This paper deals with the existence and uniqueness of periodic solutions for the first-order
functional differential equation
y′(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.
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])
y′(t)=-a(t)y(t)+f1(t,y(t-τ(t)))+f2(t,y(t-τ(t))),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 cone P⊂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 that u0,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.
Let E 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 function f: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.
Assume f: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.
Let D⊂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
t<ϕ(t,x)ψ(t,x)≤1 for x∈D and t∈(0,1),
A(tx,y)≥ϕ(t,x)A(x,y) for any t∈(0,1) and (x,y)∈D×D,
A(x,ty)≤A(x,y)/ψ(t,y) for any t∈(0,1) and (x,y)∈D×D.
Assume that I⊂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.
Let P 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:
there exists r0>0 such that u0≥r0v0;
u0≤A(u0,v0) and A(v0,u0)≤v0;
there exists ω0∈[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 Theorem 1.4 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 real T-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
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
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
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
H(t,s)=exp(∫sta(u)du)exp(∫0Ta(u)du)-1.
Furthermore, the Cauchy function G(t,s) satisfies
0<m≡lim0≤t,s≤TG(t,s)≤G(t,s)≤max0≤t,s≤TG(t,s)≡M<∞.
Now let CT(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
∥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 functions v0,ω0∈CT1(R) are said to form a pair of lower and upper quasisolutions of (1.1) if v0(t)≤ω0(t) and
v0′(t)≤-a(t)v0(t)+f1(t,v0(t-τ(t)))+f2(t,ω0(t-τ(t))),
as well as
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)))).
Let A:P×P→CT(R) be defined by
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:
for any s∈R,f1(s,x) is an increasing function of x, and f2(s,x) is a decreasing function of x;
there exist u0,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
for any s∈R,f1(s,·) is an α-concave function, f2(s,·) is a convex function;
there exist ε≥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 mapping A:P×P→CT(R) is a mixed monotone operator in view of (B1). Let
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
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
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
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
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 that A:[u1,v1]×[u1,v1]→CT(R) is a (ϕ-concave)-(-ψ-convex) operator, where
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
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
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
A(tu,v)≥ϕ(t,u)A(u,v).
Further we can prove
t<ϕ(t,u)ψ(t,u)≤1
for any t∈(0,1) and u∈[u0,v0]. Indeed, since
ϕ(t,u)ψ(t,u)=εtα1-t+εt,t∈(0,1),u∈[u0,v0],
hence, we only need to prove
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
εtα1-t+εt≤1,t∈(0,1).
On the other hand, the function
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,
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
there exist r0>0 such that u0≥r0v0;
for any s∈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:
η(t,y) is monotone in y and left lower semicontinuous in t;
for any (t,y)∈(0,1)×[u0,v0],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 that A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
ϕ(t,u)=tα,ψ(t,v)=[1+η(t,v)]tfort∈(0,1),u,v∈[u0,v0].
In fact,
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
for any s∈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:
there exists ε∈(0,1] such that A(θ,v0)≥εA(v0,u0);
for any (t,y)∈(0,1)×[u0,v0],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.
Set un=A(un-1,vn-1) and vn=A(vn-1,un-1) for n∈N. Then we know that
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
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),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
there exists r0>0 such that u0≥r0v0;
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:
(1+η(t,x))(1+ζ(t,x)) is monotone in x and left lower semicontinuous in t;
for any (t,x)∈(0,1)×[u0,v0],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 that A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
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
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
if u0≤v0, there exists r0 such that u0≥r0v0;
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:
η(t,x) is monotone in x and left lower semicontinuous in t;
for any (t,x)∈(0,1)×[u0,v0],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 that A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
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
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:
η(t,x) is monotone in x and left lower semicontinuous in t;
there exists ε∈(1/2,1) such that A(u0,v0)≥εA(v0,θ) and
(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.
Set un=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
u0≤u1≤u2≤⋯≤un≤⋯≤vn≤⋯≤v2≤v1≤v0.
From (JH2) we can see that u1≥εv1.
Next we will prove that A:[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
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
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
for any s∈R,f1(s,·) is an α1-concave function,f2(s,·) is a (-α2)-convex function; where 0≤α1+α2<1;
there exist r0>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 that A:[u0,v0]×[u0,v0]→CT(R) is a (ϕ-concave)-(-ψ-convex) mixed monotone operator, where
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.
If P is a solid cone, we have the following result.
Theorem 2.9.
Suppose that P is a solid cone of E, that condition (B1) holds, and that
for any s∈R,f1(s,·) is a α1-concave function, f2(s,·) is a (-α2)-convex function, where 0≤α1+α2<1;
there exist u0,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, from u0,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
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
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,
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, let CT(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
∥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
y(t)=∫tt+TG(t,s)[p(s)y1/3(s-τ(s))+q(s)y-1/2(s-τ(s))]ds,
where
G(t,s)=exp(∫tsa(u)du)exp(∫0Ta(u)du)-1.
Set
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.
Acknowledgment
The first author is supported by Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University, by Development Foundation of Higher Education Department of Shanxi Province, and by Science and Technology Bureau of Datong City. The second author is supported by the National Science Council of R. O. China and also by the Natural Science Foundation of Guang Dong of P. R. China under Grant number (951063301000008).
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