# The Existence and Multiplicity of Positive Solutions for Second-Order Periodic Boundary Value Problem

Authors: Feng Wang; Fang Zhang; Fuli Wang
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The existence and multiplicity of positive solutions are established for second-order periodic boundary value problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.


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## Body

## 1. Introduction

In the present paper, we discuss the existence of positive solutions of the periodic boundary value problem (PBVP) for second-order differential equation(1.1)x′′(t)=f(t,x),0<t<1,x(0)=x(1),x′(0)=x′(1),
where f:[0,1]×ℝ→ℝ is a continuous function. Our purpose here is to provide sufficient conditions for the existence of multiple positive solutions to the periodic boundary value problem (1.1). This will be done by applying the theory of a fixed point index for A-proper semilinear operators defined on cones obtained by Cremins [1].We are interested in positive solutions of (1.1), because we have been motivated by a problem from the Theory of Nonlinear Elasticity modelling radial oscillations of an elastic spherical membrane made up of a neo-Hookean material and subjected to an internal pressure. Because of wide interests in physics and engineering, second-order periodic boundary value problems have been studied widely in the literature; we refer the reader to [2–30] and references therein. In [6, 7], by using Krasnoselskii’s fixed point theorem, the existence and multiplicity of positive solutions are established to the periodic boundary value problem on(1.2)-x′′+ρ2x=f(t,x),0<t<2π,ρ>0,x(0)=x(2π),x′(0)=x′(2π),x′′+ρ2x=f(t,x),0<t<2π,0<ρ<12,x(0)=x(2π),x′(0)=x′(2π).Agarwal et al. [8] discussed the existence of positive solutions for the second-order differential equation(1.3)-x′′(t)+b(t)x(t)=g(t)f(t,x(t)),0<t<ω,x(0)=x(ω),x′(0)=x′(ω),
where b(t) and g(t) are continuous ω-periodic positive functions and f∈C(ℝ×[0,∞),[0,∞)). By employing fixed point index theory in cones, they found sufficient conditions for the existence of at least one positive solution. Recently, Torres [9] and Yao [10] obtained some results on the existence of positive solutions of a general periodic boundary value problem(1.4)x′′(t)=f(t,x(t)),0<t<2π,x(0)=x(2π),x′(0)=x′(2π).
In this case, the problem (1.4) has no Green function. In order to overcome this difficulty, their main technique is to rewrite the original PBVP (1.4) as an equivalent one, so that the Krasnoselskii fixed point theorem on compression and expansion of cones can be applied. Inspired by the above work, the aim of this paper is to consider the existence and multiplicity of positive solutions for the periodic boundary value problem (1.1). The method we used here is different in essence from other papers and the main results of this paper are also new.This paper is organized as follows. In Section2, we give some preliminaries and establish several lemmas, and the main theorems are formulated and proved in Section 3. Finally, in Section 4, we give two examples to illustrate our results.


## 2. Notation and Preliminaries

We start by introducing some basic notation relative to theory of the fixed point index for A-proper semilinear operators defined on cones established by Cremins (see [1]).LetX and Y be Banach spaces, D a linear subspace of X, {Xn}⊂D, and {Yn}⊂Y sequences of oriented finite-dimensional subspaces such that Qny→y in Y for every y and dist(x,Xn)→0 for every x∈D, where Qn:Y→Yn and Pn:X→Xn are sequences of continuous linear projections. The projection scheme Γ={Xn,Yn,Pn,Qn} is then said to be admissible for maps from D⊂X to Y.Definition 2.1 (see [1]).
A mapT:D⊂X→Y is called approximation-proper (A-proper) at a point y∈Y with respect to Γ if Tn≡PnT|D∩Xn is continuous for each n∈ℕ and whenever {xnj:xnj∈D∩Xnj} is bounded with Tnjxnj→y, then there exists a subsequence {xnjk} such that xnjk→x∈D, and Tx=y. T is said to be A-proper on a set Ω if it is A-proper at all points of Ω.
LetK be a cone in a finite-dimensional Banach space X, and let Ω⊂X be open and bounded with Ω∩K=ΩK≠∅. Let T:Ω¯K→K be continuous such that Tx≠x on ∂ΩK=∂Ω∩K, where Ω¯K and ∂ΩK denote the closure and boundary, respectively, of ΩK relative K. Let ρ:X→K be an arbitrary retraction.
The following definition of finite-dimensional index forms the basis of generalized index for A-proper mapsI-T.Definition 2.2 (see [1]).
One defines(2.1)iK(T,Ω)=deg⁡B(I-Tρ,ρ-1(Ω)∩BR,0),
where the degree is the Brouwer degree and BR is a ball containing ΩK.
Now letK be a cone in an infinite-dimensional Banach space X with projection scheme Γ such that Qn(K)⊆K for every n∈ℕ. Let ρ:X→K be an arbitrary retraction and Ω⊂X an open bounded set such that ΩK=Ω∩K≠∅. Let T:Ω¯K→K be such that I-T is A-proper at 0. Write Kn=K∩Xn=QnK and Ωn=ΩK∩Xn. Then Qnρ:Xn→Kn is a finite dimensional retraction.Definition 2.3 (see [1]).
IfTx≠x on ∂ΩK, then one defines
(2.2)indK(T,Ω)={k∈Z∪{±∞}:iKnj(QnjT,Ωnj)⟶kforsomenj⟶∞},
that is, the index is the set of limit points of iKnj(QnjT,Ωnj), where the finite dimensional index is that defined above.
LetL:domL⊂X→Y be a Fredholm map of index zero, and let P:X→X,Q:Y→Y be continuous projectors such that ImP=KerL, KerQ=ImL and X=KerL⊕ KerP, Y=ImL⊕ImQ. The restriction of L to domL∩KerP, denoted L1, is a bijection onto ImL with continuous inverse L1-1: ImL→domL∩KerP. Since dim ImQ=dim KerL, there exists a continuous bijection J: ImQ→ KerL. Let K be a cone in an infinite-dimensional Banach space X with projection scheme Γ. If we let H=L+J-1P, then H: domL⊂X→Y is a linear bijection with bounded inverse. Thus K1=H(K∩domL) is a cone in the Banach space Y.
LetΩ⊂X be open and bounded with ΩK∩domL≠∅,L:domL⊂X→Y a bounded Fredholm operator of index zero, and N:Ω¯K∩ domL→Y a bounded continuous nonlinear operator such that L-N is A-proper at 0.
We can now extend the definition of the index to A-proper maps of the formL-N acting on cones.Definition 2.4 (see [1]).
Letρ1 be a retraction from Y to K1, and assume QnK1⊂K1,P+JQN+L1-1(I-Q)N maps K∩domL to K∩domL and Lx≠Nx on ∂ΩK. One defines the fixed point index of L-N over ΩK as
(2.3)indK([L,N],Ω)=indK1(T,U),
where U=H(ΩK),T:Y→Y is defined as Ty=(N+J-1P)H-1y for each y∈Y, and the index on the right is that of Definition 2.3.
For convenience, we recall some properties ofindK.Proposition 2.5 (see [1]).
LetL:domL→Y be Fredholm of index zero, and let  Ω⊂X be open and bounded. Assume that  P+JQN+L1-1(I-Q)N maps  K to  K, and  Lx≠Nx on  ∂ΩK. Then one has
(P1)
(existence property) ifindK([L,N],Ω)≠{0}, then there exists x∈ΩK such that Lx=Nx;(P2)
(normality property) ifx0∈ΩK, then indK([L,-J-1P+ŷ0],Ω)={1}, where ŷ0=Hx0 and ŷ0(y)=y0 for every y∈H(ΩK);(P3)
(additivity property) ifLx≠Nx for x∈Ω¯K∖(Ω1∪Ω2), where Ω1 and Ω2 are disjoint relatively open subsets of ΩK, then(2.4)indK([L,N],Ω)⊆indK([L,N],Ω1)+indK([L,N],Ω2)
with equality if either of indices on the right is a singleton;(P4)
(homotopy invariance property) ifL-N(λ,x) is an A-proper homotopy on ΩK for λ∈[0,1] and (N(λ,x)+J-1P)H-1:K1→K1 and θ∉(L-N(λ,x))(∂ΩK) for λ∈[0,1], then indK([L,N(λ,x)],Ω)=indK1(Tλ,U) is independent of λ∈[0,1], where Tλ=(N(λ,x)+J-1P)H-1.The following two lemmas will be used in this paper.Lemma 2.6.
IfL:domL→Y is Fredholm of index zero,  Ω is an open bounded set, and  ΩK∩domL≠∅, and let  L-λN be A-proper for  λ∈[0,1]. Assume that  N is bounded and  P+JQN+L1-1(I-Q)N maps  K to  K. If there exists  e∈K1∖{θ}, such that
(2.5)Lx-Nx≠μe,
for every x∈∂ΩK and all μ≥0, then indK([L,N],Ω)={0}.Proof.
Choose a real numberl such that
(2.6)l>sup⁡x∈Ω‖Lx-Nx‖‖e‖,
and define N(μ,x):[0,1]×Ω¯K→Y by
(2.7)N(μ,x)=Nx+lμe.
Trivially, (N(μ,x)+J-1P)H-1:K1→K1 and from (2.5) we obtain
(2.8)Nx+lμe≠Lx,forany(μ,x)∈[0,1]×∂ΩK.
Again, by homotopy invariance property in Proposition 2.5, we have
(2.9)indK([L,N(0,x)],Ω)=indK([L,N],Ω)=indK([L,N(1,x)],Ω).
However,
(2.10)indK([L,N(1,x)],Ω)={0}.
In fact, if indK([L,N(1,x)],Ω)≠{0}, the existence property in Proposition 2.5 implies that there exists x0∈ΩK such that
(2.11)Lx0=Nx0+le.
Then
(2.12)l=‖Lx0-Nx0‖‖e‖,
which contradicts (2.6). So
(2.13)indK([L,N],Ω)={0}.Remark 2.7.
The original condition of [1, Theorem  5] was given with θ≠e∈L(K∩domL) instead of e∈K1∖{θ}. The modification is necessary since otherwise it cannot guarantee that (N+μe+J-1P)H-1:K1→K1.
We assume that there is a continuous bilinear form[y,x] on Y×X such that y∈ImL if and only if [y,x]=0 for each x∈KerL. This condition implies that if {x1,x2,…,xn} is a basis in KerL, then the linear map J:ImQ→KerL defined by Jy=β∑i=1n[y,xi]xi, β∈ℝ+ is an isomorphism and that if y=∑i=1nyixi, then [J-1y,xi]=yi/β for 1≤i≤n and [J-1x0,x0]>0 for x0∈KerL.
In [1], Cremins extended a continuation theorem related to that of Mawhin [31] and Petryshyn [32] for semilinear equations to cones; refer to [1, Corollary  1] for the details. By Lemma 2.6 and [1, Corollary  1], we obtain the following existence theorem of positive solutions to a semilinear equation in cones. It is worth mentioning that the positive or nonnegative solutions of an operator equation Lx=Nx were also discussed by a recent paper of O’Regan and Zima [33] and the earlier papers [34–38].Lemma 2.8.
IfL:domL→Y is Fredholm of index zero,  K⊂X is a cone,  Ω1 and  Ω2 are open bounded sets such that  θ∈Ω1⊂Ω¯1⊂Ω2 and  Ω2∩K∩domL≠∅. Suppose that  L-λN is A-proper for  λ∈[0,1] with  N:Ω2∩K¯→Y bounded. Assume that
(C1)(P+JQN)(K)⊂K and (P+JQN+L1-1(I-Q)N)(K)⊂K,
(C2)Lx≠λNx for x∈∂Ω2∩K,λ∈(0,1],
(C3)QNx≠0 for x∈∂Ω2∩K∩KerL,
(C4)[QNx,x]≤0,forallx∈∂Ω2∩K∩KerL,
(C5) there exists e∈K1∖{θ}, such that
(2.14)Lx-Nx≠μe,foreveryμ≥0,x∈∂Ω1∩K.
Then there exists x∈domL∩K∩(Ω¯2∖Ω1) such that Lx=Nx.Corollary 2.9.
Assume all conditions of Lemma2.8 hold except  (C2) and assume (C2)′∥Lx-Nx∥2≥∥Nx∥2-∥Lx∥2 for each x∈∂Ω2∩K. Then the same conclusion holds.Proof.
We show that(C2)′ implies (C2), that is, Lx≠λNx, for each x∈∂Ω2∩K,λ∈(0,1]. Here λ∈[0,1). Otherwise, the proof is finished. If x∈KerL∩∂Ω2∩K, then it follows from Lx=λNx=θ that Lx=Nx has a solution in domL∩K∩(Ω¯2∖Ω1), and Corollary 2.9 is proved. If x∈domL∖KerL∩∂Ω2∩K and Lx=λNx for some λ∈(0,1), then Nx=λ-1Lx and
(2.15)(λ-1)2‖Nx‖2=‖Lx-Nx‖2≥‖Nx‖2-‖Lx‖2=(1-λ2)‖Nx‖2,
by condition (C2)′; that is, (λ-1)2≥1-λ2, contradicting the fact that λ∈(0,1). This completes the proof of Corollary 2.9.The following lemma can be found by(a) of [32, Lemma  2].Lemma 2.10.
Suppose eitherN or  L1-1 is compact, then  L-λN is A-proper for  λ∈[0,1].


## 3. Main Results

The goal of this section is to apply Lemma2.8 to discuss the existence and multiplicity of positive solutions for the PBVP (1.1).LetX={x∈C[0,1]:x′′∈C[0,1],x(0)=x(1),x′(0)=x′(1)} endowed with the norm ∥x∥X=maxt∈[0,1]|x(t)|, and let Y=C[0,1] with the norm ∥y∥Y=maxt∈[0,1]|y(t)| and K={x∈X:x(t)≥0,t∈[0,1]}, then K is a cone of X.We define(3.1)dom⁡L=X,L:dom⁡L⟶Y,Lx(t)=-x′′(t),N:X⟶Y,Nx(t)=-f(t,x(t)),
then PBVP (1.1) can be written as(3.2)Lx=Nx,x∈K.It is easy to check that(3.3)Ker⁡L={x∈dom⁡L:x(t)≡con⁡[0,1],c∈R},Im⁡L={y∈Y:∫01y(s)ds=0},dim⁡Ker⁡L=codimIm⁡L=1,
so that L is a Fredholm operator of index zero.Next, define the projectionsP:X→X by(3.4)Px=∫01x(s)ds
and Q:Y→Y by(3.5)Qy=∫01y(s)ds.Furthermore, we define the isomorphismJ:ImQ→ImP as Jy=βy, where β=1/24. It is easy to verify that the inverse operator L1-1:ImL→domL∩KerP of L|domL∩KerP:domL∩KerP→ImL as (L1-1y)(t)=∫01G(s,t)y(s)ds, where(3.6)G(s,t)={s2(1-2t+s),0≤s<t≤1,12(1-s)(2t-s),0≤t≤s≤1.For notational convenience, we setH(s,t)=1/24+G(s,t)-∫01G(s,t)ds or(3.7)H(s,t)={124+s2(1-2t+s)+t22-t2+112,0≤s<t≤1,124+12(1-s)(2t-s)+t22+t2+112,0≤t≤s≤1.
By routine methods of advanced calculus, we get maxs,t∈[0,1]H(s,t)=1/8.Now we can state and prove our main results.Theorem 3.1.
Assume that there exist two positive numbersa,b such that
(H1)
f(t,x)≤x,forallt∈[0,1],x∈[min{a,b},max{a,b}](H2)
if one of the two conditions
(i)
maxt∈[0,1]f(t,a)<0,mint∈[0,1]f(t,b)>0,(ii)
mint∈[0,1]f(t,a)>0,maxt∈[0,1]f(t,b)<0
is satisfied, then the PBVP (1.1) has at least one positive solution x*∈K satisfying min{a,b}≤∥x*∥X≤max{a,b}.Proof.
It is easy to seea≠b. Without loss of generality, let a<b.
First, we note thatL, as so defined, is Fredholm of index zero, L1-1 is compact by Arzela-Ascoli theorem, and thus L-λN is A-proper for λ∈[0,1] by (a) of Lemma 2.10.
For eachx∈K, then by condition (H1)(3.8)Px+JQNx=∫01x(s)ds-124∫01f(s,x(s))ds≥2324∫01x(s)ds≥0,Px+JQNx+L1-1(I-Q)Nx=∫01x(s)ds-124∫01f(s,x(s))ds+∫01G(s,t)[-f(s,x(s))+∫01f(s,x(s))ds]ds=∫01x(s)ds-∫01H(s,t)f(s,x(s))ds≥∫01(1-H(s,t))x(s)ds≥0.
This implies that condition(C1) of Lemma 2.8 is satisfied. To apply Lemma 2.8, we should define two open bounded subsets Ω1,Ω2 of X so that (C2)–(C5) of Lemma 2.8 hold.
We prove only Case(H2)(i). In the same way, we can prove Case (H2)(ii).
Let(3.9)Ω1={x∈X:‖x‖X<a},Ω2={x∈X:‖x‖X<b}.
Clearly, Ω1 and Ω2 are bounded and open sets and
(3.10)θ∈Ω1⊂Ω¯1⊂Ω2.
Next we show that (H2)(i) implies (C2). For this purpose, suppose that there exist x0∈K∩∂Ω2 and λ0∈(0,1] such that Lx0=λ0Nx0 then x0′′(t)=λ0f(t,x0(t)) for all t∈[0,1]. Let t0∈[0,1], such that x0(t0)=maxt∈[0,1]x0(t)=b. From boundary conditions, we have t0∈[0,1). Then we have the following two cases. Case 1 (t0=0).
In this case,x0′(0)≤0,x0′(1)≥0. Since boundary condition x0′(0)=x0′(1), we have x0′(0)=x0′(1)=0. So we have x0′′(0)=λ0f(0,x0(0))=λ0f(0,b)>0. It follows from x0′′(t) being continuous in [0,1] that there exists δ∈(0,1), such that x0′′(t)>0 when t∈(0,δ]. Thus, x0′(t)=x0′(0)+∫0tx0′′(s)ds>0. Hence,
(3.11)x0(t)=x0(0)+∫0tx0′(s)ds>x0(0),t∈(0,δ],
and x0(0) is not the maximum on [0,1], a contradiction.Case 2 (t0∈(0,1)).
In this case,x0′(t0)=0,x0′′(t0)≤0. This gives
(3.12)0≥x0′′(t0)=λ0f(t0,x(t0))=λ0f(t0,b)>0,
which contradicts (H2)(i). So for each x∈∂Ω2∩K and λ∈(0,1], we have Lx≠λNx. Thus, (C2) of Lemma 2.8 is satisfied.
To prove(C4) of Lemma 2.8, we define the bilinear form [·,·]:Y×X→ℝ as
(3.13)[y,x]=∫01y(t)x(t)dt.
It is clear that [·,·] is continuous and satisfies [y,x]=0 for every x∈KerL,y∈ImL. In fact, for any x∈KerL and y∈ImL, we have x≡c, a constant, and there exists x∈X such that y(t)=-x′′(t) for each t∈[0,1]. By x′(0)=x′(1), we get
(3.14)[y,x]=∫01y(t)x(t)dt=-c∫01x′′(t)dt=0.
Let x∈KerL∩∂Ω2∩K, then x(t)≡b, so we have by condition (H2)(i)
(3.15)QNx=-∫01f(t,b)dt≠0,[QNx,x]=-∬01f(t,b)dt⋅bds<0.
Thus, (C3) and (C4) of Lemma 2.8 are verified.
Finally, we prove(C5) of Lemma 2.8 is satisfied. We may suppose that Lx≠Nx,forallx∈∂Ω1∩K. Otherwise, the proof is completed.
Lete≡1∈K1∖{θ}. We claim that
(3.16)Lx-Nx≠μe,∀x∈∂Ω1∩K,μ≥0.
In fact, if not, there exist x2∈∂Ω1∩K,μ1>0, such that
(3.17)Lx2-Nx2=μ1.
Since QL=θ, operating on both sides of the latter equation by Q, we obtain(3.18)QNx2+Qμ1=0,
that is,
(3.19)∫01(-f(t,x2)+μ1)dt=0.
For any x2∈∂Ω1∩K, we have ∥x2∥X=a. Then there exists t1∈[0,1], such that x2(t1)=a. By condition (H2)(i) and μ1>0,
(3.20)∫01(-f(t,x2(t1))+μ1)dt=∫01(-f(t,a)+μ1)dt>0,
in contradiction to (3.19). So (3.16) holds; that is, (C5) of Lemma 2.8 is verified.
Thus, all conditions of Lemma2.8 are satisfied and there exists x∈K∩(Ω¯2∖Ω1) such that Lx=Nx and the assertion follows. Thus, x*∈K and a≤∥x*∥X≤b.Let[c] be the integer part of c. The following result concerns the existence of n positive solutions.Theorem 3.2.
Assume that there existn+1 positive numbers  a1<a2<⋯<an+1 such that
(H1)′
f(t,x)≤x,forallt∈[0,1],x∈[a1,an+1],(H2)′
if one of the two conditions
(i)
maxt∈[0,1]f(t,a2i-1)<0,i=1,2,…,[(n+2)/2], mint∈[0,1]f(t,a2i)>0,i=1,2,…,[(n+1)/2](ii)
mint∈[0,1]f(t,a2i-1)>0,i=1,2,…,[(n+2)/2], mint∈[0,1]f(t,a2i)<0,i=1,2,…,[(n+1)/2]
is satisfied, then the PBVP (1.1) has at least n positive solutions xi*∈K,i=1,2,…,n satisfying ai<∥xi*∥X<ai+1.Proof.
Modeling the proof of Theorem3.1, we can prove that if there exist two positive numbers a,b such that maxt∈[0,1]f(t,a)<0,mint∈[0,1]f(t,b)>0, then PBVP (1.1) has at least one positive solution x*∈K satisfying min{a,b}<∥x*∥X<max{a,b}.
By the claim, for every pair of positive numbers{ai,ai+1},i=1,2,…,n, (1.1) has at least n positive solutions xi*∈K satisfying ai<∥xi*∥X<ai+1.We have the following existence result for two positive solutions.Corollary 3.3.
Assume that there exist three positive numbersa1<a2<a3 such that (H1)′′
f(t,x)≤x,forallt∈[0,1],x∈[a1,a3],(H2)′′
if one of the two conditions
(i)
maxt∈[0,1]f(t,a1)<0,mint∈[0,1]f(t,a2)>0, maxt∈[0,1]f(t,a3)<0,(ii)
mint∈[0,1]f(t,a1)>0,maxt∈[0,1]f(t,a2)<0,mint∈[0,1]f(t,a3)>0is satisfied, then the PBVP (1.1) has at least two positive solutions x1*,x2*∈K satisfying a1≤∥x1*∥X<a2<∥x2*∥X≤a3.We also have the following existence result for three positive solutions.Corollary 3.4.
Assume that there exist four positive numbersa1<a2<a3<a4 such that(H1)′′′
f(t,x)≤x,forallt∈[0,1],x∈[a1,a4],(H2)′′′
if one of the two conditions
(i)
maxt∈[0,1]f(t,a1)<0,mint∈[0,1]f(t,a2)>0,maxt∈[0,1]f(t,a3)<0, mint∈[0,1]f(t,a4)>0, and(ii)
mint∈[0,1]f(t,a1)>0, maxt∈[0,1]f(t,a2)<0,mint∈[0,1]f(t,a3)>0, maxt∈[0,1]f(t,a4)<0,is satisfied, then the PBVP (1.1) has at least three positive solutions x1*,x2*,x3*∈K satisfying a1≤∥x1*∥X<a2<∥x2*∥X<a3<∥x3*∥X≤a4.Remark 3.5.
From similar arguments and techniques, we can also deal with the following periodic boundary value problem (PBVP)(3.21)-x′′(t)=f(t,x),0<t<1,x(0)=x(1),x′(0)=x′(1).
We can also verify that the similar results presented in this paper are valid for PBVP (3.21); we omit the details here.


## 4. Some Examples

In this section, we give some examples to illustrate the main results of the paper.Example 4.1.
Consider the following second-order periodic boundary value problem (PBVP):(4.1)x′′(t)=45(t2-t-1)(2x3+3x2-12x+6)x,0<t<1,x(0)=x(1),x′(0)=x′(1),
where f(t,x(t))=(t2-t-1)(2x3+3x2-12x+6)x. In this case, f(t,x)≤x,x≥0,0≤t≤1.
Corresponding to the assumptions of Corollary3.3, we set a1=1/2,a2=1, and a3=2. It is easy to check that the other conditions of Corollary 3.3 are satisfied; hence, PBVP (4.1) has at least two positive solutions x1*,x2* satisfying 1/2≤∥x1*∥X<1<∥x2*∥X≤2.Example 4.2.
Consider the periodic boundary value problem (PBVP)(4.2)x′′(t)=5t+210sin⁡x,0<t<1,x(0)=x(1),x′(0)=x′(1).
Now, let f(t,x)=((5t+2)/10)sinx; thus, f(t,x)≤x,x≥0,0≤t≤1. Set a1=π/2,a2=3π/2,a3=5π/2,a4=7π/2. Then Corollary 3.4 ensures that there exist at least three positive solutions x1*,x2*,x3* satisfying π/2≤∥x1*∥X<3π/2<∥x2*∥X<5π/2<∥x3*∥X≤7π/2.



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*Source: 725646-2012-05-07.xml*