# Multiple Positive Solutions for First-Order Impulsive Integrodifferential Equations on the Half Line in Banach Spaces

Authors: Dajun Guo
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.


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## Body

## 1. Introduction

LetE be a real Banach space and P a cone in E which defines a partial ordering in E by x≤y if and only if y-x∈P. P is said to be normal if there exists a positive constant N such that θ≤x≤y implies ||x||≤N||y||, where θ denotes the zero element of E and the smallest N is called the normal constant of P. If x≤y and x≠y, we write x<y. For details on cone theory, see [1].In paper [2], we considered the infinite boundary value problem (IBVP) for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line in E: (1.1)u′(t)=f(t,u(t),(Tu)(t),(Su)(t)),∀t∈J′,Δu|t=tk=Ik(u(tk))(k=1,2,3,…),u(∞)=βu(0),
where J=[0,∞), 0<t1<⋯<tk<…,tk→∞, J′=J∖{t1,…,tk,…}, f∈C[J×P×P×P,P], Ik∈C[P,P] (k=1,2,3,…), β>1, u(∞)=limt→∞u(t), and (1.2)(Tu)(t)=∫0tK(t,s)u(s)ds,(Su)(t)=∫0∞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 nonnegative numbers. Δu|t=tk denotes the jump of u(t) at t=tk, that is, (1.3)Δu|t=tk=u(tk+)-u(tk-),
where u(tk+) and u(tk-) represent the right and left limits of u(t) at t=tk, respectively. By using the fixed point index theory, we discussed the multiple positive solutions of IBVP(1.1). But the discussion dealt with sublinear equations, that is, we assume that there exists c∈C[J,R+]∩L[J,R+] such that (1.4)‖f(t,u,v,w)‖c(t)(‖u‖+‖v‖+‖w‖)⟶0asu,v,w∈P,‖u‖+‖v‖+‖w‖⟶∞
uniformly for t∈J (see condition (H5) in [2]).Now, in this paper, we discuss the multiple positive solutions of an infinite three-point boundary value problem (which includes IBVP(1.1) as a special case) for superlinear case by means of different method, that is, by using the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [3] (see also [1]), and the key point is to introduce a new cone Q.Consider the infinite three-point boundary value problem for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line inE: (1.5)u′(t)=f(t,u(t),(Tu)(t),(Su)(t)),∀t∈J′,Δu|t=tk=Ik(u(tk))(k=1,2,3,…),u(∞)=γu(η)+βu(0),
where 0≤γ<1,β+γ>1, and tm-1<η≤tm (for some m). It is clear that IBVP(1.5) includes IBVP(1.1) as a special case when γ=0.LetPC[J,E]={u:u is a map from J into E such that u(t) is continuous at t≠tk, left continuous at t=tk, and u(tk+) exists, k=1,2,3,…} and BPC[J,E]={u∈PC[J,E]:supt∈J||u(t)||<∞}. It is clear that BPC[J,E] is a Banach space with norm (1.6)‖u‖B=sup⁡t∈J‖u(t)‖.
Let BPC[J,P]={u∈BPC[J,E]:u(t)≥θ,∀t∈J} and Q={u∈BPC[J,P]:u(t)≥β-1(1-γ)u(s),∀t,s∈J}. Obviously, BPC[J,P] and Q are two cones in space BPC[J,E] and Q⊂BPC[J,P]. u∈BPC[J,P]∩C1[J′,E] is called a positive solution of IBVP(1.5) if u(t)>θ for t∈J and u(t) satisfies (1.5).


## 2. Several Lemmas

Let us list some conditions.(H1)
supt∈J∫0tK(t,s)ds<∞,supt∈J∫0∞H(t,s)ds<∞, and
(2.1)lim⁡t′→t∫0∞|H(t′,s)-H(t,s)|ds=0,∀t∈J.
In this case, let
(2.2)k*=sup⁡t∈J∫0tK(t,s)ds,h*=sup⁡t∈J∫0∞H(t,s)ds.(H2)
There exista∈C[J,R+] and g∈C[R+×R+×R+,R+] such that
(2.3)‖f(t,u,v,w)‖≤a(t)g(‖u‖,‖v‖,‖w‖),∀t∈J,u,v,w∈P,a*=∫0∞a(t)dt<∞.(H3)
There existγk≥0 (k=1,2,3,…) and F∈C[R+,R+] such that
(2.4)‖Ik(u)‖≤γkF(‖u‖),∀u∈P(k=1,2,3,…),γ*=∑k=1∞γk<∞.(H4)
For anyt∈J and r>0,f(t,Pr,Pr,Pr)={f(t,u,v,w):u,v,w∈Pr} and Ik(Pr)={Ik(u):u∈Pr} (k=1,2,3,…) are relatively compact in E, where Pr={u∈P:||u||≤r}.Remark 2.1.
Obviously, condition (H4) is satisfied automatically when E is finite dimensional.Remark 2.2.
It is clear that if condition(H1) is satisfied, then the operators T and S defined by (1.2) are bounded linear operators from BPC[J,E] into BPC[J,E] and ||T||≤k*,||S||≤h*; moreover, we have T(BPC[J,P])⊂BPC[J,P] and S(BPC[J,P])⊂BPC[J,P].We shall reduce IBVP(1.5) to an impulsive integral equation. To this end, we consider the operator A defined by (2.5)(Au)(t)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))}+∫0tf(s,u(s),(Tu)(s),(Su)(s))ds+∑0<tk<tIk(u(tk)),∀t∈J.
In what follows, we write J1=[0,t1],Jk=(tk-1,tk] (k=2,3,4,…).Lemma 2.3.
If conditions (H1)–(H4) are satisfied, then operator A defined by (2.5) is a completely continuous (i.e., continuous and compact) operator from BPC[J,P] into Q.Proof.
Letr>0 be given. Let
(2.6)Mr=max⁡{g(x,y,z):0≤x≤r,0≤y≤k*r,0≤z≤h*r},(2.7)Nr=max⁡{F(x):0≤x≤r}.
For u∈BPC[J,P],||u||B≤r, we see that by virtue of condition (H2) and (2.6),
(2.8)‖f(t,u(t),(Tu)(t),(Su)(t))‖≤Mra(t),∀t∈J,
which implies the convergence of the infinite integral
(2.9)∫0∞f(t,u(t),(Tu)(t),(Su)(t))dt,(2.10)‖∫0∞f(t,u(t),(Tu)(t),(Su)(t))dt‖≤∫0∞‖f(t,u(t),(Tu)(t),(Su)(t))‖dt≤Mra*.
On the other hand, condition (H3) and (2.7) imply the convergence of the infinite series
(2.11)∑k=1∞Ik(u(tk)),(2.12)‖∑k=1∞Ik(u(tk))‖≤∑k=1∞‖Ik(u(tk))‖≤Nrγ*.
It follows from (2.5), (2.10), and (2.12) that
(2.13)||(Au)(t)||≤1β+γ-1{∫η∞‖f(s,u(s),(Tu)(s),(Su)(s))‖ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0η‖f(s,u(s),(Tu)(s),(Su)(s))‖ds+∑k=m∞‖Ik(u(tk))‖+(1-γ)∑k=1m-1‖Ik(u(tk))‖}+∫0t‖f(s,u(s),(Tu)(s),(Su)(s))‖ds+∑0<tk<t‖Ik(u(tk))‖≤1β+γ-1{∫0∞‖f(s,u(s),(Tu)(s),(Su)(s))‖ds+∑k=1∞||Ik(u(tk))||}+∫0∞‖f(s,u(s),(Tu)(s),(Su)(s))‖ds+∑k=1∞||Ik(u(tk))||=β+γβ+γ-1{∫0∞‖f(s,u(s),(Tu)(s),(Su)(s))‖ds+∑k=1∞||Ik(u(tk))||}≤β+γβ+γ-1(Mra*+Nrγ*),∀t∈J,
which implies that Au∈BPC[J,P] and
(2.14)‖Au‖B≤β+γβ+γ-1(Mra*+Nrγ*).
Moreover, by (2.5), we have
(2.15)(Au)(t)≥1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))},∀t∈J,(2.16)(Au)(t)≤1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))}+∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1∞Ik(u(tk)),∀t∈J.
It is clear that
(2.17)∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1∞Ik(u(tk))≤11-γ{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))},
so, (2.16) and (2.17) imply
(2.18)(Au)(t)≤{1β+γ-1+11-γ}×{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))},∀t∈J.
It follows from (2.15) and (2.18) that
(2.19)(Au)(t)≥1β+γ-1(1β+γ-1+11-γ)-1(Au)(s)=β-1(1-γ)(Au)(s),∀t,s∈J.
Hence, Au∈Q. That is, A maps BPC[J,P] into Q.
Now, we are going to show thatA is continuous. Let un,u̅∈BPC[J,P],||un-u̅||B→0 (n→∞). Then r=supn||un||B<∞ and ||u̅||B≤r. Similar to (2.14), it is easy to get
(2.20)‖Aun-Au̅‖B≤β+γβ+γ-1{+∑k=1∞‖Ik(un(tk))-Ik(u̅(tk))‖∫0∞‖f(s,un(s),(Tun)(s),(Sun)(s))-f(s,u̅(s),(Tu̅)(s),(Su̅)(s))ds‖+∑k=1∞‖Ik(un(tk))-Ik(u̅(tk))‖}(n=1,2,3,…).
It is clear that
(2.21)f(t,un(t),(Tun)(t),(Sun)(t))⟶f(t,u̅(t),(Tu̅)(t),(Su̅)(t))asn⟶∞,∀t∈J.
Moreover, we see from (2.8) that
(2.22)‖f(t,un(t),(Tun)(t),(Sun)(t))-f(t,u̅(t),(Tu̅)(t),(Su̅)(t))‖≤2Mra(t)=σ(t),∀t∈J(n=1,2,3,…);σ∈L[J,R+].
It follows from (2.21), (2.22) and the dominated convergence theorem that
(2.23)lim⁡n→∞∫0∞‖f(t,un(t),(Tun)(t),(Sun)(t))-f(t,u̅(t),(Tu̅)(t),(Su̅)(t))‖dt=0.
On the other hand, for any ϵ>0, we can choose a positive integer j such that
(2.24)Nr∑k=j+1∞γk<ϵ.
And then, choose a positive integer n0 such that
(2.25)∑k=1j‖Ik(un(tk))-Ik(u̅(tk))‖<ϵ,∀n>n0.
From (2.24) and (2.25), we get
(2.26)∑k=1∞‖Ik(un(tk))-Ik(u̅(tk))‖<ϵ+2Nr∑k=j+1∞γk<3ϵ,∀n>n0,
hence
(2.27)lim⁡n→∞∑k=1∞‖Ik(un(tk))-Ik(u̅(tk))‖=0.
It follows from (2.20), (2.23), and (2.51) that ||Aun-Au̅||B→0asn→∞, and the continuity of A is proved.
Finally, we prove thatA is compact. Let V={un}⊂BPC[J,P] be bounded and ||un||B≤r (n=1,2,3,…). Consider Ji=(ti-1,ti] for any fixed i. By (2.5) and (2.8), we have
(2.28)‖(Aun)(t′)-(Aun)(t)‖≤∫tt′‖f(s,un(s),(Tun)(s),(Sun)(s))‖ds≤Mr∫tt′a(s)ds,∀t,t′∈Ji,t′>t(n=1,2,3,…),
which implies that the functions {wn(t)} (n=1,2,3,…) defined by
(2.29)wn(t)={(Aun)(t),∀t∈Ji=(ti-1,ti],(Aun)(ti-1+),∀t=ti-1(n=1,2,3,…)
((Aun)(ti-1+) denotes the right limit of (Aun)(t) at t=ti-1) are equicontinuous on J̅i=[ti-1,ti]. On the other hand, for any ϵ>0, choose a sufficiently large τ>η and a sufficiently large positive integer j>m such that
(2.30)Mr∫τ∞a(s)ds<ϵ,Nr∑k=j+1∞γk<ϵ.
We have, by (2.29), (2.5), (2.8), (2.30), and condition (H3),
(2.31)wn(t)=1β+γ-1{+∑k=j+1∞Ik(un(tk))+(1-γ)∑k=1m-1Ik(un(tk))∫ητf(s,un(s),(Tun)(s),(Sun)(s))ds+∫τ∞f(s,un(s),(Tun)(s),(Sun)(s)ds)+(1-γ)×∫0ηf(s,un(s),(Tun)(s),(Sun)(s))ds+∑k=mjIk(un(tk))+∑k=j+1∞Ik(un(tk))+(1-γ)∑k=1m-1Ik(un(tk))}+∫0tf(s,un(s),(Tun)(s),(Sun)(s))ds+∑k=1i-1Ik(un(tk)),∀t∈J̅i(n=1,2,3,…),(2.32)‖∫τ∞f(s,un(s),(Tun)(s),(Sun)(s))ds‖<ϵ(n=1,2,3,…),(2.33)‖∑k=j+1∞Ik(un(tk))‖<ϵ(n=1,2,3,…).
It follows from (2.31), (2.32), (2.33), (2.8), and [4, Theorem 1.2.3] that
(2.34)α(W(t))≤1β+γ-1{2∫ητα(f(s,V(s),(TV)(s),(SV)(s)))ds∑k=m∞Ik(u(tk))+(1-γ)+2ϵ+2(1-γ)∫0ηα(f(s,V(s),(TV)(s),(SV)(s)))ds+∑k=mjα(Ik(V(tk)))+2ϵ+(1-γ)∑k=1m-1α(Ik(V(tk)))}+2∫0tα(f(s,V(s),(TV)(s),(SV)(s)))ds+∑k=1i-1α(Ik(V(tk))),∀t∈J̅i,
where W(t)={wn(t):n=1,2,3,…}, V(s)={un(s):n=1,2,3,…}, (TV)(s)={(Tun)(s):n=1,2,3,…}, (SV)(s)={(Sun)(s):n=1,2,3,…} and α(U) denotes the Kuratowski measure of noncompactness of bounded set U⊂E (see [4, Section 1.2]). Since V(s),(TV)(s),(SV)(s)⊂Pr* for s∈J, where r*=max{r,k*r,h*r}, we see that, by condition (H4),
(2.35)α(f(s,V(s),(TV)(s),(SV)(s)))=0,∀t∈J,(2.36)α(Ik(V(tk)))=0(k=1,2,3,…).
It follows from (2.34)–(2.36) that
(2.37)α(W(t))≤4ϵβ+γ-1,∀t∈J̅i,
which implies by virtue of the arbitrariness of ϵ that α(W(t))=0 for t∈J̅i.
By Ascoli-Arzela theorem (see [4, Theorem 1.2.5]), we conclude that W={wn:n=1,2,3,…} is relatively compact in C[J̅i,E]; hence, {wn(t)} has a subsequence which is convergent uniformly on J̅i, so, {(Aun(t)} has a subsequence which is convergent uniformly on Ji. Since i may be any positive integer, so, by diagonal method, we can choose a subsequence {(Auni)(t)} of {(Aun)(t)} such that {(Auni)(t)} is convergent uniformly on each Jk (k=1,2,3,…). Let
(2.38)lim⁡i→∞(Auni)(t)=v(t),∀t∈J.
It is clear that v∈PC[J,P]. By (2.14), we have
(2.39)‖Auni‖B≤β+γβ+γ-1(Mra*+Nrγ*),(i=1,2,3,…),
which implies that v∈BPC[J,P] and
(2.40)‖v‖B≤β+γβ+γ-1(Mra*+Nrγ*).
Let ϵ>0 be arbitrarily given and choose a sufficiently large positive number τ such that
(2.41)Mr∫τ∞a(s)ds+Nr∑tk≥τγk<ϵ.
For any τ<t<∞, we have, by (2.5),
(2.42)(Auni)(t)-(Auni)(τ)=∫τtf(s,uni(s),(Tuni)(s),(Suni)(s))ds+∑τ≤tk<tIk(uni(t)),(i=1,2,3,…),
which implies by virtue of (2.8), condition (H3) and (2.41) that
(2.43)‖(Auni)(t)-(Auni)(τ)‖≤Mr∫τta(s)ds+Nr∑τ≤tk<tγk<ϵ,(i=1,2,3,…).
Letting i→∞ in (2.43), we get
(2.44)‖v(t)-v(τ)‖≤ϵ,∀t>τ.
On the other hand, since {(Auni)(t)} converges uniformly to v(t) on [0,τ] as i→∞, there exists a positive integer i0 such that
(2.45)‖(Auni)(t)-v(t)‖<ϵ,∀t∈[0,τ],i>i0.
It follows from (2.43)–(2.45) that
(2.46)‖(Auni)(t)-v(t)‖≤‖(Auni)(t)-(Auni)(τ)‖+‖(Auni)(τ)-v(τ)‖+‖v(τ)-v(t)‖<3ϵ,∀t>τ,i>i0.
By (2.45) and (2.46), we have
(2.47)‖Auni-v‖B≤3ϵ,∀i>i0,
hence ||Auni-v||B→0 as i→∞, and the compactness of A is proved.Lemma 2.4.
Let conditions (H1)–(H4) be satisfied. Then u∈BPC[J,P]∩C1[J′,E] is a solution of IBVP(1.5) if and only if u∈Q is a solution of the following impulsive integral equation:
(2.48)u(t)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))}+∫0tf(s,u(s),(Tu)(s),(Su)(s))ds+∑0<tk<tIk(u(tk)),∀t∈J.
that is, u is a fixed point of operator A defined by (2.5) in Q.Proof.
Foru∈PC[J,E]∩C1[J′,E], it is easy to get the following formula:
(2.49)u(t)=u(0)+∫0tu′(s)ds+∑0<tk<t[u(tk+)-u(tk)],∀t∈J.
Letu∈BPC[J,P]∩C1[J′,E] be a solution of IBVP(1.5). By (1.5) and (2.49), we have
(2.50)u(t)=u(0)+∫0tf(s,u(s),(Tu)(s),(Su)(s))ds+∑0<tk<tIk(u(tk)),∀t∈J.
We have shown in the proof of Lemma 2.3 that the infinite integral (2.9) and the infinite series (2.11) are convergent, so, by taking limits as t→∞ in both sides of (2.50), we get
(2.51)u(∞)=u(0)+∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1∞Ik(u(tk)).
On the other hand, by (1.5) and (2.50), we have
(2.52)u(∞)=γu(η)+βu(0),(2.53)u(η)=u(0)+∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1m-1Ik(u(tk)).
It follows from (2.51)–(2.53) that
(2.54)u(0)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))},
and, substituting it into (2.50), we see that u(t) satisfies (2.48), that is, u=Au. Since Au∈Q by virtue of Lemma 2.3, we conclude that u∈Q.
Conversely, assume thatu∈Q is a solution of (2.48). We have, by (2.48),
(2.55)u(0)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))},(2.56)u(η)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)×∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))}+∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1m-1Ik(u(tk)).
Moreover, by taking limits as t→∞ in (2.33), we see that u(∞) exists and
(2.57)u(∞)=1β+γ-1{∫η∞f(s,u(s),(Tu)(s),(Su)(s))ds∑k=m∞Ik(u(tk))+(1-γ)+(1-γ)∫0ηf(s,u(s),(Tu)(s),(Su)(s))ds+∑k=m∞Ik(u(tk))+(1-γ)∑k=1m-1Ik(u(tk))}+∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1∞Ik(u(tk)).
It follows from (2.55)–(2.57) that
(2.58)γu(η)+βu(0)=u(∞).
On the other hand, direct differentiation of (2.48) gives
(2.59)u′(t)=f(t,u(t),(Tu)(t),(Su)(t)),∀t∈J′,
and, it is clear, by (2.48),
(2.60)Δu|t=tk=Ik(u(tk))(k=1,2,3,…).
Hence, u∈C1[J′,E] and u(t) satisfies (1.5).Corollary 2.5.
Let coneP be normal. If u is a fixed point of operator A defined by (1.5) in Q and ||u||B>0, then u(t)>θ for t∈J, so, u is a positive solution of IBVP(1.5).Proof.
Foru∈Q, we have
(2.61)u(t)≥β-1(1-γ)u(s)≥θ,∀t,s∈J,
so,
(2.62)‖u(t)‖≥N-1β-1(1-γ)‖u‖B,∀t∈J,
where N denotes the normal constant of P. Since ||u||B>0, (2.61) and (2.62) imply that u(t)>θ for t∈J.Lemma 2.6 (Fixed point theorem of cone expansion and compression with norm type, see [3, Theorem  3] or [1, Theorem  2.3.4]).
LetP be a cone in real Banach space E and Ω1,Ω2 two bounded open sets in E such that θ∈Ω1, Ω̅1⊂Ω2, where θ denotes the zero element of E and Ω̅2 denotes the closure of Ω2. Let operator A:P∩(Ω̅2∖Ω1)→P be completely continuous. Suppose that one of the following two conditions is satisfied: (a)
(2.63)‖Ax‖≤‖x‖,∀x∈P∩∂Ω1;‖Ax‖≥‖x‖,∀x∈P∩∂Ω2,
where ∂Ωi denotes the boundary of  Ωi (i=1,2).(b)
(2.64)‖Ax‖≥‖x‖,∀x∈P∩∂Ω1,‖Ax‖≤‖x‖,∀x∈P∩∂Ω2.Then A has at least one fixed point in P∩(Ω̅2∖Ω1).


## 3. Main Theorems

Let us list more conditions.(H5)
There existu0∈P∖{θ},b∈C[J,R+], and τ∈C[P,R+] such that
(3.1)f(t,u,v,w)≥b(t)τ(u)u0,∀t∈J,u,v,w∈P,τ(u)‖u‖⟶∞asu∈P,‖u‖⟶∞,b*=∫0∞b(t)dt<∞.Remark 3.1.
Condition (H5) means that f(t,u,v,w) is superlinear with respect to u.(H6)
There existu1∈P∖{θ},c∈C[J,R+], and σ∈C[P,R+] such that
(3.2)f(t,u,v,w)≥c(t)σ(u)u1,∀t∈J,u,v,w∈P,σ(u)‖u‖⟶∞asu∈P,‖u‖⟶0,c*=∫0∞c(t)dt<∞.Theorem 3.2.
Let coneP be normal and conditions (H1)–(H6) satisfied. Assume that there exists a ξ>0 such that
(3.3)N(β+γ)β+γ-1(Mξa*+Nξγ*)<ξ,
where
(3.4)Mξ=max⁡{g(x,y,z):0≤x≤ξ,0≤y≤k*ξ,0≤z≤h*ξ},Nξ=max⁡{F(x):0≤x≤ξ}.
(for g(x,y,z),F(x),a* and γ*, see conditions (H2) and (H3)). Then IBVP(1.5) has at least two positive solutions u*,u**∈Q∩C1[J′,E] such that 0<||u*||B<ξ<||u**||B.Proof.
By Lemmas2.3, 2.4, and Corollary 2.5, operator A defined by (2.5) is completely continuous from Q into Q, and we need to prove that A has two fixed points u* and u** in Q such that 0<||u*||B<ξ<||u**||B.
By condition (H5), there exists an r1>0 such that
(3.5)τ(u)≥β(β+γ-1)N2(1-γ)2b*‖u0‖‖u‖,∀u∈P,‖u‖≥r1,
where N denotes the normal constant of P, so,
(3.6)f(t,u,v,w)≥β(β+γ-1)N2‖u‖(1-γ)2b*‖u0‖b(t)u0,∀t∈J,u,v,w∈P,||u||≥r1.
Choose
(3.7)r2>max⁡{Nβ(1-γ)-1r1,ξ}.
For u∈Q,||u||B=r2; we have by (2.62) and (3.7),
(3.8)‖u(t)‖≥N-1β-1(1-γ)‖u‖B=N-1β-1(1-γ)r2>r1,∀t∈J,
so, (2.5), (3.8), (3.6), and (2.62) imply
(3.9)(Au)(t)≥1-γβ+γ-1(∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds)≥βN2(1-γ)b*‖u0‖(∫0∞‖u(s)‖b(s)ds)u0≥N‖u‖Bb*‖u0‖(∫0∞b(s)ds)u0=N‖u‖B‖u0‖u0,∀t∈J,
and consequently,
(3.10)||Au||B≥||u||B,∀u∈Q,||u||B=r2.
Similarly, by condition (H6), there exists r3>0 such that
(3.11)σ(u)≥β(β+γ-1)N2(1-γ)2c*||u1||||u||,∀u∈P,0<||u||<r3,
so,
(3.12)f(t,u,v,w)≥β(β+γ-1)N2||u||(1-γ)2c*||u1||c(t)u1,∀t∈J,u,v,w∈P,0<||u||<r3.
Choose
(3.13)0<r4<min⁡{r3,ξ}.
For u∈Q,||u||B=r4, we have by (3.13) and (2.62),
(3.14)r3>||u(t)||≥N-1β-1(1-γ)‖u‖B=N-1β-1(1-γ)r4>0,
so, similar to (3.9), we get by (2.5), (3.12), and (3.14)
(3.15)(Au)(t)≥1-γβ+γ-1(∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds)≥βN2(1-γ)c*||u1||(∫0∞||u(s)||c(s)ds)u1≥N‖u‖Bc*||u1||(∫0∞c(s)ds)u1=N‖u‖B||u1||u1,∀t∈J;
hence
(3.16)‖Au‖B≥‖u‖B,∀u∈Q,||u||=r4.
On the other hand, for u∈Q, ||u||B=ξ, by condition (H2), condition (H3), (3.4), we have
(3.17)‖f(t,u(t),(Tu)(t),(Su)(t))‖≤Mξa(t),∀t∈J,(3.18)‖Ik(u(tk))‖≤Nξγk(k=1,2,3,…).
It is clear that
(3.19)(Au)(t)≤β+γβ+γ-1(∫0∞f(s,u(s),(Tu)(s),(Su)(s))ds+∑k=1∞Ik(u(tk)))∀t∈J.
It follows from (3.17)–(3.19) that
(3.20)‖Au‖B≤N(β+γ)β+γ-1(Mξa*+Nξγ*).
Thus, (3.20) and (3.3) imply
(3.21)‖Au‖B<‖u‖B,∀u∈Q,‖u‖B=ξ.
From (3.7) and (3.13), we know 0<r4<ξ<r2; hence, (3.10), (3.16), (3.21), and Lemma 2.6 imply that A has two fixed points u*,u**∈Q such that r4<||u*||B<ξ<||u**||B<r2. The proof is complete.Theorem 3.3.
Let coneP be normal and conditions (H1)–(H5) satisfied. Assume that
(3.22)g(x,y,z)x+y+z⟶0asx+y+z⟶0+,F(x)x⟶0asx⟶0+.
(for g(x,y,z) and F(x), see conditions (H2) and (H3)). Then IBVP(1.5) has at least one positive solution u*∈Q∩C1[J′,E].Proof.
As in the proof of Theorem3.2, we can choose r2>0 such that (3.10) holds (in this case, we only choose r2>Nβ(1-γ)-1r1 instead of (3.7)). On the other hand, by (3.22), there exists r5>0 such that
(3.23)g(x,y,z)≤ϵ0(x+y+z),∀0<x+y+z<r5,F(x)≤ϵ0x,∀0<x<r5,
where
(3.24)ϵ0=β+γ-1N(β+γ)[(1+k*+h*)a*+γ*].
Choose
(3.25)0<r6<min⁡{r51+k*+h*,r2}.
For u∈Q,||u||B=r6, we have by (2.62) and (3.25),
(3.26)0<N-1β-1(1-γ)r6≤||u(t)||≤r6<r5,∀t∈J,0<N-1β-1(1-γ)r6≤||u(t)||+||(Tu)(t)||+||(Su)(t)||≤(1+k*+h*)r6<r5,∀t∈J,
so, (3.23) imply
(3.27)g(||u(t)||,||(Tu)(t)||,||(Su)(t)||)≤ϵ0(||u(t)||+||(Tu)(t)||+||(Su)(t)||)≤ϵ0(1+k*+h*)r6,∀t∈J,F(||u(tk)||)≤ϵ0||u(tk)||≤ϵ0r6,(k=1,2,3,…).
It follows from (3.19), condition (H2), condition (H3), (3.27), and (3.24) that
(3.28)||(Au)(t)||≤N(β+γ)β+γ-1{ϵ0(1+k*+h*)r6∫0∞a(s)ds+ϵ0r6∑k=1∞γk}=N(β+γ)ϵ0r6β+γ-1{(1+k*+h*)a*+γ*}=r6,∀t∈J,
and consequently,
(3.29)‖Au‖B≤‖u‖B,∀u∈Q,‖u‖B=r6.
Since 0<r6<r2 by virtue of (3.25), we conclude from (3.10), (3.29), and Lemma 2.6 that A has a fixed point u*∈Q such that r6≤||u*||B≤r2. The theorem is proved.Example 3.4.
Consider the infinite system of scalar first-order impulsive integrodifferential equations of mixed type on the half line:(3.30)un′(t)=18n2e-5t([un+1(t)+∑m=1∞um(t)]2+3u2n(t)+∑m=1∞um(t))+19n3e-6t{(∫0te-(t+1)sun(s)ds)2+(∫0∞un+2(s)ds(1+t+s)2)3},∀0≤t<∞,t≠k(k=1,2,3,…;n=1,2,3,…),Δun|t=k=16n23-k([un(k)]2+[un+2(k)]2),(k=1,2,3,…;n=1,2,3,…),u(∞)=12un(92)+6un(0),(n=1,2,3,…).
Evidently, un(t)≡0(n=1,2,3,…) is the trivial solution of infinite system (3.30).Conclusion.
Infinite system (3.30) has at least two positive solutions {un*(t)} (n=1,2,3,…) and {un**(t)} (n=1,2,3,…) such that
(3.31)0<inf⁡0≤t<∞∑n=1∞un*(t)≤sup⁡0≤t<∞∑n=1∞un*(t)<1<sup⁡0≤t<∞∑n=1∞un**(t),inf⁡0≤t<∞∑n=1∞un**(t)>0.Proof.
LetE=l1={u=(u1,…,un,…):∑n=1∞|un|<∞} with norm ||u||=∑n=1∞|un| and P=(u1,…,un,…):un≥0,n=1,2,3,…}. Then P is a normal cone in E with normal constant N=1, and infinite system (3.30) can be regarded as an infinite three-point boundary value problem of form (1.5). In this situation, u=(u1,…,un,…), v=(v1,…,vn,…), w=(w1,…,wn,…), tk=k(k=1,2,3,…), K(t,s)=e-(t+1)s, H(t,s)=(1+t+s)-2, η=9/2, γ=1/2, β=6, f=(f1,…,fn,…), and Ik=(Ik1,…,Ikn…), in which
(3.32)fn(t,u,v,w)=18n2e-5t([un+1+∑m=1∞um]2+3u2n+∑m=1∞um)+19n3e-6t(vn2+wn+23),∀t∈J=[0,∞),u,v,w∈P(n=1,2,3,…),(3.33)Ikn(u)=16n23-k(un2+u2n+12),∀u∈P(k=1,2,3,…;n=1,2,3,…).
It is easy to see that f∈C[J×P×P×P,P], Ik∈C[P,P]  (k=1,2,3,…), and condition (H1) is satisfied and k*≤1,h*≤1. We have, by (3.32),
(3.34)0≤fn(t,u,v,w)≤18n2e-5t([2||u||]2+4||u||)+19n3e-6t(‖v‖2+‖w‖3)≤1n2e-5t(12‖u‖2+14||u||+19‖v‖2+19‖w‖3),∀t∈J,u,v,w∈P(n=1,2,3,…),
so, observing the inequality ∑n=1∞(1/n2)<2, we get
(3.35)‖f(t,u,v,w)‖=∑n=1∞fn(t,u,v,w)≤e-5t(‖u‖2+12‖u‖+29‖v‖2+29‖w‖3),∀t∈J,u,v,w∈P,
which implies that condition (H2) is satisfied for a(t)=e-5t(*=1/5) and
(3.36)g(x,y,z)=x2+12x+29y2+29z3.
By (3.33), we have
(3.37)0≤Ikn(u)≤16n23-k‖u‖2,∀u∈P(k=1,2,3,…;n=1,2,3,…),
so, condition (H3) is satisfied for γk=3-k-1(γ*=1/6) and
(3.38)F(x)=x2.
On the other hand, (3.32) implies
(3.39)fn(t,u,v,w)≥18n2e-5t‖u‖2,∀t∈J,u,v,w∈P(n=1,2,3,…),fn(t,u,v,w)≥18n2e-5t‖u‖,∀t∈J,u,v,w∈P(n=1,2,3,…),
so, we see that condition (H5) is satisfied for b(t)=(1/8)e-5t(b*=1/40), τ(u)=||u||2 and u0=(1,…,1/n2,…), and condition (H6) is satisfied for c(t)=(1/8)e-5t(c*=1/40), σ(u)=||u||, and let u1=(1,…,1/n2,…). Now, we check that condition (H4) is satisfied. Let t∈J and r>0 be fixed, and {z(m)} be any sequence in f(t,Pr,Pr,Pr), where z(m)=(z1(m),…,zn(m),…). Then, we have, by (3.34),
(3.40)0≤zn(m)≤1n2(1118r2+14r+19r3),(n,m=1,2,3,…).
So, {zn(m)} is bounded, and, by diagonal method, we can choose a subsequence {mi}⊂{m} such that
(3.41)z(mi)⟶z̅nasi⟶∞(n=1,2,3,…),
which implies by virtue of (3.40) that
(3.42)0≤z̅n≤1n2(1118r2+14r+19r3),(n=1,2,3,…).
Consequently, z̅=(z̅1,…,z̅n,…)∈l1=E. Let ϵ>0 be given. Choose a positive integer n0 such that
(3.43)(∑n=n0+1∞1n2)(1118r2+14r+19r3)<ϵ3.
By (3.41), we see that there exists a positive integer i0 such that
(3.44)|zn(mi)-z̅n|<ϵ3n0,∀i>i0(n=1,2,…,n0).
It follows from (3.40)–(3.44) that
(3.45)‖z(mi)-z̅‖=∑n=1∞|zn(mi)-z̅n|≤∑n=1n0|zn(mi)-z̅n|+∑n=n0+1∞|zn(mi)|+∑n=n0+1∞|z̅n|<ϵ3+ϵ3+ϵ3=ϵ,∀i>i0.
Thus, we have proved that f(t,Pr,Pr,Pr) is relatively compact in E. Similarly, by using (3.37), we can prove that Ik(Pr) is relatively compact in E. Hence, condition (H4) is satisfied. Finally, it is easy to check that inequality (3.3) is satisfied for ξ=1 (in this case, Mξ≤17/36 and Nξ=1). Hence, our conclusion follows from Theorem 3.2.Example 3.5.
Consider the infinite system of scalar first-order impulsive integrodifferential equations of mixed type on the half line:(3.46)un′(t)=1n3(1+t)3(un(t)+2un+1(t)+∑m=1∞um(t))3+1n4(1+t)4(∫0tu2n(s)ds1+ts+s2)4+1n5(1+t)5(∫0∞e-ssin2(t-s)u3n(s)ds)5,∀0≤t<∞,t≠2k(k=1,2,3,…;n=1,2,3,…),Δun∣t=2k=1n2e-k[un(2k)]3+1n32-k[u2n+1(2k)]4,(k=1,2,3,…;n=1,2,3,…),4un(∞)=3un(7)+2un(0),(n=1,2,3,…).
Evidently, un(t)≡0(n=1,2,3,…) is the trivial solution of infinite system (3.46).Conclusion.
Infinite system (3.46) has at least one positive solution {un*(t)} (n=1,2,3,…) such that
(3.47)inf⁡0≤t<∞∑n=1∞un*(t)>0.Proof.
LetE=l1=({u=(u1,…,un,…):∑n=1∞|un|<∞} with norm ||u||=∑n=1∞|un| and P={u=(u1,…,un,…)∈l1:un≥0,n=1,2,3,…}. Then P is a normal cone in E with normal constant N=1, and infinite system (3.46) can be regarded as an infinite three-point boundary value problem of form (1.5) in E. In this situation, u=(u1,…,un,…), v=(v1,…,vn,…), w=(w1,…,wn,…), tk=2k (k=1,2,3,…),  K(t,s)=(1+ts+s2)-1, H(t,s)=e-ssin2(t-s), η=7, γ=3/4, β=1/2, f=(f1,…,fn,…), and Ik=(Ik1,…,Ikn,…), in which
(3.48)fn(t,u,v,w)=1n3(1+t)-3(un+2un+1+∑m=1∞um)3+1n4(1+t)-4v2n4+1n5(1+t)-5w3n5,∀t∈J=[0,∞),u,v,w∈P(n=1,2,3,…),Ikn(u)=1n2e-kun3+1n32-ku2n+14,(k=1,2,3,…;n=1,2,3,…).
It is clear that f∈C[J×P×P×P,P],Ik∈C[P,P] (k=1,2,3,…), and condition (H1) is satisfied and k*≤π/2,h*≤1. We have
(3.49)0≤fn(t,u,v,w)≤1n3(1+t)-3((3‖u‖)3+‖v‖4+‖w‖5),∀t∈J,u,v,w∈P(n=1,2,3,…),0≤Ikn(u)≤1n22-k(‖u‖3+‖u‖4),∀u∈P(k=1,2,3,…,n=1,2,3,…),
so, condition (H2) is satisfied for a(t)=(1+t)-3(a*=(1/2)) and
(3.50)g(x,y,z)=54x3+2y4+2z5,
and (H3) is satisfied for γk=2-k(γ*=1) and
(3.51)F(x)=2x3+2x4.
From
(3.52)fn(t,u,v,w)≥1n3(1+t)-3‖u‖3,∀t∈J,u,v,w∈P(n=1,2,3,…),
we see that condition (H5) is satisfied for b(t)=(1+t)-3(b*=1/2),τ(u)=||u||3, and u0=(1,…,1/n3,…). Moreover, it is clear that (3.22) are satisfied. Similar to the discussion in Example 3.4, we can prove that f(t,Pr,Pr,Pr) and Ik(Pr) (for fixed t∈J and r>0;k=1,2,3,…) are relatively compact in E=l1; so, condition (H4) is satisfied. Hence, our conclusion follows from Theorem 3.3.



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