# Multiple Positive Solutions for Singular Periodic Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

Authors: Hongxia Fan; Yongxiang Li
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second-order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.


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## Body

## 1. Introduction

The theory of impulsive differential equations describes processes that experience a sudden change of their state at certain moments. In recent years, a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [1]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [2–14]. Meanwhile, the theory of ordinary differential equations in abstract spaces has become a new important branch (see [15–18]). So it is interesting and important to discuss the existence of positive solutions for impulsive boundary value problem in a Banach space.Let(E,∥·∥) be a real Banach space, J=[0,2π], 0=t0<t1<t2<⋯<tm<tm+1=2π, J0=[0,t1], and Ji=(ti,ti+1], i=1,…,m. Note that PC[J,E]={u:u is a map fromJintoEsuch that u(t)is continuous at t≠tk and left continuous att=tk and u(tk+)exist, k=1,2,…,m}, and it is also a Banach space with norm
(1.1)‖u‖PC=sup⁡t∈J‖u(t)‖.Let the Banach spaceE be partially ordered by a cone P of E; that is, x≤y if and only if y-x∈P, and PC[J,E] is partially ordered by K={u∈PC[J,E]:u(t)∈P,t∈J}:u≤v if and only if v-u∈K; that is, u(t)≤v(t) for all t∈J.In this paper, we consider the following singular periodic boundary value problem with impulsive effects in BanachE(1.2)-u′′(t)+M2u(t)=f(t,u(t)),t∈J,t≠tk,Δu|t=tk=Ik(u(tk)),Δu′|t=tk=-I̅k(u(tk)),k=1,2,…,m,u(0)=u(2π),u′(0)=u′(2π),
where M>0 is constant, f(t,u) may be singular at t=0and/ort=2π, f∈C[(0,2π)×P,P], Ik,I̅k∈C[P,P], Δu|t=tk=u(tk+)-u(tk-), Δu′|t=tk=u′(tk+)-u′(tk-), k=1,2,…,m, and ui(tk+) (resp., ui(tk-)) denote the right limit (resp., left limit) of ui(t) at t=tk, i=0,1.In the special case whereE=ℝ+=[0,+∞), and Ik=I¯k=0, k=1,2,…,m, problem (1.2) is reduced to the usual second-order periodic boundary value problem. For example, in [19], the periodic boundary value problem: (1.3)-u′′(t)+Mu(t)=f(t,u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π),
was proved to have at least one positive solution, by Jiang [19].In [20], the authors  studied the multiplicity of positive solutions for IBVP(1.2) in E=ℝ+; the main tool is the theory of fixed point index.In [21], the author considers the following periodic boundary value problem of second-order integrodifferential equations of mixed type in Banach space:
(1.4)-u′′=f(t,u,Tu,Su),t∈(0,2π),Δu|t=tk=Ik(u(tk)),Δu′|t=tk=-I̅k(u(tk)),k=1,2,…,m,u(0)=u(2π),u′(0)=u′(2π),
where f∈C[J×E×E×E,E], Ik,I̅k∈C[E,E], and the operators T, S are given by
(1.5)Tu(t)=∫0tk(t,s)u(s)ds,Su(t)=∫02πk1(t,s)u(s)ds,
with k∈C[D,ℝ], D={(t,s)∈ℝ2:0≤s≤t≤2π}, k1∈C[J×J,ℝ]. By applying the monotone iterative technique and cone theory based on a comparison result, the author obtained an existence theorem of minimal and maximal solutions for the IBVP(1.4).Motivated by the above facts, our aim is to study the multiplicity of positive solutions for IBVP(1.2) in a Banach space. By means of the fixed point index theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions for IBVP(1.2). Moreover, an application is given to illustrate the main result.The rest of this paper is organized as follows. In Section2, we present some basic lemmas and preliminary facts which will be needed in the sequel. Our main result and its proof are arranged in Section 3. An example is given to show the application of the result in Section 4.


## 2. Preliminaries

LetTr={x∈E:∥x∥≤r}, Br={u∈PC[J,E]:∥u∥PC≤r} (r>0); for D⊂PC[J,E], we denote D(t)={u(t):u∈D}⊂E(t∈J).α denotes the Kuratowski measure of noncompactness.LetPC1[J,E]={u∣ube a map fromJintoEsuch that u(t)is continuously differentiable at t≠tk and left continuous att=tk andu(tk+), u′(tk-), u′(tk+)exist, k=1,2,…,m}. Evidently, PC1[J,E] is a Banach space with norm
(2.1)‖u‖PC1=max⁡⁡{‖u‖PC,‖u′‖PC}.LetJ′=J∖{t1,t2,…,tm}; a map u∈PC1[J,E]∩C2[J′,E] is a solution of IBVP(1.2) if it satisfies (1.2).Now, we first give the following lemmas in order to prove our main result.Lemma 2.1 (see [17]).
LetK be a cone in real Banach space E, and let Ω be a nonempty bounded open convex subset of K. Suppose that A:Ω¯→K is a strict set contraction and A(Ω¯)⊂K. Then the fixed-point index i(A,Ω,K)=1.Lemma 2.2 (see [21]).
u∈PC1[J,E]∩C2[J′,E] is a solution of IBVP (1.2) if and only if u∈PC[J,E] is a solution of the impulsive integral equation:
(2.2)u(t)=∫02πG(t,s)f(s,u(s))ds+∑k=1m[G(t,tk)I̅k(u(tk))+H(t,tk)Ik(u(tk))],
where
(2.3)G(t,s)=(2M(e2πM-1))-1{eM(2π-t+s)+eM(t-s),0≤s≤t≤2π,eM(2π+t-s)+eM(s-t),0≤t≤s≤2π,H(t,s)=(2(e2πM-1))-1{eM(2π-t+s)-eM(t-s),0≤s≤t≤2π,eM(s-t)-eM(2π+t-s),0≤t<s≤2π.
By simple calculations, we obtain that for (t,s)∈J×J,
(2.4)l0:=eπMM(e2πM-1)≤G(t,s)≤e2πM+12M(e2πM-1):=l1,(2.5)|H(t,s)|≤12,MG(t,s)+H(t,s)>0.To establish the existence of multiple positive solutions inPC1[J,E]∩C2[J′,E] of IBVP(1.2), let us list the following assumptions:(A1)
∥f(t,x)∥≤g(t)∥h(x)∥, t∈(0,2π), x∈P, where g:(0,2π)→(0,∞) is continuous andh:P→P is bounded and continuous and satisfies ∫02πg(s)ds<+∞.(A2)
h(x) in (A1) satisfies(2.6)cl1∫02πg(s)ds+l1∑k=1mhk+12∑k=1mck<1,dl1∫02πg(s)ds+l1∑k=1mek+12∑k=1mdk<1,
where (2.7)c=lim⁡‖x‖→0¯‖h(x)‖‖x‖,d=lim⁡‖x‖→+∞¯‖h(x)‖‖x‖,ck=lim⁡‖x‖→0¯‖Ik‖‖x‖,dk=lim⁡‖x‖→+∞¯‖Ik‖‖x‖,hk=lim⁡‖x‖→0¯‖I̅k‖‖x‖,ek=lim⁡‖x‖→+∞¯‖I̅k‖‖x‖.(A3)
For anyr>0 and [a,b]⊂(0,2π), f is uniformly continuous on [a,b]×Tr.(A4)
There existL,Lk,Hk≥0 such that α(f(t,D))≤Lα(D), α(Ik(D))≤Lkα(D), α(I̅k(D))≤Hkα(D) (k=1,…,m), and 4πLl1+l1∑k=1mHk+(1/2)∑k=1mLk<1,for t∈(0,2π), and D⊂P is bounded.(A5)
For anyx∈P, I̅k(x)≥MIk(x);(A6)
P is a solid cone, and there exist u0∈P∘, J0′=[a′,b′]⊂J such that t∈J′0, x≥u0implyf(t,x)≥h̅(t)u0, h̅∈C(J0′,[0,+∞)), and l:=l0∫a′b′h̅(s)ds>1.Define an operatorA as follows:
(2.8)(Au)(t)=∫02πG(t,s)f(s,u(s))ds+∑k=1m[G(t,tk)I̅k(u(tk))+H(t,tk)Ik(u(tk))],t∈J.Lemma 2.3.
Assuming (A1) and (A4) hold, then, for anyr>0, A:PC[J,P]∩Br→PC[J,P] is bounded and continuous.Proof.
According to (A1) and (A4), we obtain thatA is a bounded operator. In the following, we will show that A is continuous.
Let{un},{u}⊂PC[J,P]∩Br, and ∥un-u∥PC→0. Next we show that ∥Aun-Au∥PC→0. By (A1), {(Aun)(t)} is equicontinuous on each Ji(i=0,…,m). By the Lebesgue dominated convergence theorem and (2.4), we have
(2.9)‖Aun(t)-Au(t)‖≤‖∫02πG(t,s)(f(s,un(s))-f(s,u(s)))ds‖+∑k=1mG(t,tk)‖I̅k(un(tk))-I̅k(u(tk))‖+∑k=1m|H(t,tk)|‖Ik(un(tk))-Ik(u(tk))‖≤l1∫02π‖f(s,un(s))-f(s,u(s))‖ds+l1∑k=1m‖I̅k(un(tk))-I̅k(u(tk))‖+12∑k=1m‖Ik(un(tk))-Ik(u(tk))‖⟶0(n⟶∞).
In view of the Ascoli-Arzela theorem, {Aun} is a relatively compact set in PC[J,E]. In the following we will verify that ∥Aun-Au∥PC→0(n→∞).
If this is not true, then there areε0>0 and {uni}⊂{un} such that ∥Auni-Au∥PC≥ε0(i=1,2,…). Since {Aun} is a relatively compact set, there exists a subsequence of {Auni} which converges to v∈PC[J,P], without loss of generality, and we assume that limi→∞Auni=v, that is, limi→∞∥Auni-v∥PC=0, so v=Au, which imply a contradiction. Therefore A is continuous.Lemma 2.4.
Assuming (A1), (A3), and (A4) hold, then, for anyR>0, A:PC[J,P]∩BR→PC[J,P] is a strict set contraction operator.Proof.
For anyR>0, S⊂PC[J,P]∩BR, by (A1), AS is bounded and equicontinuous on each Ji, i=0,…,m,  and by [17],
(2.10)αPC(AS)=sup⁡t∈Jα((AS)(t)),
where (AS)(t)={Au(t):u∈S,t∈J}.
Let(2.11)D={∫02πG(t,s)f(s,u(s))ds:u∈S},Dδ={∫δ2π-δG(t,s)f(s,u(s))ds:u∈S},0<δ<min⁡⁡{π,t1,2π-tm}.
By (A1) and (2.4), for any u∈S,
(2.12)‖∫δ2π-δG(t,s)f(s,u(s))ds-∫02πG(t,s)f(s,u(s))ds‖≤l1max⁡x∈TR‖h(u)‖∫0δg(s)ds+l1max⁡u∈TR‖h(x)‖∫2π-δ2πg(s)ds.
In view of (2.12) and (A1), we have dH(Dδ,D)→0(δ→0+), where dH(Dδ,D) denotes the Hausdorff distance of D and Dδ.
Therefore,(2.13)lim⁡δ→0+α(Dδ)=α(D).
Next we will estimate α(Dδ). Since
(2.14)∫δ2π-δG(t,s)f(s,u(s))ds∈(2π-2δ)co⁡̅({G(t,s)f(s,u(s)):s∈[δ,2π-δ]}),
thus
(2.15)α(Dδ)=α({∫δ2π-δG(t,s)f(s,u(s))ds:u∈S})≤2(π-δ)α(co⁡̅{G(t,s)f(s,u(s)):s∈[δ,2π-δ],u∈S})≤2πα({G(t,s)f(s,u(s)):s∈[δ,2π-δ],u∈S})≤2πl1α(f(Iδ×S(Iδ))),
where Iδ=[δ,2π-δ], S(Iδ)={u(t):t∈Iδ,u∈S}.
By (A3) and (A4), it is not difficult to prove that(2.16)α(f(Iδ×S(Iδ)))=max⁡t∈Iδα(f(t,S(Iδ)))≤Lα(S(Iδ))≤Lα(S(J)).
By [17], we have
(2.17)Lα(S(J))≤2LαPC(S).
Let δ→0+, and making use of the fact that limδ→0+α(Dδ)=α(D), we obtain
(2.18)α(D)≤2πl12LαPC(S)=4πLl1αPC(S).
It is clear that
(2.19)α({∑k=1mG(t,tk)I̅k(u(tk)):u∈S})≤(l1∑k=1mHk)αPC(S),(2.20)α({∑k=1mH(t,tk)Ik(u(tk)):u∈S})≤(12∑k=1mLk)αPC(S).
Hence, according to (2.18)–(2.20), we have
(2.21)αPC(AS)≤(4πLl1+l1∑k=1mHk+12∑k=1mLk)αPC(S).
By (A4) and Lemma2.3, A is a strict set contraction operator from PC[J,P] into PC[J,P].


## 3. Main Result

Theorem 3.1.
Assuming that (A1)–(A6) hold, then the IBVP (1.2) has at least two positive solutions u1 and u2 satisfying
(3.1)u1(t)≥lu0(t),fort∈J0′=[a′,b′],l>1,
where l was specified in (A6).Proof.
First we verify that there existsδ>0 such that ∥v∥≥δ for v≥u0. If this is not true, then there exists {vn}⊂Ewhich satisfies vn≥u0 and ∥vn∥<(1/n)(n=1,2,…), so we have u0≤θ, which is a contradiction with u0∈P∘.
By (A2), there existc′>c, ck′>ck, d′>d, dk′>dk, hk′>hk, and ek′>ek, and
(3.2)0<r1<δ,r2>max⁡⁡{δ,l‖u0‖}
satisfy
(3.3)c′l1∫02πg(s)ds+l1∑k=1mhk′+12∑k=1mck′<1,(3.4)b:=d′l1∫02πg(s)ds+l1∑k=1mek′+12∑k=1mdk′<1.
For x∈Tr1∩P,
(3.5)‖h(x)‖≤c′‖x‖,‖Ik(x)‖≤ck′‖x‖,‖I̅k(x)‖≤hk′‖x‖.
For ∥x∥≥r2 and x∈P,
(3.6)‖h(x)‖≤d′‖x‖,‖Ik(x)‖≤dk′‖x‖,‖I̅k(x)‖≤ek′‖x‖.
Therefore, for any x∈P, we have
(3.7)‖h(x)‖≤d′‖x‖+M′,‖Ik(x)‖≤dk′‖x‖+M′,‖I̅k(x)‖≤ek′‖x‖+M′,
where
(3.8)M′=max⁡⁡{M0,M1,…,Mm,K1,…,Km},M0=sup⁡⁡{‖h(x)‖:x∈Tr2∩P},Mk=sup⁡⁡{‖Ik(x)‖:x∈Tr2∩P},Kk=sup⁡⁡{‖I̅k(x)‖:x∈Tr2∩P}(k=1,2,…,m).
Letr3=r2+(1-b)-1G, G=M′[l1∫02πg(s)ds+ml1+m/2], U1={u∈PC[J,P]:∥u∥PC<r1}, U2={u∈PC[J,P]:∥u∥PC<r3}, U3={u∈PC[J,P]:∥u∥PC<r3, u(t)≥lu0 for t∈J0′ and l>1}. It is clear that U1,U2,U3 are nonempty, bounded, and convex open sets in PC[J,P], and U¯1=PC[J,P]∩Br1, U¯2=PC[J,P]∩Br3, and U¯3={u∈U¯2:u(t)≥lu0,t∈J0′}.
From (3.2), we obtain
(3.9)U1⊂U2,U3⊂U2,U1∩U3=∅.
According to Lemma 2.4, A:U¯2→PC[J,P] is a strict set contraction operator, and for u∈U¯2, by (2.4) and (3.7), we obtain
(3.10)‖(Au)(t)‖=‖∫02πG(t,s)f(s,u(s))ds+∑k=1m[G(t,tk)I̅k(u(tk))+H(t,tk)Ik(u(tk))]‖≤l1∫02πg(s)ds‖h(u)‖+∑k=1m(l1‖I̅k‖+12‖Ik‖)≤l1∫02πg(s)ds(d′‖u‖+M′)+l1∑k=1m(ek′‖u‖+M′)+12∑k=1m(dk′‖u‖+M′)=[d′l1∫02πg(s)ds+l1∑k=1mek′+12∑k=1mdk′]‖u‖+M′[l1∫02πg(s)ds+ml1+m2]=b‖u‖+G≤br3+G<r3.
Hence
(3.11)A(U¯2)⊂U2.
Similarly,A:U¯1→PC[J,P] is a strict set contraction operator, and for u∈U¯1, by (3.3) and (3.5), we obtain
(3.12)‖(Au)(t)‖=‖∫02πG(t,s)f(s,u(s))ds+∑k=1m[G(t,tk)I̅k(u(tk))+H(t,tk)Ik(u(tk))]‖≤c′l1∫02πg(s)ds‖u‖+∑k=1m(l1hk′‖u‖+12ck′‖u‖)=[c′l1∫02πg(s)ds+l1∑k=1mhk′+12∑k=1mck′]‖u‖<‖u‖≤r1,
so
(3.13)A(U¯1)⊂U1.
Letu∈U¯3, by (3.11), we have ∥Au∥PC<r3.
By (2.5), (A5), and (A6), for t∈J0′,
(3.14)(Au)(t)=∫02πG(t,s)f(s,u(s))ds+∑k=1m[G(t,tk)I̅k(u(tk))+H(t,tk)Ik(u(tk))]≥∫02πG(t,s)f(s,u(s))ds+∑k=1m[MG(t,tk)+H(t,tk)]Ik(u(tk))≥∫02πG(t,s)f(s,u(s))ds≥l0∫a′b′f(s,u(s))ds≥l0∫a′b′h̅(s)dsu0=lu0.
So Au∈U3, and
(3.15)A(U¯3)⊂U3.
According to (3.11)–(3.15) and Lemma 2.1, we have
(3.16)i(A,Uj,PC[J,P])=1(j=1,2,3).
Hence
(3.17)i(A,U2∖(U¯1∪U¯3),PC[J,P])=i(A,U2,PC[J,P])-i(A,U1,PC[J,P])-i(A,U3,PC[J,P])=-1.
Thus,A has two fixed points u1 and u2 in U3 and U2∖(U¯1∪U¯3), respectively, which means u1(t) and u2(t) are positive solution of the IBVP (1.2), where u1(t)≥lu0, for t∈J0′ and l>1.


## 4. Example

To illustrate how our main result can be used in practice, we present an example.Example 4.1.
Consider the following problem:(4.1)-xn′′(t)+4xn=1t[3(2+999πt)ln⁡(1+xn+12)+xn3],t∈J,Δxn|t=1/3=18xn(13),Δxn′|t=1/3=-14xn(13),xn(0)=xn(2π),xn′(0)=xn′(2π),
where xm+n=xn (n=1,2,…,m).Conclusion
IBVP (4.1) has at least two positive solutions {x1n(t)} and {x2n(t)} such that x1n(t)>1 for t∈[π,2π], n=1,2,…,m.Proof.
LetJ=[0,2π], E=ℝm={x=(x1,x2,…,xm):xn∈ℝ,n=1,2,…,m}; then, E is a Banach space with norm ∥x∥=max1≤n≤m|xn|. Let P={(x1,x2,…,xm):xn≥0,n=1,2,…,m}; then, P is a solid cone in E. Compared to IBVP (1.2), f(t,x)=(f1,f2,…,fm), fn(t,x)=(1/t)[3(2+(999/π)t)ln(1+xn+12)+xn/3)] is singular at t=0. I(x)=(I1(x),I2(x),…,Im(x)), and In(x)=(1/8)xn(1/3). I̅(x)=(I̅1(x),…,I̅m(x)),  and I̅n(x)=(1/4)xn(1/3), n=1,2,…,m.
Next we will verify that the conditions in Theorem3.1 are satisfied.
Letg(t)=1/t, h(x)=(h1(x),h2(x),…,hm(x)), and hn(x)=6000ln(1+xn+12)+xn/3. It is clear that ∥f(t,x)∥≤g(t)∥h(x)∥, for t∈(0,2π) and x∈E, so (A1) is satisfied.
By simple calculations, we haveM=2, c=d=1/3, c1=d1=1/8, h1=e1=1/4, ∫02πg(s)ds=5.01326, l0=0.00093, and l1=0.25. Hence, l1∫02πg(s)dsc+l1h1+(1/2)c1<1; that is, (A2) is satisfied.
SinceE is a finite-dimensional space, it is obvious that (A3) and (A4) are satisfied.
It is clear thatI̅n(x)=(1/4)xn(1/3) and MIn(x)=2×(1/8)xn(1/3)=(1/4)xn(1/3), so I̅n(x)=MIn(x); that is, (A5) is satisfied.
Letu0=(1,1,…,1)∈P∘ and J0′=[π,2π]⊂[0,2π]; for t∈J0′ and x≥u0, we have
(4.2)fn(t,x)=1t[(3(2+999πt)ln⁡⁡(1+xn+12)+xn3)]>3000ln⁡2t.
Let h̅(t)=3000ln2/t; then, for t∈J0′ and x≥u0, we obtain that f(t,x)≥h̅(t)u0 and l0∫π2πh̅(s)ds>1. Therefore (A6) is satisfied.
By Theorem3.1, IBVP (4.1) has at least two positive solutions {x1n(t)} and {x2n(t)} and satisfies x1n(t)>1, n=1,2,…,m.



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