(s)≥β–u_B.

On the other hand, for any  <t<t, we have

u(t) t ≥β

–_u

S,

so,

u() = lim t→+

u(t) –u() t =t→lim+

u(t) t ≥β

–_u_S,

hence,

u_B=sup s∈R+

u(s)≥u()≥β–uS.

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

t

K(t,s)s ds<∞,supt∈J

∞

 H(t,s)s ds<∞and

lim t→t

∞



Ht,s–H(t,s)s ds= , ∀t∈R+.

In this case, let

k∗=sup t∈R+

t



K(t,s)s ds, h∗=sup t∈R+

∞



H(t,s)s ds.

(H) There exista,b∈C[R++,R+],g∈C[R++,R+] andG∈C[R++×R+×R+,R+] such that

f(t,u,v,w,z)≤a(t)g(u) +b(t)G(v,w,z), ∀t,u,v∈R++,w,z∈R+

and

a∗_r=

∞



a(t)gr(t)dt<∞

for anyr> , where

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and

b∗=

∞



b(t)dt<∞.

(H)Ik(v)≤tk¯Ik(v),∀v∈R++(k= , , , . . .), and there existγk∈R+(k= , , , . . .) and

F∈C[R++,R+] such that

¯

Ik(v)≤γkF(v), ∀v∈R++(k= , , , . . .)

and

¯

γ = ∞

k=

tkγk<∞,

and, consequently,

γ∗= ∞

k=

γk≤t– γ¯<∞.

It is clear: if condition (H) is satisﬁed, then () implies ().

(H) There existsc∈C[R++,R++] such that

f(t,u,v,w,z)

c(t)v → ∞ asv→ ∞

uniformly fort,u∈R++,w,z∈R+, and

c∗=

_∞



c(t)dt<∞.

(H) There existsd∈C[R++,R++] such that

d(t)–f(t,u,v,w,z)→ ∞ asv→+

uniformly fort,u∈R++,w,z∈R+, and

d∗=

∞



d(t)dt<∞.

Remark  It is clear: if condition (H) is satisﬁed, then the operatorsT andSdeﬁned

by () are bounded linear operators fromDPC[R+,R] intoBC[R+,R] (the Banach space

of all bounded continuous functionsu(t) onR+with the normuB=supt∈R+|u(t)|) and

T ≤k∗,S ≤h∗; moreover, we haveT(DPC_[R

+,R+])⊂BC[R+,R+] (BC[R+,R+] ={u∈

BC[R+,R] :u(t)≥,∀t∈R+}) andS(DPC[R+,R+])⊂BC[R+,R+].

Remark  Condition (H) means that the functionf(t,u,v,w,z) is superlinear with

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Remark  Condition (H) means that the functionf(t,u,v,w,z) is singular atv=  and it

is stronger than ().

Remark  In what follows, we need the following two formulas (see [], Lemma ):

(a) Ifu∈PC[R+,R]∩C[R++,R], then

u(t) =u() +

t



u(s)ds+

<tk<t

ut_k+–ut–_k, ∀t∈R+. ()

(b) Ifu∈PC_[R

+,R]∩C[R++,R], then

u(t) =u() +tu() +

t



(t–s)u(s)ds

+

<tk<t

ut_k+–ut–_k+ (t–tk)ut+_k–ut_k–, ∀t∈R+. ()

We shall reduce IBVP () to an impulsive integral equation. To this end, we ﬁrst consider operatorAdeﬁned by

(Au)(t) = t β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ik

ut–_k

+

t



(t–s)fs,u(s),u(s), (Tu)(s), (Su)(s)ds

+

<tk<t

Ik

ut_k–+ (t–tk)¯Ik

ut–_k, ∀t∈R+. ()

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

Lemma  If conditions(H)-(H)are satisﬁed,then operator A deﬁned by()is a

contin-uous operator from Q+into Q;moreover,for any q>p> ,A(Qpq)is relatively compact.

Proof Letu∈Q+anduB=r. Thenr>  and, by () and Remark (a),

β–rt≤u(t)≤rt, β–r≤u(t)≤r, ∀t∈R+. ()

By conditions (H), (H) and (), we have (fork∗,h∗,a(t),g(u),b(t),G(v,w,z),gr(t) and

a∗_r,b∗, see conditions (H) and (H))

ft,u(t),u(t), (Tu)(t), (Su)(t)≤a(t)gr(t) +Grb(t), ∀t∈R++, ()

where

Gr=maxg(v,w,z) :β–r≤v≤r, ≤w≤k∗r, ≤z≤h∗r, which implies the convergence of the inﬁnite integral

_∞



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and

∞



ft,u(t),u(t), (Tu)(t), (Su)(t)dt≤a∗_r+Grb∗. ()

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

¯

Ikut_k–≤Nrγk (k= , , , . . .), ()

where

Nr=maxF(v) :β–r≤v≤r,

which implies the convergence of the inﬁnite series

∞

k=

¯

Ikut_k– ()

and

∞

k=

¯

Ik

ut_k–≤Nrγ∗. ()

In addition, from () we get

(Au)(t) t ≥

 β– 

∞



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

+ ∞

k=

¯

Ikut–_k

, ∀t∈R++. ()

Moreover, by condition (H), we have

Ik(v)≤tkI¯k(v), ∀v∈R++(k= , , , . . .),

so, () gives

(Au)(t) t ≤

 β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut–_k

+

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut–_k

= β

β– 

∞



+ ∞

k=

¯

Ikut–_k

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On the other hand, by (), we have

(Au)(t) =  β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ik

ut_k–

+

t



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

<tk<t

¯

Ik

ut–_k, ∀t∈R+, ()

so,

(Au)(t)≥  β– 

∞



+ ∞

k=

¯

Ikut–_k

, ∀t∈R+ ()

and

(Au)(t)≤ β β– 

∞



+ ∞

k=

¯

Ikut–_k

, ∀t∈R+. ()

It follows from (), ()-() thatAu∈Q,i.e. Au∈DPC_[R

+,R] and

inf t∈R++

(Au)(t) t ≥β

–_Au_S,

inf t∈R+

(Au)(t)≥β–(Au)_B,

and, by (), (), () and (),

AuS≤

β β– 

a∗_r+Grb∗+Nrγ∗, ()

(Au)_B≤ β β– 

a∗_r+Grb∗+Nrγ∗

. ()

Thus, we have proved thatAmapsQ+intoQ.

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

¯

r≤ unD≤r¯ (n= , , , . . .).

So, () and () imply

β–¯r≤un(t)

t ≤¯r, β

–_¯_r_≤u(t)¯

t ≤r,¯ ∀t∈R++(n= , , , . . .) ()

and

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By (), we have

|(Aun)(t) – (Au)(t)¯ | t

≤ 

β– 

∞



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

–fs,u(s),¯ u¯(s), (Tu)(s), (S¯ u)(s)¯ ds+ ∞

k=

Ik¯u_nt_k––¯Iku¯t_k–

+

t



fs,un(s),u_n(s), (Tun)(s), (Sun)(s)–fs,u(s),¯ u¯(s), (Tu)(s), (T¯ u)(s)¯ ds

+ t

<tk<t

Iku_nt–_k–Iku¯t–_k+

<tk<t

¯Iku_nt–_k–¯Iku¯t_k–,

∀t∈R++(n= , , , . . .). ()

When  <t≤t, we have

<tk<t

Iku_nt_k––Iku¯t_k–= ,

so,

sup t∈R++

 t

<tk<t

Ik

u_nt_k––Ik

¯

ut_k–

= sup t<t<∞

 t

<tk<t

Iku_nt–_k–Iku¯t–_k

≤ 

t

∞

k=

_Ik

u_nt–_k–Iku¯t–_k. ()

It follows from () and () that

Aun–Au¯S = sup

t∈R++

|(Aun)(t) – (Au)(t)¯ | t

≤ 

t

∞

k=

Iku_nt_k––Iku¯t_k–

+ β β– 

∞



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

–fs,u(s),¯ u¯(s), (Tu)(s), (S¯ u)(s)¯ ds +

∞

k=

¯Iku_nt–_k–¯Iku¯t–_k

(n= , , , . . .). ()

It is clear that

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

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and, similar to () and observing (), we have

_f

t,un(t),u_n(t), (Tun)(t), (Sun)(t)–ft,u(t),¯ u¯(t), (Tu)(t), (S¯ u)(t)¯

≤a(t)g(t) +¯ Gb(t)¯ =σ(t), ∀t∈R++(n= , , , . . .), ()

where

¯

g(t) =maxg(u) :β–¯rt≤u≤rt¯,

¯

G(t) =maxg(v,w,z) :β–¯r≤v≤r, ¯ ≤w≤k∗r, ¯ ≤z≤h∗¯r. It is easy to see that condition (H) implies

a∗_pq=

_∞



a(t)gpq(t)dt<∞ ()

for anyq>p> , where

gpq(t) =max

g(u) :β–pt≤u≤qt. () So,

∞



a(t)g(t)dt¯ <∞,

and therefore,

∞



σ(t)dt<∞. ()

It follows from (), (), () and the dominated convergence theorem that

lim n→∞

∞



ft,un(t),u_n(t), (Tun)(t), (Sun)(t)–ft,u(t),¯ u¯(t), (Tu)(t), (S¯ u)(t)¯ dt

= . ()

On the other hand, similar to () and observing (), we have

¯

Iku_nt_k–≤ ¯Nrγk, Ik¯u¯t_k–≤ ¯Nrγk (k,n= , , , . . .), () where

¯

Nr=maxF(v) :β–r¯≤v≤r¯.

For any given> , by () and condition (H), we can choose a positive integerksuch

that

∞

k=k+

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and

∞

k=k+

tkI¯k

¯

ut_k–<,

so,

∞

k=k+

Iku_nt_k–< (n= , , , . . .), () ∞

k=k+

Iku¯t_k–<, ()

∞

k=k+

¯

Iku_nt_k–≤  t

∞

k=k+

tkIk¯u_nt_k–<t_– ()

and

∞

k=k+

¯

Ik

¯

ut_k–≤  t

∞

k=k+

tkI¯k

¯

ut_k–<t–_ . ()

It is clear that

Iku_nt_k–→Iku¯t–_k asn→ ∞(k= , , , . . .) and

¯

Iku_nt_k–→ ¯Iku¯t–_k asn→ ∞(k= , , , . . .), so, we can choose a positive integernsuch that

k

k=

Iku_nt–_k–Iku¯t–_k<, ∀n>n ()

and

k

k=

¯Iku_nt–_k–¯Iku¯t–_k<, ∀n>n. ()

From ()-(), we get

∞

k=

Iku_nt–_k–Iku¯t–_k< , ∀n>n

and

∞

k=

(12)

hence

lim n→∞

∞

k=

Iku_nt_k––Iku¯t_k–=  ()

and

lim n→∞

∞

k=

I¯k

u_nt_k––¯Ik

¯

ut_k–= . ()

It follows from (), (), () and () that

lim

n→∞Aun–Au¯S= . ()

On the other hand, from () it is easy to get

(Aun)– (Au)¯ _B≤ β β– 

∞



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

–fs,u(s),¯ u¯(s), (Tu)(s), (S¯ u)(s)¯ ds +

∞

k=

¯Iku_nt–_k–¯Iku¯t–_k

. ()

So, (), () and () imply

lim n→∞(Aun)

_{– (A}_u)_¯

B= . ()

It follows from () and () thatAun–Au¯D→ asn→ ∞, and the continuity ofAis proved.

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

β–pt≤ ¯un(t)≤qt, β–p≤ ¯un(t)≤q, ∀t∈R+(n= , , , . . .). ()

Similar to (), (), () and observing (), we have

ft,un(t),¯ u¯_n(t), (Tun)(t), (S¯ un)(t)¯

≤a(t)gpq(t) +Gpqb(t), ∀t∈R++(n= , , , . . .), ()

¯

Iku¯_nt_k–≤Npqγk (k,n= , , , . . .) () and

Au¯nS≤ β β– 

a∗_pq+Gpqb∗+Npqγ∗

(n= , , , . . .), ()

wheregpq(t) anda∗_pqare deﬁned by () and (), respectively, and

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From () we see that functions{(Aun)(t)¯ }(n= , , , . . .) are uniformly bounded on [,r] for anyr> . On the other hand, by () and ()-(), we have

≤(Aun)¯ t– (Aun)(t)¯ = t

_–_t

β– 

∞



fs,un(s),¯ u¯_n(s), (Tun)(s), (S¯ un)(s)¯ ds+ ∞

k=

¯

Iku¯_nt_k–

+t–t t



fs,un(s),¯ u¯_n(s), (Tun)(s), (S¯ un)(s)¯ ds

+

t t

t–sfs,u¯n(s),u¯n(s), (Tu¯n)(s), (Su¯n)(s)

ds

+t–t

<tk<t

¯

Ik

¯

u_nt–_k

≤ t–t

β– 

+t–ta∗_pq+Gpqb∗

+ (tk–tk–)

t t

a(s)gpq(s) +Gpqb(s)

ds+t–tNpqγ∗,

∀t,t∈Jk,t>t(k,n= , , , . . .),

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

wn(t) =

(Aun)(t),¯ ∀t∈Jk= (tk–,tk],

(Aun)(t¯ +

k–), ∀t=tk–

(n= , , , . . .)

((Aun¯ )(t+

k–) denotes the right limit of (Aun¯ )(t) at t=tk–) are equicontinuous on ¯Jk=

[tk–,tk]. Consequently, by the Ascoli-Arzela theorem,{wn(t)}has a subsequence which is convergent uniformly onJ¯k. So, functions{Au¯n(t)}(n= , , , . . .) have a subsequence which is convergent uniformly on Jk. Now, by the diagonal method, we can choose a subsequence{(Au¯ni)(t)} (i= , , , . . .) of{(Au¯n)(t)} (n= , , , . . .) such that {(Au¯ni)(t)} (i= , , , . . .) is convergent uniformly on eachJk(k= , , , . . .). Let

lim

i→∞(Au¯ni)(t) =w(t),¯ ∀t∈R+. ()

Similarly, we can discuss{(Aun)¯ (t)}(n= , , , . . .). Similar to () and by (), we have

(Au¯n)_B≤ β β– 

(n= , , , . . .) ()

and

≤(Aun)¯ t– (Aun)¯ (t) =

t t

fs,un(s),¯ u¯_n(s), (Tun)(s), (S¯ un)(s)¯ ds

≤ t

t

a(s)gpq(s) +Gpqb(s)ds, ∀t,t∈Jk,t>t(n= , , , . . .),

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no-tation, we may assume that{(Auni¯ )(t)}(i= , , , . . .) itself converges uniformly on eachJk (k= , , , . . .). Let

lim i→∞(Au¯ni)

_{(t) =}_y(t). _()

By (), () and the uniformity of convergence, we have

¯

w(t) =y(t), ∀t∈R+, ()

and so,w¯ ∈PC[R+,R]. From () and (), we get

¯wS≤ β β– 

and

w_¯_B≤ β β– 

a∗_pq+Gpqb∗+Npqγ∗.

Consequently,w¯ ∈DPC_[R

+,R] and

¯wD≤ β β– 

a∗_pq+Gpqb∗+Npqγ∗.

Let>  be arbitrarily given. Choose a suﬃciently large positive numberηsuch that

∞

η

a(t)gpq(t)dt+Gpq

∞

η

b(t)dt+Npq tk≥η

γk<. ()

For anyη<t<∞, we have, by (), () and (),

≤(Auni¯ )(t) – (Auni¯ )(η)

=

t

η

fs,uni¯ (s),u¯_ni(s), (Tuni¯ )(s), (Suni¯ )(s)ds+

η≤tk<t

¯

Iku¯_nit–_k

≤

t

η

a(s)gpq(s)ds+Gpq

t

η

b(s)ds+Npq

η≤tk<t

γk (i= , , , . . .),

which implies by virtue of () that

≤(Auni¯ )(t) – (Auni¯ )(η) <, ∀t>η(i= , , , . . .). ()

Lettingi→ ∞in () and observing () and (), we get

≤ ¯w(t) –w¯(η)≤, ∀t>η. ()

On the other hand, since{(Au¯ni)(t)}converges uniformly tow¯(t) on [,η] asi→ ∞, there exists a positive integerisuch that

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It follows from ()-() that

(Auni¯ )(t) –w¯(t)≤(Auni¯ )(t) – (Auni¯ )(η)+(Auni¯ )(η) –w¯(η)

+w¯(η) –w¯(t)< , ∀t>η,i>i. ()

By () and (), we have

(Au¯ni)–w¯_B≤, ∀i>i,

hence

lim i→∞(Auni¯ )

_–_w_¯

B= . ()

It is clear that () implies

(Auni¯ )t_k+– (Auni¯ )t_k–=Iku¯_nit_k– (k,i= , , , . . .). ()

By virtue of the uniformity of convergence of{(Auni¯ )(t)}, we see that

lim i→∞(Au¯ni)

t–_k=w¯t–_k, lim i→∞(Au¯ni)

t_k+=w¯t_k+ (k= , , , . . .),

so, () implies that

lim i→∞Ik

¯

u_nit_k– (k= , , , . . .)

exist and

¯

wt+_k–w¯t_k–=lim i→∞Ik

¯

u_n i

t_k– (k= , , , . . .).

Let

lim i→∞Ik

¯

u_nit_k–=αk (k= , , , . . .).

Thenαk≥ (k= , , , . . .) and

¯

wt+_k–w¯t_k–=αk (k= , , , . . .). ()

By () and condition (H), we have

Ik

¯

u_nit–_k≤Npqtkγk (k,i= , , , . . .), ()

so,

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For any given> , choose a suﬃciently large positive integerksuch that

Npq ∞

k=k+

tkγk<, ()

and then, choose another suﬃciently large integerisuch that

Iku¯_nit_k––αk< k

, ∀i>i(k= , , . . . ,k). ()

It follows from ()-() that

∞

k=

Iku¯_nit_k––αk≤ k

k=

Iku¯_nit–_k–αk

+ ∞

k=k+

Iku¯_nit–_k+ ∞

k=k+

αk< , ∀i>i,

hence

lim i→∞

∞

k=

Ik

¯

u_nit–_k–αk= . ()

By formula () and (), (), we have

(Auni¯ )(t) =

t



(Auni¯ )(s)ds+

<tk<t

Iku¯_nit_k–, ∀t∈R+(i= , , , . . .)

and

¯

w(t) =

t

 ¯

w(s)ds+

<tk<t

αk, ∀t∈R+,

which imply

(Auni¯ )(t) –w(t)¯ ≤t(Auni¯ )–w¯_B

+

<tk<t

Iku¯_nit_k––αk, ∀t∈R+(i= , , , . . .). ()

Since

<tk<t

Iku¯_nit_k––αk= , ∀ <t≤t,

() implies

Auni¯ –w¯S≤(Auni¯ )–w¯_B+t–

∞

k=

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By (), () and (), we have

lim

i→∞Auni¯ –w¯S= . ()

It follows from () and () thatAu¯ni–w¯D→ asi→ ∞, and the relative compactness

ofA(Qpq) is proved.

Lemma  Let conditions (H)-(H)be satisﬁed.Then u∈Q+∩C[R++,R]is a positive

solution of IBVP()if and only if u∈Q+is a solution of the following impulsive integral

equation:

u(t) = t β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut–_k

+

t



+

<tk<t

Ikut–_k+ (t–tk)¯Ikut–_k, ∀t∈R+, ()

i.e. u is a ﬁxed point of operator A deﬁned by().

Proof Ifu∈Q+∩C[R++,R] is a positive solution of IBVP (), then, by () and formula

(), we have

u(t) =tu() +

t



+

<tk<t

Ikut–_k+ (t–tk)¯Ikut–_k, ∀t∈R+. ()

Diﬀerentiation of () gives

u(t) =u() +

t



<tk<t

¯

Ik

ut–

k

, ∀t∈R+. ()

Under conditions (H)-(H), 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 () and using the relationu(∞) =βu(), we get

u() =  β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut–_k

. ()

Now, substituting () into (), we see thatu(t) satisﬁes equation ().

Conversely, ifu∈Q+ is a solution of equation (), then direct diﬀerentiation of ()

twice gives

u(t) =  β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut_k–

+

t



<tk<t

¯

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and

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

So,u∈C_[R

++,R] and

u|t=tk=Ik

ut_k–, u|t=tk=¯Ik

ut_k– (k= , , , . . .).

Moreover, taking limits ast→ ∞in (), we see thatu(∞) exists and

u(∞) = β β– 

∞



fs,u(s),u(s), (Tu)(s), (Su)(s)ds+ ∞

k=

¯

Ikut–_k

=βu().

Hence,u(t) is a positive solution of IBVP ().

Lemma (Fixed point theorem of cone expansion and compression with norm type, see Corollary  in [] or Theorem  in [] or Theorem .. in [], see also [, ]) Let P be a cone in a real Banach space E and,be two bounded open sets in E such that

θ ∈,¯⊂,whereθ denotes the zero element of E and¯idenotes the closure ofi (i= , ).Let the operator A:P∩(¯\)→P be completely continuous(i.e. continuous

and compact).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∩(¯\).

3 Main theorem

Theorem Let conditions(H)-(H)be satisﬁed.Assume that there exists r> such that

β β– 

a∗_r+Grb∗+Nrγ∗<r, ()

where a∗_r,b∗andγ∗are deﬁned in conditions(H)and(H),and,Grand Nrare deﬁned

by two equalities below()and(),respectively.Then IBVP()has at least two positive solutions u∗,u∗∗∈Q+∩C[R++,R]such that

 < inf t∈R++

u∗(t) t ≤t∈Rsup++

u∗(t) t <r,

 < inf t∈R+

u∗(t)≤sup t∈R+

u∗(t) <r,

β–r< inf t∈R++

u∗∗(t) t ≤t∈Rsup++

u∗∗(t) t <∞

and

β–r< inf t∈R+

u∗∗(t)≤sup t∈R+

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

Q, and we need to prove thatAhas two ﬁxed pointsu∗andu∗∗inQ+such that  <u∗D< r<u∗∗D.

By condition (H), there existsr>  such that

f(t,u,v,w,z)≥β

₍_β_{– )}

c∗ c(t)v, ∀t,u∈R++,v≥r,w,z∈R+. ()

Choose

r>max

βr,r

. ()

Foru∈Q,uD=r, we have, by () and (),

u(t)≥β–r>r, ∀t∈R+,

so, () and () imply

(Au)(t)≥  β– 

∞



≥β

c∗

_∞



c(s)u(s)ds≥r c∗

_∞



c(s)ds=r, ∀t∈R+,

and consequently,

(Au)_B≥r,

hence

AuD≥ uD, ∀u∈Q,uD=r. ()

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

f(t,u,v,w,z)≥(β– )r

d∗ d(t), ∀t,u∈R++,  <v<r,w,z∈R+. ()

Choose

 <r<min{r,r}. ()

Foru∈Q,uD=r, we have, by () and (),

r>r≥u(t)≥β–r> ,

so, we get, by () and (),

(Au)(t)≥  β– 

∞



≥ r

d∗

_∞



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hence

(Au)_B>r,

and consequently,

AuD>uD, ∀u∈Q,uD=r. ()

On the other hand, foru∈Q,uD=r, () and () imply

AuD≤

β β– 

a∗_r+Grb∗+Nrγ∗. ()

Thus, from () and (), we get

AuD<uD, ∀u∈Q,uD=r. ()

By () and () we know  <r<r<r, and, by Lemma , operatorAis completely

con-tinuous fromQrrintoQ. Hence, (), (), () and Lemma  imply thatAhas two ﬁxed

pointsu∗,u∗∗∈Qrr such thatr<u∗D<r<u∗∗D≤r. The proof is complete.

Example Consider the inﬁnite boundary value problem for second order impulsive sin-gular integro-diﬀerential equation of mixed type on the half line:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(t) = e–t

t

(  [u(t)]

+ 

u(t)+ [u(t)] _{) +} e–t

t{

(_te–(t+)s_u(s)_ds)

+ (_∞_(+u(s_t)₊ds_s₎)}, ∀ <t<∞,t=k(k= , , , . . .),

u|t=k= –k– 

u(k–₎₊√_u₍_k–₎ (k= , , , . . .),

u|t=k=k––k–√

u(k–₎ (k= , , , . . .),

u() = , u(∞) = u().

()

Conclusion IBVP () has at least two positive solutionsu∗,u∗∗∈PC[R+,R]∩C[R++,R]

such that

 < inf

<t<∞ u∗(t)

t ≤<supt<∞ u∗(t)

t < ,

 < inf

≤t<∞

u∗(t)≤ sup

≤t<∞

u∗(t) < , 

<<inft<∞ u∗∗(t)

t ≤<supt<∞ u∗∗(t)

t <∞

and

 <≤tinf<∞

u∗∗(t)≤ sup

≤t<∞

u∗∗(t) <∞.

Proof System () is an IBVP of form (). In this situation,tk=k(k= , , , . . .),K(t,s) = e–(t+)s_,_H(t,_{s) = ( +}_t₊_s)–_,_β_{= , and}

f(t,u,v,w,z) = e

–t

t



u

+ v+v



+ e

–t

t

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Ik(v) = –k– 

v+√v, ∀v∈R++(k= , , , . . .),

¯

Ik(v) =k––k–√

v, ∀v∈R++(k= , , , . . .).

It is clear that ()-() are satisﬁed, so, () is a singular problem. It is easy to see that condition (H) is satisﬁed andk∗≤,h∗≤. We have

f(t,u,v,w,z)≤ e

–t

t



u

+e

–t

t

 

 v+v



+  

w+z,

so, condition (H) is satisﬁed for

a(t) = e

–t

t

, g(u) =  u

, b(t) =e

–t

t

and

g(v,w,z) =  

 v+v



+  

w+z

with

gr(t) =max u– _:rt

 ≤u≤rt

=  r  

t–_,

a∗_r=

∞



a(t)gr(t)dt=    r   ∞ 

e–t

t

dt<∞

()

and

b∗=

∞



e–t

t

dt<∞. ()

It is obvious that condition (H) is satisﬁed forγk=k––k– (γ∗= _ ) andF(v) =v–

 .

From

f(t,u,v,w,z)≥ e

–t

t

v, ∀t,u,v∈R++,w,z∈R+

and

f(t,u,v,w,z)≥ e

–t

t



v, ∀t,u,v∈R++,w,z∈R+,

we see that conditions (H) and (H) are satisﬁed for

c(t) = e

–t

t

c∗=b∗, see ()

and

d(t) =e

–t

t

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respectively. Finally, we check that inequality () is satisﬁed forr= ,i.e.

a∗_+Gb∗+Nγ∗

< . ()

By () and (), we have

a∗_ =

 



∞



e–t

t

dt<

     dt

t

+

∞



e–tdt

=     +  e – <      +   =  , and

b∗<

 

dt

t

+

∞



e–tdt= +

 e

–_<

.

Moreover, it is easy to get

G<



, N= .

Hence

a∗_+Gb∗+Nγ∗

<   ,+  ,+   = , ,< .

Consequently, () holds, and our conclusion follows from the theorem.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author thanks the reviewers for valuable suggestions.

Received: 8 December 2014 Accepted: 21 April 2015 References

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