AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation16289110.1155/2009/162891162891Research ArticleExistence and Uniqueness of Periodic Solutions of Mixed Monotone Functional Differential EquationsKangShugui1ChengSui Sun2PetersonAllan1Institute of Applied MathematicsShanxi Datong UniversityDatongShanxi 037009Chinasxdtdx.edu.cn2Department of MathematicsTsing Hua UniversityHsinchu 30043Taiwan tsinghua.edu.cn200917082009200921042009030720092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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(tτ(t)))+f2(t,y(tτ(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., ) 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,f2C(R2,R)   and T-periodic with respect to the first variable, and that a(t)>0 for tR.

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., ). 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 PE is called a cone if it satisfies the following two conditions: (i) xP and λ0λxP; (ii) xP and -xPx=θ.

Every cone PE induces an ordering in E given by xy, if and only if y-xP. A cone P is called normal if there is M>0 such that x,yE and θxyxMy. P is said to be solid if the interior P0 of P is nonempty.

Assume that u0,v0E and u0v0. The set {xE:u0xv0} is denoted by [u0,v0]. Assume that h>θ. Let Ph={xE:λ,μ>0  such  that  λhxμh}. Obviously if P is a solid cone and hP0, then Ph=P0.

Definition 1.1.

Let E be an ordered Banach space, and let DE. An operator is called mixed monotone on D×D if A:D×DE and A(x1,y1)A(x2,y2) for any x1,x2,y1,y2D that satisfy x1x2 and y2y1.Also,  x*D is called a fixed point of A if A(x*,x*)=x*.

A function f:IRR 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,yI. We say that the function f is a concave function if -f is a convex function.

Definition 1.2.

Assume f:IRR and 0α<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 xI and t(0,1).

Definition 1.3.

Let DE, and let A:D×DE. 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 xD 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 IR and x0I. Recall that a function f:IR is said to be left lower semicontinuous at x0 if liminfnf(xn)f(x0) for any monotonically increasing sequence {xn}I that converges to x0.

The proof of the following theorem can be found in .

Theorem 1.4.

Let P be a normal cone of E. Let u0,v0E such that u0v0, 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 u0r0v0;

u0A(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 nN, then limnxn=x* and limnyn=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:RR is said to be a T-periodic solution of (1.1) if substitution of it into (1.1) yields an identity for all tR.

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<mlim0t,sTG(t,s)G(t,s)max0t,sTG(t,s)M<.

Now let CT(R) be the Banach space of all real T-periodic continuous functions y:RR endowed with the usual linear structure as well as the norm y=supt[0,T]|y(t)|. Then P={ϕCT(R):ϕ(x)0,xR} is a normal cone of CT(R).

Definition 2.1.

The functions v0,ω0CT1(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×PCT(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 sR,f1(s,x) is an increasing function of x, and f2(s,x) is a decreasing function of x;

there exist u0,v0P 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 sR,f1(s,·) is an α-concave function, f2(s,·) is a convex function;

there exist ε1/(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 limnxn=x* and limnyn=x*.

Proof.

The mapping A:P×PCT(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 z0R such that m(z0)=minzRm(z)<0. Then m(z0)-a(z0)m(z0)>0, which is a contradiction since m(z0)=minzRm(z). Thus u0A(u0,v0). Similarly, we can prove A(v0,u0)v0. Then we have u1A(u1,v1),A(v1,u1)v1,u0u1u2unvnv2v1v0. From condition (C2), we know that u1εv1. Since u1v1, we must have 0<ε1.

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 zG, 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)))dszz+TG(z,s)f1(s,u(s-τ(s)))ds+tzz+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)))dstαzz+TG(z,s)f1(s,u(s-τ(s)))ds+zz+TG(z,s)f2(s,v(s-τ(s)))dstαA(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+εt1,t(0,1). From 0<ε1, we know that εtα-εt+ttα1 for any 0<t<1, therefore εtα1-t+εt1,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 ε1/(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 limnxn=x* and limnyn=x*. The proof is complete.

Theorem 2.3.

Suppose that conditions (B1) and (B2) hold, and

there exist r0>0 such that u0r0v0;

for any sR,f1(s,·) is an α-concave function and f2(s,ty)[(1+η)t]-1f2(s,y) for any yP 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)  and  yn=A(yn-1,xn-1) for nN, then limnxn and limnyn=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 sR,f1(s,·) is a concave function; f2(s,ty)[(1+η)t]-1f2(s,y) for any yP 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-11t2+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)  and  yn=A(yn-1,xn-1) for nN, then limnxn=x* and limnyn=x*.

Proof.

Set un=A(un-1,vn-1) and vn=A(vn-1,un-1) for nN. Then we know that u1A(u1,v1),A(v1,u1)v1,u0u1u2unvnv2v1v0. From (EH2) we have u1εv1. 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 u0r0v0;

f1(s,x)>0 and f2(s,x)>0 for any s,xR, and there exist e>0,f1(s,tx)(1+η)tf1(s,x) for any xPe and t(0,1), where Pe={xE:λ,μ>0 such that λexμe},f2(s,tx)[(1+ζ)t]-1f2(s,x) for any xP 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)  and  yn=A(yn-1,xn-1) for nN, then limnxn=x* and limnyn=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 u0v0, there exists r0 such that u0r0v0;

f1(s,x)>0 and f2(s,x)>0 for any s,xR; there exist e>0 and η=η(t,x) such that f1(s,tx)(1+η)tf1(s,x) for any xPe and t(0,1), where Pe={xE:λ,μ>0 such that λexμe}; for any sR, 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)  and  yn=A(yn-1,xn-1) for nN, then limnxn=x* and limnyn=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,xR;f1(s,tx)(1+η)tf1(s,x) for any xPe and t(0,1), where Pe={xE:λ,μ>0 such that λexμe}; for any sR,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)  and  yn=A(yn-1,xn-1) for nN, then limnxn=x* and limnyn=x*.

Proof.

Set un=A(un-1,vn-1) and vn=A(vn-1,un-1) for nN. Then we have u1A(u1,v1),A(v1,u1)v1, and u0u1u2unvnv2v1v0. From (JH2) we can see that u1εv1.

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 sR,f1(s,·) is an α1-concave function,f2(s,·) is a (-α2)-convex function; where 0α1+α2<1;

there exist r0>0 such that u0r0v0.

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)  and  yn=A(yn-1,xn-1) for nN, then limnxn=x* and limnyn=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 sR,f1(s,·) is a α1-concave function, f2(s,·) is a (-α2)-convex function, where 0α1+α2<1;

there exist u0,v0P0 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)  and  yn=A(yn-1,xn-1), then xnx*,ynx*(n).

Indeed, from u0,v0P0, we know that there exists r0>0 such that u0r0v0. 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+103/2qmaxa(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+1000qmax100pmin+1000qmin. Then (3.1) will have a unique solution y=y*(t) that satisfies 10-3y*(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))]dsnN,ωn(t)=tt+TG(t,s)[p(s)ωn-11/3(s-τ(s))+q(s)vn-1-1/2(s-τ(s))]dsnN, 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={ϕCT(R):ϕ(x)0,xR} 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 nN, then limnvn=y* and limnωn=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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