# Existence and Uniqueness of Periodic Solutions of Mixed Monotone Functional Differential Equations

Authors: Shugui Kang; Sui Sun Cheng
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

This paper deals with the existence and uniqueness of periodic solutions for the first-order
functional differential equationy′(t)=−a(t)y(t)+f1(t,y(t−τ(t)))+f2(t,y(t−τ(t)))
with periodic coefficients and delays. We choose the mixed monotone operator theory to approach
our problem because such methods, besides providing the usual existence results,
may also sometimes provide uniqueness as well as additional numerical schemes for the computation
of solutions.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence and uniqueness of periodic solutions for the first-order functional differential equation (cf., e.g., [1–5])
(1.1)y′(t)=-a(t)y(t)+f1(t,y(t-τ(t)))+f2(t,y(t-τ(t))),(1.2)x′(t)=a(t)x(t)-f1(t,x(t-τ(t)))-f2(t,x(t-τ(t))),
where we will assume that a=a(t) and τ=τ(t) are continuous T-periodic functions, that T>0, that f1,f2∈C(R2,R) and T-periodic with respect to the first variable, and that a(t)>0 for t∈R.Functional differential equations with periodic delays such as those stated above appear in a number of ecological, economical, control and physiological, and other models. One important question is whether these equations can support periodic solutions, and whether they are unique. The existence question has been studied extensively by many authors (see, e.g., [1–5]). The uniqueness problem seems to be more difficult, and less studies are known.We will tackle the existence and uniqueness question by fixed point theorems for mixed monotone operators. We choose this approach because such fixed point methods, besides providing the usual existence and uniqueness results, sometimes may also provide additional numerical schemes for the computation of solutions.We first recall some useful terminologies (see [6, 7]). Let E be a real Banach space with zero element θ. A nonempty closed convex set P⊂E is called a cone if it satisfies the following two conditions: (i) x∈P and λ≥0⇒λx∈P; (ii) x∈P and -x∈P⇒x=θ.Every coneP⊂E induces an ordering in E given by x≤y, if and only if y-x∈P. A cone P is called normal if there is M>0 such that x,y∈E and θ≤x≤y⇒∥x∥≤M∥y∥. P is said to be solid if the interior P0 of P is nonempty.Assume thatu0,v0∈E and u0≤v0. The set {x∈E:u0≤x≤v0} is denoted by [u0,v0]. Assume that h>θ. Let Ph={x∈E:∃λ,μ>0suchthatλh≤x≤μh}. Obviously if P is a solid cone and h∈P0, then Ph=P0.Definition 1.1.
LetE be an ordered Banach space, and let D⊂E. An operator is called mixed monotone on D×D if A:D×D→E and A(x1,y1)≤A(x2,y2) for any x1,x2,y1,y2∈D that satisfy x1≤x2 and y2≤y1.Also,x*∈D is called a fixed point of A if A(x*,x*)=x*.A functionf:I⊂R→R is said to be convex in I if f(tx+(1-t)y)≤tf(x)+(1-t)f(y) for any t∈[0,1] and any x,y∈I. We say that the function f is a concave function if -f is a convex function.Definition 1.2.
Assumef:I⊂R→R and 0≤α<1.Then,f is said to be an α-concave or -α-convex function if f(tx)≥tαf(x) or, respectively, f(tx)≤t-αf(x) for x∈I and t∈(0,1).Definition 1.3.
LetD⊂E, and let A:D×D→E. The operator A is called (ϕ-concave)-(-ψ-convex) if there exist functions ϕ:(0,1]×D→(0,∞) and ψ:(0,1]×D→(0,∞) such that(H0)
t<ϕ(t,x)ψ(t,x)≤1 for x∈D and t∈(0,1),(H1)
A(tx,y)≥ϕ(t,x)A(x,y) for any t∈(0,1) and (x,y)∈D×D,(H2)
A(x,ty)≤A(x,y)/ψ(t,y) for any t∈(0,1) and (x,y)∈D×D.
Assume thatI⊂R and x0∈I. Recall that a function f:I→R is said to be left lower semicontinuous at x0 if liminfn→∞f(xn)≥f(x0) for any monotonically increasing sequence {xn}⊂I that converges to x0.
The proof of the following theorem can be found in [7].Theorem 1.4.
LetP be a normal cone of E. Let u0,v0∈E such that u0≤v0, and let A:[u0,v0]×[u0,v0]→E be a mixed monotone operator. If A is a (ϕ-concave)-(-ψ-convex) operator and satisfies the following three conditions:(A1)
there existsr0>0 such that u0≥r0v0;(A2)
u0≤A(u0,v0) and A(v0,u0)≤v0;(A3)
there existsω0∈[u0,v0] such that minx∈[u0,v0]ϕ(t,x)ψ(t,x)=ϕ(t,ω0)ψ(t,ω0) for each t∈(0,1), and ϕ(t,ω0)ψ(t,ω0) is left lower semicontinuous at any t∈(0,1),then A has a unique fixed point x*∈[u0,v0], that is, x*=A(x*,x*), and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1) and yn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Remark 1.5.
Condition (A3) in Theorem1.4 can be replaced by (A3') ϕ(t,x)ψ(t,x) is monotone in x and left lower semicontinuous at any t∈(0,1).


## 2. Main Results

A realT-periodic continuous function y:R→R is said to be a T-periodic solution of (1.1) if substitution of it into (1.1) yields an identity for all t∈R.It is well known (see, e.g., [1, 2]) that  (1.1) has a T-periodic solution y(t) if, and only if, y(t) is a T-periodic solution of the equation
(2.1)y(t)=∫tt+TG(t,s)f1(s,y(s-τ(s)))ds+∫tt+TG(t,s)f2(s,y(s-τ(s)))ds,
where
(2.2)G(t,s)=exp(∫tsa(u)du)exp(∫0Ta(u)du)-1,
and (1.2) has a T-periodic solution x(t) if, and only if, x(t) is a T-periodic solution of the equation
(2.3)x(t)=∫t-TtH(t,s)f1(s,x(s-τ(s)))ds+∫t-TtH(t,s)f2(s,x(s-τ(s)))ds,
where
(2.4)H(t,s)=exp(∫sta(u)du)exp(∫0Ta(u)du)-1.
Furthermore, the Cauchy function G(t,s) satisfies
(2.5)0<m≡lim0≤t,s≤TG(t,s)≤G(t,s)≤max0≤t,s≤TG(t,s)≡M<∞.Now letCT(R) be the Banach space of all real T-periodic continuous functions y:R→R endowed with the usual linear structure as well as the norm
(2.6)∥y∥=supt∈[0,T]|y(t)|.
Then P={ϕ∈CT(R):ϕ(x)≥0,x∈R} is a normal cone of CT(R).Definition 2.1.
The functionsv0,ω0∈CT1(R) are said to form a pair of lower and upper quasisolutions of (1.1) if v0(t)≤ω0(t) and
(2.7)v0′(t)≤-a(t)v0(t)+f1(t,v0(t-τ(t)))+f2(t,ω0(t-τ(t))),
as well as
(2.8)ω0′(t)≥-a(t)ω0(t)+f1(t,ω0(t-τ(t)))+f2(t,v0(t-τ(t))).
We remark that the term quasi is used in the above definition to remind us that they are different from the traditional concept of lower and upper solutions (cf. (2.7) with v0′(t)≤-a(t)v0(t)+f1(t,v0(t-τ(t)))+f2(t,v0(t-τ(t)))).
LetA:P×P→CT(R) be defined by
(2.9)A(u,v)(t)=∫tt+TG(t,s)f1(s,u(s-τ(s)))ds+∫tt+TG(t,s)f2(s,v(s-τ(s)))ds.
We need two basic assumptions in the main results:(B1)
for anys∈R,f1(s,x) is an increasing function of x, and f2(s,x) is a decreasing function of x;(B2)
there existu0,v0∈P such that u0 and v0 form a respective pair of lower and upper quasisolutions for (1.1).Theorem 2.2.
Suppose that conditions (B1) and  (B2) hold, and(C1)
for anys∈R,f1(s,·) is an α-concave function, f2(s,·) is a convex function;(C2)
there existε≥1/(2-α) such that A(u0,v0)εA(v0,θ).Then (1.1) has a unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1) and yn=A(yn-1,xn-1), then limn→∞xn=x* and limn→∞yn=x*.Proof.
The mappingA:P×P→CT(R) is a mixed monotone operator in view of (B1). Let
(2.10)u1(z)=∫zz+TG(z,s)f1(s,u0(s-τ(s)))ds+∫zz+TG(z,s)f2(s,v0(s-τ(s)))ds.
Then
(2.11)u1′(z)=-a(z)u1(z)+G(z,z+T)f1(z+T,u0(z+T-τ(z+T)))-G(z,z)f1(z,u0(z-τ(z)))+G(z,z+T)f2(z+T,v0(z+T-τ(z+T)))-G(z,z)f2(z,v0(z-τ(z)))=-a(z)u1(z)+G(z,z+T)f1(z,u0(z-τ(z)))-G(z,z)f1(z,u0(z-τ(z)))+G(z,z+T)f2(z,v0(z-τ(z)))-G(z,z)f2(z,v0(z-τ(z)))=-a(z)u1(z)+f1(z,u0(z-τ(z)))+f2(z,v0(z-τ(z))).
Set m(z)=u1(z)-u0(z). Then
(2.12)m′(z)=u1′(z)-u0′(z)-a(z)m(z).
Next, we will prove that m(z)0. Suppose to the contrary that there exists z0∈R such that
(2.13)m(z0)=minz∈Rm(z)<0.
Then m′(z0)≥-a(z0)m(z0)>0, which is a contradiction since m(z0)=minz∈Rm(z). Thus u0≤A(u0,v0). Similarly, we can prove A(v0,u0)≤v0. Then we have
(2.14)u1≤A(u1,v1),A(v1,u1)≤v1,u0≤u1≤u2≤⋯≤un≤⋯≤vn≤⋯≤v2≤v1≤v0.
From condition (C2), we know that u1≥εv1. Since u1≤v1, we must have 0<ε≤1.
We will prove thatA:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator, where
(2.15)ϕ(t,u)=tα,ψ(t,v)=ε1-(1-ε)t,t∈(0,1),u,v∈[u0,v0].
In fact, for any u,v∈[u0,v0],t∈(0,1), and z∈G, we have
(2.16)A(u,tv)(z)=A(u,tv+(1-t)θ)(z)=∫zz+TG(z,s)f1(s,u(s-τ(s)))ds+∫zz+TG(z,s)f2(s,(tv+(1-t)θ)(s-τ(s)))ds≤∫zz+TG(z,s)f1(s,u(s-τ(s)))ds+t∫zz+TG(z,s)f2(s,v(s-τ(s)))ds+(1-t)∫zz+TG(z,s)f2(s,θ(s-τ(s)))ds=tA(u,v)(z)+(1-t)A(u,θ)(z)≤tA(u,v)(z)+(1-t)A(v0,θ)(z)≤tA(u,v)(z)+1-tεA(u0,v0)(z)≤tA(u,v)(z)+1-tεA(u,v)(z)=1ψ(t,v)A(u,v)(z),
thus
(2.17)A(u,tv)≤1ψ(t,v)A(u,v),A(tu,v)(z)=∫zz+TG(z,s)f1(s,tu(s-τ(s)))ds+∫zz+TG(z,s)f2(s,v(s-τ(s)))ds≥tα∫zz+TG(z,s)f1(s,u(s-τ(s)))ds+∫zz+TG(z,s)f2(s,v(s-τ(s)))ds≥tαA(u,v)(z)=ϕ(t,u)A(u,v)(z),
so that
(2.18)A(tu,v)≥ϕ(t,u)A(u,v).
Further we can prove
(2.19)t<ϕ(t,u)ψ(t,u)≤1
for any t∈(0,1) and u∈[u0,v0]. Indeed, since
(2.20)ϕ(t,u)ψ(t,u)=εtα1-t+εt,t∈(0,1),u∈[u0,v0],
hence, we only need to prove
(2.21)t<εtα1-t+εt≤1,t∈(0,1).
From 0<ε≤1, we know that εtα-εt+t≤tα≤1 for any 0<t<1, therefore
(2.22)εtα1-t+εt≤1,t∈(0,1).
On the other hand, the function
(2.23)g(t)=εtα-1+(1-ε)t-1,t∈[0,1]
satisfies g(1)=0 and g′(t)=ε(α-1)tα-2+1-ε. From ε≥1/(2-α), we have ε(1-α)/(1-ε)≥1. Then t2-α<ε(1-α)/(1-ε) for 0<t<1. Thus ε(α-1)tα-2+1-ε<0, that is, g′(t)<0. Therefore, g(t)>0 for any 0<t<1. Finally,
(2.24)t<εtα1-t+εt,t∈(0,1).
Therefore, A:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator. From (2.20), ϕ(t,u)ψ(t,u) is monotone in u and is left lower semicontinuous at t. By Theorem 1.4, we know that A has a unique fixed point x*∈[u1,v1]⊂[u0,v0]. Hence (1.1) has a unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1) and yn=A(yn-1,xn-1), then limn→∞xn=x* and limn→∞yn=x*. The proof is complete.Theorem 2.3.
Suppose that conditions (B1) and  (B2) hold, and(D1)
there existr0>0 such that u0≥r0v0;(D2)
for anys∈R,f1(s,·) is an α-concave function and f2(s,ty)≤[(1+η)t]-1f2(s,y) for any y∈P and t∈[0,1], where η=η(t,y) satisfies the following conditions:
(DH1)
η(t,y) is monotone in y and left lower semicontinuous in t;(DH2)
for any(t,y)∈(0,1)×[u0,v0],(2.25)1tα-1<η(t,y)≤1t-1<1t1+α-1.Then (1.1) has a unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn and limn→∞yn=x*.Proof.
We assert thatA:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
(2.26)ϕ(t,u)=tα,ψ(t,v)=[1+η(t,v)]tfort∈(0,1),u,v∈[u0,v0].
In fact,
(2.27)A(tu,v)≥tαA(u,v)=ϕ(t,u)A(u,v),A(u,tv)≤1t[1+η(t,v)]A(u,v)=1ψ(t,v)A(u,v)
for any u,v∈[u0,v0] and t∈(0,1). From (2.25), we know that t<ϕ(t,u)ψ(t,u)≤1. Thus A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator. We may now complete our proof by Theorem 1.4.Theorem 2.4.
Suppose that conditions (B1) and  (B2) hold, and(E1)
for anys∈R,f1(s,·) is a concave function; f2(s,ty)≤[(1+η)t]-1f2(s,y) for any y∈P and t∈[0,1], and η=η(t,y) satisfies the following conditions:
(EH1)
there existsε∈(0,1] such that A(θ,v0)≥εA(v0,u0);(EH2)
for any(t,y)∈(0,1)×[u0,v0],(2.28)1t+ε(1-t)-1<η(t,y)≤1t-1≤1t2+εt(1-t)-1.Then (1.1) has unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Proof.
Setun=A(un-1,vn-1) and vn=A(vn-1,un-1) for n∈N. Then we know that
(2.29)u1≤A(u1,v1),A(v1,u1)≤v1,u0≤u1≤u2≤⋯≤un≤⋯≤vn≤⋯≤v2≤v1≤v0.
From (EH2) we have u1≥εv1. Next we will prove that A:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator, where
(2.30)ϕ(t,u)=t+ε(1-t),ψ(t,v)=[1+η(t,v)]tfort∈(0,1),u,v∈[u0,v0].
In fact, for any u,v∈[u0,v0] and t∈(0,1),(2.31)A(tu,v)=A(tu+(1-t)θ,v)≥tA(u,v)+(1-t)A(θ,v)≥tA(u,v)+(1-t)A(θ,v0)≥tA(u,v)+ε(1-t)A(v0,u0)≥tA(u,v)+ε(1-t)A(u,v)=ϕ(t,u)A(u,v),A(u,tv)≤1[1+η(t,v)]tA(u,v)=1ψ(t,v)A(u,v).
From (2.28), we know that t<ϕ(t,u)ψ(t,u)≤1. Thus A:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator. We may now complete our proof by Theorem 1.4.Theorem 2.5.
Suppose that conditions (B1) and (B2) hold, and(F1)
there existsr0>0 such that u0≥r0v0;(F2)
f1(s,x)>0 and f2(s,x)>0 for any s,x∈R, and there exist e>0,f1(s,tx)≥(1+η)tf1(s,x) for any x∈Pe and t∈(0,1), where Pe={x∈E:∃λ,μ>0 such that λe≤x≤μe},f2(s,tx)≤[(1+ζ)t]-1f2(s,x) for any x∈P and t∈[0,1];η=η(t,x),ζ=ζ(t,x) satisfies the following conditions:
(FH1)
(1+η(t,x))(1+ζ(t,x)) is monotone in x and left lower semicontinuous in t;(FH2)
for any(t,x)∈(0,1)×[u0,v0],(2.32)1+η(t,x)≤1t,1+ζ(t,x)≤1t,1t-1<η(t,x)+ζ(t,x)+η(t,x)ζ(t,x)≤1t2-1.Then (1.1) has a unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Proof.
We may easily prove thatA:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
(2.33)ϕ(t,u)=[1+η(t,u)]t,ψ(t,v)=[1+ζ(t,v)]tfort∈(0,1),u,v∈[u0,v0].
And from (FH2) we know that
(2.34)t<ϕ(t,u)ψ(t,u)≤1
for any t∈(0,1) and u∈[u0,v0]. Now the proof can be completed by means of Theorem 1.4.Theorem 2.6.
Suppose that conditions (B1) and  (B2) hold, and(G1)
ifu0≤v0, there exists r0 such that u0≥r0v0;(G2)
f1(s,x)>0 and f2(s,x)>0 for any s,x∈R; there exist e>0 and η=η(t,x) such that f1(s,tx)≥(1+η)tf1(s,x) for any x∈Pe and t∈(0,1), where Pe={x∈E:∃λ,μ>0 such that λe≤x≤μe}; for any s∈R, f2(s,·) is a (-α)-convex function, and η=η(t,x) satisfies the following conditions:
(GH1)
η(t,x) is monotone in x and left lower semicontinuous in t;(GH2)
for any(t,x)∈(0,1)×[u0,v0],(2.35)1+η(t,x)≤1t,1tα-1<η(t,x)≤1t-1<1t1+α-1.Then (1.1) has a unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Proof.
It is easily seen thatA:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
(2.36)ϕ(t,u)=[1+η(t,u)]t,ψ(t,v)=tαfort∈(0,1),u,v∈[u0,v0].
From (GH2), we know that t<ϕ(t,u)ψ(t,u)≤1. Then A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator. The proof may now be completed by means of Theorem 1.4.Theorem 2.7.
Suppose that conditions (B1) and  (B2) hold, and(J1)
f1(s,x)>0 and f2(s,x)>0 for any s,x∈R;f1(s,tx)≥(1+η)tf1(s,x) for any x∈Pe and t∈(0,1), where Pe={x∈E:∃λ,μ>0 such that λe≤x≤μe}; for any s∈R,f2(s,·) is a convex function; η=η(t,x) satisfies the following conditions:
(JH1)
η(t,x) is monotone in x and left lower semicontinuous in t;(JH2)
there existsε∈(1/2,1) such that A(u0,v0)≥εA(v0,θ) and
(2.37)(1-t)(1-ε)ε<η(t,x)≤1t-1<1-tεt for any (t,x)∈(0,1)×[u0,v0].
Then (1.1) has unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Proof.
Setun=A(un-1,vn-1) and vn=A(vn-1,un-1) for n∈N. Then we have u1≤A(u1,v1),A(v1,u1)≤v1, and
(2.38)u0≤u1≤u2≤⋯≤un≤⋯≤vn≤⋯≤v2≤v1≤v0.
From (JH2) we can see that u1≥εv1.
Next we will prove thatA:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator. We need only to verify that A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator, where
(2.39)ϕ(t,u)=[1+η(t,u)]t,ψ(t,v)=ε1-(1-εt)tfort∈(0,1),u,v∈[u0,v0].
In fact, for any u,v∈[u0,v0] and t∈(0,1), we have
(2.40)A(tu,v)≥[1+η(t,u)]tA(u,v)=ϕ(t,u)A(u,v),A(u,tv)=A(u,tv+(1-t)θ)≤tA(u,v)+(1-t)A(u,θ)≤tA(u,v)+(1-t)A(v0,θ)≤tA(u,v)+1-tεA(u0,v0)≤tA(u,v)+1-tεA(u,v)=1ψ(t,v)A(u,v).
From (JH2), we have t<ϕ(t,u)ψ(t,u)≤1. Then A:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator. The rest of the proof follows from Theorem 1.4.Theorem 2.8.
Suppose that conditions (B1) and (B2) hold, and(K1)
for anys∈R,f1(s,·) is an α1-concave function,f2(s,·) is a (-α2)-convex function; where 0≤α1+α2<1;(K2)
there existr0>0 such that u0≥r0v0.Then (1.1) has unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1) for n∈N, then limn→∞xn=x* and limn→∞yn=x*.Indeed, it is easily seen thatA:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
(2.41)ϕ(t,u)=tα1,ψ(t,v)=tα2fort∈(0,1),u,v∈[u0,v0].
The rest of the proof now follows from Theorem 1.4.IfP is a solid cone, we have the following result.Theorem 2.9.
Suppose thatP is a solid cone of E, that condition (B1) holds, and that(L1)
for anys∈R,f1(s,·) is a α1-concave function, f2(s,·) is a (-α2)-convex function, where 0≤α1+α2<1;(L2)
there existu0,v0∈P0 such that u0(t) and v0(t) form a pair of lower and upper quasisolutions for (1.1).Then (1.1) has unique solution x*∈[u0,v0], and for any x0,y0∈[u0,v0], if we set xn=A(xn-1,yn-1)andyn=A(yn-1,xn-1), then xn→x*,yn→x*(n→∞).Indeed, fromu0,v0∈P0, we know that there exists r0>0 such that u0≥r0v0. The rest of the proof is similar to that of Theorem 2.7.


## 3. An Example

As an example, consider the equation(3.1)y′(t)=-a(t)y(t)+[p(t)y1/3(t-τ(t))+q(t)y-1/2(t-τ(t))],
where p(t) and q(t) are nonnegative continuous T-periodic functions; a(t) and τ(t) are continuous T-periodic functions and satisfy
(3.2)pmax+103/2qmax≤a(t)≤102pmin+103qmin,
where pmax=maxt∈[0,T]p(t),  pmin=mint∈[0,T]p(t),  qmax=maxt∈[0,T]q(t),  qmin=mint∈[0,T]q(t), and pmax+1000qmax≤100pmin+1000qmin. Then (3.1) will have a unique solution y=y*(t) that satisfies 10-3≤y*(t)≤1. Furthermore, if we set v0(t)=10-3,ω0(t)=1,
(3.3)vn(t)=∫tt+TG(t,s)[p(s)vn-11/3(s-τ(s))+q(s)ωn-1-1/2(s-τ(s))]dsn∈N,ωn(t)=∫tt+TG(t,s)[p(s)ωn-11/3(s-τ(s))+q(s)vn-1-1/2(s-τ(s))]dsn∈N,
then {vn} and {ωn} converge uniformly to y*.Indeed, letCT(R) be the Banach space of all real T-periodic continuous functions defined on R and endowed with the usual linear structure as well as the norm
(3.4)∥y∥=supt∈[0,1]|y(t)|.
The set P={ϕ∈CT(R):ϕ(x)≥0,x∈R} is a normal cone of CT(R). Equation (3.1) has a T-periodic solution y(t), if and only if, y(t) is a T-periodic solution of the equation
(3.5)y(t)=∫tt+TG(t,s)[p(s)y1/3(s-τ(s))+q(s)y-1/2(s-τ(s))]ds,
where
(3.6)G(t,s)=exp(∫tsa(u)du)exp(∫0Ta(u)du)-1.
Set
(3.7)A(x,y)=∫tt+TG(t,s)[p(s)x1/3(s-τ(s))+q(s)y-1/2(s-τ(s))]ds,v0(t)=10-3,ω0(t)=1,α1=1/3, and α2=1/2. Then v0(t) and ω0(t) form a pair of lower and upper quasisolutions for (3.1). By Theorem 2.8, we know that (3.1) has a unique solution y*∈[10-3,1], and if we set vn=A(vn-1,ωn-1),ωn=A(ωn-1,vn-1) for n∈N, then limn→∞vn=y* and limn→∞ωn=y*.Other examples can be constructed to illustrate the other results in the previous section.



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*Source: 162891-2009-09-07.xml*