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

Abstract

This paper deals with the existence and uniqueness of periodic solutions for the first-order functional differential equation ^{y[variant prime]} (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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Shugui Kang 1 and Sui Sun Cheng 2

Recommended by Allan Peterson

1, Institute of Applied Mathematics, Shanxi Datong University, Datong, Shanxi 037009, China
2, Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan

Received 21 April 2009; Accepted 3 July 2009

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]) [figure omitted; refer to PDF] [figure omitted; refer to PDF] 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[implies]λx∈P; (ii) x∈P and -x∈P[implies]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[implies]||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>0 such that λh≤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[arrow right]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[arrow right]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[arrow right]R and 0≤α<1. Then, f is said to be an α -concave or -α -convex function if f(tx)≥^tα f(x) or, respectively, f(tx)≤^t-α f(x) for x∈I and t∈(0,1).

Definition 1.3.

Let D⊂E , and let A:D×D[arrow right]E. The operator A is called ( [varphi] -concave)-( -ψ -convex) if there exist functions [varphi]:(0,1]×D[arrow right](0,∞) and ψ:(0,1]×D[arrow right](0,∞) such that

( _H0 ): t<[varphi](t,x)ψ(t,x)≤1 for x∈D and t∈(0,1) ,

( _H1 ): A(tx,y)≥[varphi](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 that I⊂R and _x0 ∈I. Recall that a function f:I[arrow right]R is said to be left lower semicontinuous at _x0 if _{lim inf n[arrow right]∞} 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 ] [arrow right]E be a mixed monotone operator. If A is a ( [varphi] -concave)-( -ψ -convex) operator and satisfies the following three conditions:

(A1) there exists _r0 >0 such that _u0 ≥_r0v0 ;

(A2) _u0 ≤A(_u0 ,_v0 ) and A(_v0 ,_u0 )≤_v0 ;

(A3) there exists _ω0 ∈[_u0 ,_v0 ] such that _{min x∈[u0 ,v0 ]} [varphi](t,x)ψ(t,x)=[varphi](t,_ω0 )ψ(t,_ω0 ) for each t∈(0,1) , and [varphi](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 _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Remark 1.5.

Condition (A3) in Theorem 1.4 can be replaced by (A3') [varphi](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[arrow right]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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and (1.2) has a T -periodic solution x(t) if, and only if, x(t) is a T -periodic solution of the equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Furthermore, the Cauchy function G(t,s) satisfies [figure omitted; refer to PDF]

Now let _CT (R) be the Banach space of all real T -periodic continuous functions y:R[arrow right]R endowed with the usual linear structure as well as the norm [figure omitted; refer to PDF] Then P={[varphi]∈_CT (R):[varphi](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 [figure omitted; refer to PDF] as well as [figure omitted; refer to PDF]

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[variant prime]} (t)≤-a(t)_v0 (t)+_f1 (t,_v0 (t-τ(t)))+_f2 (t,_v0 (t-τ(t))) ).

Let A:P×P[arrow right]_CT (R) be defined by [figure omitted; refer to PDF]

We need two basic assumptions in the main results:

( _B1 ): for any s∈R, _f1 (s,x) is an increasing function of x , and _f2 (s,x) is a decreasing function of x;

( _B2 ): 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

(C1) for any s∈R, _f1 (s,·) is an α -concave function, _f2 (s,·) is a convex function;

(C2) there exist [straight epsilon]≥1/(2-α) such that A(_u0 ,_v0 )[straight epsilon]A(_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 _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Proof.

The mapping A:P×P[arrow right]_CT (R) is a mixed monotone operator in view of (B1). Let [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Set m(z)=_u1 (z)-_u0 (z) . Then [figure omitted; refer to PDF] Next, we will prove that m(z)0. Suppose to the contrary that there exists _z0 ∈R such that [figure omitted; refer to PDF] Then ^{m[variant prime]} (_z0 )≥-a(_z0 )m(_z0 )>0, which is a contradiction since m(_z0 )=min z∈R m(z). Thus _u0 ≤A(_u0 ,_v0 ). Similarly, we can prove A(_v0 ,_u0 )≤_v0 . Then we have [figure omitted; refer to PDF] From condition (C2), we know that _u1 ≥[straight epsilon]_v1 . Since _u1 ≤_v1 , we must have 0<[straight epsilon]≤1.

We will prove that A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) operator, where [figure omitted; refer to PDF] In fact, for any u,v∈[_u0 ,_v0 ], t∈(0,1) , and z∈G, we have [figure omitted; refer to PDF] thus [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] Further we can prove [figure omitted; refer to PDF] for any t∈(0,1) and u∈[_u0 ,_v0 ]. Indeed, since [figure omitted; refer to PDF] hence, we only need to prove [figure omitted; refer to PDF] From 0<[straight epsilon]≤1, we know that [straight epsilon]^tα -[straight epsilon]t+t≤^tα ≤1 for any 0<t<1, therefore [figure omitted; refer to PDF] On the other hand, the function [figure omitted; refer to PDF] satisfies g(1)=0 and ^{g[variant prime]} (t)=[straight epsilon](α-1)^tα-2 +1-[straight epsilon]. From [straight epsilon]≥1/(2-α), we have [straight epsilon](1-α)/(1-[straight epsilon])≥1. Then ^t2-α <[straight epsilon](1-α)/(1-[straight epsilon]) for 0<t<1. Thus [straight epsilon](α-1)^tα-2 +1-[straight epsilon]<0, that is, ^{g[variant prime]} (t)<0. Therefore, g(t)>0 for any 0<t<1. Finally, [figure omitted; refer to PDF] Therefore, A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) operator. From (2.20), [varphi](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 _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* . The proof is complete.

Theorem 2.3.

Suppose that conditions ( _B1 ) and ( _B2 ) hold, and

(D1) there exist _r0 >0 such that _u0 ≥_r0v0 ;

(D2) for any s∈R,_f1 (s,·) is an α -concave function and _f2 (s,ty)≤^[(1+η)t]-1_f2 (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 ], [figure omitted; refer to PDF]

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 n∈N, then _{lim n[arrow right]∞xn} and _{lim n[arrow right]∞yn} =^x* .

Proof.

We assert that A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) mixed monotone operator, where [figure omitted; refer to PDF] In fact, [figure omitted; refer to PDF] for any u,v∈[_u0 ,_v0 ] and t∈(0,1). From (2.25), we know that t<[varphi](t,u)ψ(t,u)≤1. Thus A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -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 any s∈R, _f1 (s,·) is a concave function; _f2 (s,ty)≤^[(1+η)t]-1_f2 (s,y) for any y∈P and t∈[0,1], and η=η(t,y) satisfies the following conditions:

( _EH1 ): there exists [straight epsilon]∈(0,1] such that A(θ,_v0 )≥[straight epsilon]A(_v0 ,_u0 );

( _EH2 ): for any (t,y)∈(0,1)×[_u0 ,_v0 ], [figure omitted; refer to PDF]

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 n∈N, then _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Proof.

Set _un =A(_un-1 ,_vn-1 ) and _vn =A(_vn-1 ,_un-1 ) for n∈N. Then we know that [figure omitted; refer to PDF] From ( _EH2 ) we have _u1 ≥[straight epsilon]_v1 . Next we will prove that A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) operator, where [figure omitted; refer to PDF] In fact, for any u,v∈[_u0 ,_v0 ] and t∈(0,1), [figure omitted; refer to PDF] From (2.28), we know that t<[varphi](t,u)ψ(t,u)≤1. Thus A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -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 exists _r0 >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)t_f1 (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]-1_f2 (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 ], [figure omitted; refer to PDF]

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 n∈N, then _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Proof.

We may easily prove that A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) mixed monotone operator, where [figure omitted; refer to PDF] And from ( _FH2 ) we know that [figure omitted; refer to PDF] 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) if _u0 ≤_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)t_f1 (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 ], [figure omitted; refer to PDF]

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 n∈N, then _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Proof.

It is easily seen that A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) mixed monotone operator, where [figure omitted; refer to PDF] From ( _GH2 ), we know that t<[varphi](t,u)ψ(t,u)≤1. Then A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -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)t_f1 (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 exists [straight epsilon]∈(1/2,1) such that A(_u0 ,_v0 )≥[straight epsilon]A(_v0 ,θ) and [figure omitted; refer to PDF]

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 n∈N, then _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞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 [figure omitted; refer to PDF] From ( _JH2 ) we can see that _u1 ≥[straight epsilon]_v1 .

Next we will prove that A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) operator. We need only to verify that A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) operator, where [figure omitted; refer to PDF] In fact, for any u,v∈[_u0 ,_v0 ] and t∈(0,1), we have [figure omitted; refer to PDF] From ( _JH2 ), we have t<[varphi](t,u)ψ(t,u)≤1. Then A:[_u1 ,_v1 ]×[_u1 ,_v1 ][arrow right]_CT (R) is a ( [varphi] -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 any s∈R, _f1 (s,·) is an _α1 -concave function ,_f2 (s,·) is a (-_α2 ) -convex function; where 0≤_α1 +_α2 <1;

(K2) 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 ) and _yn =A(_yn-1 ,_xn-1 ) for n∈N, then _{lim n[arrow right]∞xn} =^x* and _{lim n[arrow right]∞yn} =^x* .

Indeed, it is easily seen that A:[_u0 ,_v0 ]×[_u0 ,_v0 ][arrow right]_CT (R) is a ( [varphi] -concave)-( -ψ -convex) mixed monotone operator, where [figure omitted; refer to PDF] 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

(L1) for any s∈R,_f1 (s,·) is a _α1 -concave function, _f2 (s,·) is a (-_α2 ) -convex function, where 0≤_α1 +_α2 <1;

(L2) 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 ) and _yn =A(_yn-1 ,_xn-1 ), then _xn [arrow right]^x* , _yn [arrow right]^x* ( n[arrow right]∞ ) .

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 [figure omitted; refer to PDF] where p(t) and q(t) are nonnegative continuous T -periodic functions; a(t) and τ(t) are continuous T -periodic functions and satisfy [figure omitted; refer to PDF] where _pmax =_{max t∈[0,T]} p(t) , _pmin =_{min t∈[0,T]} p(t) , _qmax =_{max t∈[0,T]} q(t) , _qmin =_{min t∈[0,T]} q(t) , and _pmax +1000_qmax ≤100_pmin +1000_qmin . Then (3.1) will have a unique solution y=^y* (t) that satisfies 1^0-3 ≤^y* (t)≤1 . Furthermore, if we set _v0 (t)=1^0-3 , _ω0 (t)=1 , [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] The set P={[varphi]∈_CT (R):[varphi](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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] _v0 (t)=1^0-3 , _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* ∈[1^0-3 ,1], and if we set _vn =A(_vn-1 ,_ωn-1 ), _ωn =A(_ωn-1 ,_vn-1 ) for n∈N, then _{lim n[arrow right]∞vn} =^y* and _{lim n[arrow right]∞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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