LOTKA-VOLTERRA SYSTEMS

GLOBAL ATTRACTOR OF COUPLED DIFFERENCE EQUATIONS AND APPLICATIONS TO

LOTKA-VOLTERRA SYSTEMS

C. V. PAO

Received 22 April 2004

This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to in- vestigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.

1. Introduction

Difference equations appear as discrete phenomena in nature as well as discrete analogues of differential equations which model various phenomena in ecology, biology, physics, chemistry, economics, and engineering. There are large amounts of works in the literature that are devoted to various qualitative properties of solutions of difference equations, such as existence-uniqueness of positive solutions, asymptotic behavior of solutions, stability and attractor of equilibrium solutions, and oscillation or nonoscillation of solutions (cf.

[1,4,11,13] and the references therein). In this paper, we investigate some of the above qualitative properties of solutions for a coupled system of nonlinear difference equations in the form

u_n=u_n−1+k f⁽¹⁾u_n,v_n,u_n−s¹,v_n−s²

, vn=vn−1+k f⁽²⁾un,vn,un−s1,vn−s2

(n=1, 2,...), un=φn

n∈I1

, vn=ψn n∈I2

,

(1.1)

where f⁽¹⁾and f⁽²⁾are, in general, nonlinear functions of their respective arguments,kis a positive constant,s1ands2are positive integers, andI1andI2are subsets of nonpositive

Copyright©2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:1 (2005) 57–79 DOI:10.1155/ADE.2005.57

integers given by I1≡

−s1,−s1+ 1,..., 0, I2≡

−s2,−s2+ 1,..., 0. (1.2) System (1.1) is a backward (or left-sided) difference approximation of the delay differen- tial system

du

dt ⁼f⁽¹⁾u,v,uτ1,vτ2

, dv

dt ⁼f⁽²⁾u,v,uτ1,vτ2

(t >0), u(t)=φ(t) −τ1≤t≤0, v(t)=ψ(t) −τ2≤t≤0,

(1.3) whereuτ1=u(t−τ1),vτ2=v(t−τ2), andτ1 andτ2 are positive constants representing the time delays. In relation to the above differential system, the constantkin (1.1) plays the role of the time increment∆tin the difference approximation and is chosen such that s1≡τ1/kands2≡τ2/kare positive integers.

Our consideration of the difference system (1.1) is motivated by some Lotka-Volterra models in population dynamics where the effect of time delays in the opposing species is taken into consideration. The equations for the difference approximations of these model problems, referred to as cooperative, competition, and prey-predator, respectively, involve three distinct quasimonotone reaction functions, and are given as follows (cf.

[7,11,12,15,20]):

(a) the cooperative system:

u_n=u_n−1+kα⁽¹⁾u_n1−a⁽¹⁾u_n+b⁽¹⁾v_n+c⁽¹⁾v_n−s2

vn=vn−1+kα⁽²⁾vn

1 +a⁽²⁾un−b⁽²⁾vn+c⁽²⁾un−s1

(n=1, 2,...), u_n=φ_n n∈I1

, v_n=ψ_n n∈I2

;

(1.4)

(b) the competition system:

u_n=u_n−1+kα⁽¹⁾1−a⁽¹⁾u_n−b⁽¹⁾v_n−c⁽¹⁾v_n−s2

v_n=v_n−1+kα⁽²⁾1−a⁽²⁾u_n−b⁽²⁾v_n−c⁽²⁾u_n−s1

(n=1, 2,...), u_n=φ_n n∈I1

, v_n=ψ_n n∈I2

;

(1.5)

(c) the prey-predator system:

u_n=u_n−1+kα⁽¹⁾1−a⁽¹⁾u_n−b⁽¹⁾v_n−c⁽¹⁾v_n−s2

v_n=v_n−1+kα⁽²⁾1 +a⁽²⁾u_n−b⁽²⁾v_n+c⁽²⁾u_n−s1

(n=1, 2,...), u_n=φ_n n∈I1

, v_n=ψ_n n∈I2

.

(1.6)

In the systems (1.4), (1.5), and (1.6),u_nandv_nrepresent the densities of the two popula- tion species at timenk(≡n∆t),kis a small time increment, and for eachl=1, 2,α^(l),a^(l), b^(l), andc^(l)are positive constants representing the various reaction rates.

There are huge amounts of works in the literature that dealt with the asymptotic be- havior of solutions for differential and difference systems with time delays, and much of

the discussions in the earlier work are devoted to differential systems, including various Lotka-Volterra-type equations (cf. [2,3,5,7,8,12,15,19,20]). Later development leads to various forms of difference equations, and many of them are discrete analogues of dif- ferential equations (cf. [2,3,4,5,6,8,9,10,11,19]). In recent years, attention has also been given to finite-difference equations which are discrete approximations of differential equations with the effect of diffusion (cf. [14,15,16,17,18]). In this paper, we consider the coupled difference system (1.1) for a general class of reaction functions (f⁽¹⁾,f⁽²⁾), and our aim is to show the existence and uniqueness of a global positive solution and the asymptotic behavior of the solution with particular emphasis on the global attraction of a positive equilibrium solution. The results for the general system are then applied to each of the three Lotka-Volterra models in (1.4)–(1.6) where some easily verifiable conditions on the rate constantsa^(l),b^(l), and c^(l),l=1, 2, are obtained so that a unique positive equilibrium solution exists and is a global attractor of the system.

The plan of the paper is as follows. InSection 2, we show the existence and uniqueness of a positive global solution to the general system (1.1) for arbitrary Lipschitz continu- ous functions (f⁽¹⁾,f⁽²⁾).Section 3is concerned with some comparison theorems among solutions of (1.1) for three different types of quasimonotone functions. The asymptotic behavior of the solution is treated inSection 4where sufficient conditions are obtained for ensuring the global attraction of a positive equilibrium solution. This global attrac- tion property is then applied inSection 5 to the Lotka-Volterra models in (1.4), (1.5), and (1.6) which correspond to the three types of quasimonotone functions in the general system.

2. Existence and uniqueness of positive solution

Before discussing the asymptotic behavior of the solution of (1.1) we show the existence and uniqueness of a positive solution under the following basic hypothesis on the func- tion (f⁽¹⁾,f⁽²⁾)≡(f⁽¹⁾(u,v,u_s,v_s),f⁽²⁾(u,v,u_s,v_s)).

(H1) (i) The function (f⁽¹⁾,f⁽²⁾) satisfies the local Lipschitz condition f^(l)u,v,us,vs

−f^(l)u,v,u_s,v_s

≤K^(l)|u−u|+|v−v|+us−u_s+vs−v_s foru,v,us,vs

,u,v,u_s,v_s∈᏿×᏿, (l=1, 2).

(2.1)

(ii) There exist positive constants (M⁽¹⁾,M⁽²⁾), (δ⁽¹⁾,δ⁽²⁾) with (M⁽¹⁾,M⁽²⁾)≥ (δ⁽¹⁾,δ⁽²⁾) such that for all (us,vs)∈᏿,

f⁽¹⁾M⁽¹⁾,v,us,vs

≤0≤ f⁽¹⁾δ⁽¹⁾,v,us,vs

whenδ⁽²⁾≤v≤M⁽²⁾, f⁽²⁾u,M⁽²⁾,us,vs

≤0≤f⁽²⁾u,δ⁽²⁾,us,vs

whenδ⁽¹⁾≤u≤M⁽¹⁾. (2.2) In the above hypothesis,᏿is given by

᏿≡

(u,v)∈R²;δ⁽¹⁾,δ⁽²⁾≤(u,v)≤

M⁽¹⁾,M⁽²⁾. (2.3)

To ensure the uniqueness of the solution, we assume that the time incrementksatisfies the condition

kK⁽¹⁾+K⁽²⁾<1, (2.4)

whereK⁽¹⁾andK⁽²⁾are the Lipschitz constants in (2.1).

Theorem2.1. Let hypothesis (H1) hold. Then system (1.1) has at least one global solution (u_n,v_n)in᏿. If, in addition, condition (2.4) is satisfied, then the solution(u_n,v_n)is unique in᏿.

Proof. Given anyW_n≡(w_n,z_n)∈᏿, we letU_n≡(u_n,v_n) be the solution of the uncoupled initial value problem

1 +kK⁽¹⁾un=un−1+kK⁽¹⁾wn+f⁽¹⁾wn,zn,un−s1,vn−s2

, 1 +kK⁽²⁾vn=vn−1+kK⁽²⁾zn+ f⁽²⁾wn,zn,un−s1,vn−s2

(n=1, 2,...), un=φn

n∈I1

, vn=ψn n∈I2

,

(2.5)

whereK⁽¹⁾andK⁽²⁾are the Lipschitz constants in (2.1). Define a solution operatorᏼ:

᏿→R²by

ᏼWn≡

P⁽¹⁾W_n,P⁽²⁾W_n≡

u_n,v_n W_n∈᏿. (2.6)

Then system (1.1) may be expressed as Un=ᏼUn, Un=

un,vn

(n=1, 2,...). (2.7)

To prove the existence of a global solution to (1.1) it suffices to show thatᏼhas a fixed point in᏿for everyn. It is clear from hypothesis (H1) thatᏼis a continuous map on᏿ which is a closed bounded convex subset ofR². We show thatᏼmaps᏿into itself by a marching process.

Given anyW_n≡(w_n,z_n)∈᏿, relation (2.6) and conditions (2.1), (2.2) imply that 1 +kK⁽¹⁾M⁽¹⁾−P⁽¹⁾Wn

=

1 +kK⁽¹⁾M⁽¹⁾−

u_n−1+kK⁽¹⁾w_n+f⁽¹⁾w_n,z_n,u_n−s1,v_n−s2

≥

M⁽¹⁾−u_n−1

+kK⁽¹⁾M⁽¹⁾−w_n+f⁽¹⁾M⁽¹⁾,z_n,u_n−s1,v_n−s2

−f⁽¹⁾w_n,z_n,u_n−s¹,v_n−s²

≥M⁽¹⁾−u_n−1, 1 +kK⁽²⁾M⁽²⁾−P⁽²⁾W_n

=

1 +kK⁽²⁾M⁽²⁾−

vn−1+kK⁽²⁾zn+f⁽²⁾wn,zn,un−s1,vn−s2

≥

M⁽²⁾−vn−1

+kK⁽²⁾M⁽²⁾−zn

+ f⁽²⁾wn,M⁽²⁾,un−s1,vn−s2

−f⁽²⁾wn,zn,un−s1,vn−s2

≥M⁽²⁾−vn−1 (n=1, 2,...),

(2.8)

whenever (u_n−s1,v_n−s2)∈᏿. This leads to the relation M⁽¹⁾−P⁽¹⁾W_n≥M⁽¹⁾−un−1

1 +kK⁽¹⁾ M⁽²⁾−P⁽²⁾Wn≥M⁽²⁾−v_n−1

1 +kK⁽²⁾ (n=1, 2,...).

(2.9)

A similar argument using the second inequalities in (2.2) gives P⁽¹⁾Wn−δ⁽¹⁾≥u_n−1−δ⁽¹⁾

1 +kK⁽¹⁾ , P⁽²⁾Wn−δ⁽²⁾≥vn−1−δ⁽²⁾

1 +kK⁽²⁾ (n=1, 2,...),

(2.10)

whenever (un−s1,vn−s2)∈᏿. Consider the casen=1. Since (u1₋s1,v1₋s2)=(φ1₋s1,ψ1₋s2) and (u0,v0)=(φ0,ψ0) are in᏿, relations (2.9), (2.10) imply that (δ⁽¹⁾,δ⁽²⁾)≤(P⁽¹⁾W1, P⁽²⁾W1)≤(M⁽¹⁾,M⁽²⁾). By Brower’s fixed point theorem,ᏼ≡(P⁽¹⁾,P⁽²⁾) has a fixed point U1≡(u1,v1) in᏿. This shows that (u1,v1) is a solution of (1.1) forn=1, and (u1,v1) and (u2−s1,v2−s2) are in᏿. Using this property in (2.9), (2.10) forn=2, the same argument shows thatᏼhas a fixed pointU2≡(u2,v2) in᏿, and (u2,v2) is a solution of (1.1) for n=2 and (u3₋s1,v3₋s2)∈᏿. A continuation of the above argument shows thatᏼhas a fixed pointU_n≡(u_n,v_n) in᏿for everyn, and (u_n,v_n) is a global solution of (1.1) in᏿.

To show the uniqueness of the solution, we consider any two solutions (u_n,v_n), (u_n,v_n) in᏿and let (wn,zn)=(un−u_n,vn−v_n). By (1.1),

wn=wn−1+kf⁽¹⁾un,vn,un−s1,vn−s2

−f⁽¹⁾u_n,v_n,u_n−s1,v_n−s2

, zn=zn−1+kf⁽²⁾un,vn,un−s1,vn−s2

−f⁽²⁾u_n,v_n,u_n−s1,v_n−s2

(n=1, 2,...), wn=0 n∈I1

, zn=0 n∈I2

.

(2.11)

The above relation and condition (2.1) imply that

wn≤wn−1+kK⁽¹⁾wn+zn+wn−s1+zn−s2,

z_n≤z_n−1+kK⁽²⁾w_n+z_n+w_n−s1+z_n−s2. (2.12) Addition of the above inequalities leads to

w_n+z_n≤w_n−1+z_n−1

+kK⁽¹⁾+K⁽²⁾w_n+z_n+w_n−s1+z_n−s2 (n=1, 2,...). (2.13) Sincew_n=z_n=0 forn=0,−1,−2,..., the above inequality forn=1 yields

w1+z1≤kK⁽¹⁾+K⁽²⁾w1+z1. (2.14) In view of condition (2.4), this is possible only when|w1| = |z1| =0. Usingw1=z1=0 in (2.13) forn=2 yields

w2+z2≤kK⁽¹⁾+K⁽²⁾w2+z2. (2.15)

It follows again from (2.4) that|w2| = |z2| =0. The conclusion|w_n| = |z_n| =0 for every nfollows by an induction argument. This proves (u_n,v_n)=(u_n,v_n), and therefore (u_n,v_n)

is the unique solution of (1.1) in᏿.

Remark 2.2. (a) Since problem (1.3) may be considered as an equivalent system of the scalar second-order differential equation

u=fu,u,uτ1,u_τ2

(t >0),

u(t)=φ(t) −τ1≤t≤0, u(t)=ψ(t) −τ2≤t≤0, (2.16) the conclusion inTheorem 2.1and all the results obtained in later sections are directly applicable to the difference approximation of (2.16) with (un,vn)=(un,u_n) and (f⁽¹⁾,

f⁽²⁾)=(v_n,f(u_n,v_n,u_n−s¹,v_n−s²)).

(b) System (1.1) is a difference approximation of (1.3) by the backward (or left-sided) approximation of the time derivative (du/dt,dv/dt), and this approximation preserves the nonlinear nature of the differential system. If the forward (or right-sided) approximation for (du/dt,dv/dt) is used, then the resulting difference system gives an explicit formula for (un+1,vn+1) which can be computed by a marching process for everyn=0, 1, 2,...and for any continuous function (f⁽¹⁾,f⁽²⁾). From a view point of differential equations, the forward approximation may lead to misleading information about the solution of the differential system. One reason is that a global solution to the differential system may fail to exist while the difference solution (un+1,v_n+1) exists for everyn.

(c) The uniqueness result inTheorem 2.1is in the set᏿, and it does not rule out the possibility of existence of positive solutions outside of᏿.

3. Comparison theorems

To investigate the asymptotic behavior of the solution we consider a class of quasimono- tone functions which depend on the monotone property of (f⁽¹⁾,f⁽²⁾). Specifically, we make the following hypothesis.

(H2) (f⁽¹⁾,f⁽²⁾) is aC¹-function in᏿×᏿and possesses the property∂ f⁽¹⁾/∂u_s≥0,

∂ f⁽²⁾/∂vs≥0 and one of the following quasimonotone properties for (u,v,us,vs)∈

᏿×᏿:

(a) quasimonotone nondecreasing:

∂ f⁽¹⁾

∂v ^≥0, ∂ f⁽¹⁾

∂v_s ^≥0, ∂ f⁽²⁾

∂u ^≥0, ∂ f⁽²⁾

∂u_s ^≥0; (3.1)

(b) quasimonotone nonincreasing:

∂ f⁽¹⁾

∂v ^≤0, ∂ f⁽¹⁾

∂v_s ^≤0, ∂ f⁽²⁾

∂u ^≤0, ∂ f⁽²⁾

∂u_s ^≤0; (3.2)

(c) mixed quasimonotone:

∂ f⁽¹⁾

∂v ^≤0, ∂ f⁽¹⁾

∂vs ≤0, ∂ f⁽²⁾

∂u ^≥0, ∂ f⁽²⁾

∂us ≥0. (3.3)

Notice that if (f⁽¹⁾,f⁽²⁾)≡(f⁽¹⁾(u,v),f⁽²⁾(u,v)) is independent of (u_s,v_s), then the above conditions are reduced to those required for the standard three types of quasimonotone functions (cf. [15,18]).

It is easy to see from (H2) that for quasimonotone functions the conditions on (M⁽¹⁾, M⁽²⁾), (δ⁽¹⁾,δ⁽²⁾) in (2.2) are reduced to the following.

(a) For quasimonotone nondecreasing functions:

f⁽¹⁾M⁽¹⁾,M⁽²⁾,M⁽¹⁾,M⁽²⁾≤0≤ f⁽¹⁾δ⁽¹⁾,δ⁽²⁾,δ⁽¹⁾,δ⁽²⁾,

f⁽²⁾M⁽¹⁾,M⁽²⁾,M⁽¹⁾,M⁽²⁾≤0≤ f⁽²⁾δ⁽¹⁾,δ⁽²⁾,δ⁽¹⁾,δ⁽²⁾. (3.4) (b) For quasimonotone nonincreasing functions:

f⁽¹⁾M⁽¹⁾,δ⁽²⁾,M⁽¹⁾,δ⁽²⁾≤0≤f⁽¹⁾δ⁽¹⁾,M⁽²⁾,δ⁽¹⁾,M⁽²⁾,

f⁽²⁾δ⁽¹⁾,M⁽²⁾,δ⁽¹⁾,M⁽²⁾≤0≤f⁽²⁾M⁽¹⁾,δ⁽²⁾,M⁽¹⁾,δ⁽²⁾. (3.5) (c) For mixed quasimonotone functions:

f⁽¹⁾M⁽¹⁾,δ⁽²⁾,M⁽¹⁾,δ⁽²⁾≤0≤f⁽¹⁾δ⁽¹⁾,M⁽²⁾,δ⁽¹⁾,M⁽²⁾,

f⁽²⁾M⁽¹⁾,M⁽²⁾,M⁽¹⁾,M⁽²⁾≤0≤ f⁽²⁾δ⁽¹⁾,δ⁽²⁾,δ⁽¹⁾,δ⁽²⁾. (3.6) In this section, we show some comparison results among solutions with different initial functions for each of the above three types of quasimonotone functions. The comparison results for the first two types of quasimonotone functions are based on the following positivity lemma for a function (w_n,z_n) satisfying the relation

γ_n⁽¹⁾wn≥wn−1+a⁽¹⁾_n zn+b⁽¹⁾_n wn−s1+c⁽¹⁾_n zn−s2, γ_n⁽²⁾zn≥zn−1+a⁽²⁾_n wn+b⁽²⁾_n wn−s1+c⁽²⁾_n zn−s2 (n=1, 2,...),

wn≥0 n∈I1

, zn≥0 (n∈I),

(3.7)

where for eachl=1, 2, andn=1, 2,...,γn^(l)is positive, anda⁽n^l),b⁽n^l), andcn^(l)are nonnega- tive.

Lemma3.1. Let(w_n,z_n)satisfy (3.7), and let

a⁽¹⁾_n a⁽²⁾_n < γ⁽¹⁾_n γ_n⁽²⁾ (n=1, 2,...). (3.8) Then(wn,zn)≥(0, 0)for everyn=1, 2,....

Proof. Consider the casen=1. Sincewn≥0 forn∈I1andzn≥0 forn∈I2, the inequal- ities in (3.7) yield

γ1⁽¹⁾w1≥a⁽¹⁾1 z1, γ1⁽²⁾z1≥a⁽²⁾1 w1. (3.9)

The positivity ofγ⁽¹⁾1 ,γ1⁽²⁾implies that w1≥

a⁽¹⁾1

γ⁽¹⁾1

z1≥

a⁽¹⁾1 a⁽²⁾1

γ1⁽¹⁾γ⁽²⁾1

w1, z1≥

a⁽²⁾1

γ1⁽²⁾

w1≥

a⁽¹⁾1 a⁽²⁾1

γ1⁽¹⁾γ⁽²⁾1

z1.

(3.10)

In view of (3.8), the above inequalities can hold only if (w1,z1)≥(0, 0). Assume, by in- duction, that (w_n,z_n)≥(0, 0) forn=1, 2,...,m−1 for somem >1. Then by (3.7),

γ⁽¹⁾_m wm≥a⁽¹⁾_m zm, γ⁽²⁾_m zm≥a⁽²⁾_m wm. (3.11) This leads to

wm≥

a⁽¹⁾m a⁽²⁾m

γ⁽¹⁾m γm⁽²⁾

, zm≥

a⁽¹⁾m a⁽²⁾m

γ⁽¹⁾m γm⁽²⁾

zm. (3.12)

It follows again from (3.8) that (wm,zm)≥(0, 0). The conclusion of the lemma follows by

the principle of induction.

The above positivity lemma can be extended to a function (w_n,z_n,w_n,z_n) satisfying the relation

γ_n⁽¹⁾wn≥wn−1+a⁽¹⁾_n z_n+b⁽¹⁾_n wn−s1+c⁽¹⁾_n z_n−s2, γ_n⁽²⁾zn≥zn−1+a⁽²⁾_n wn+b⁽²⁾_n wn−s1+c⁽²⁾_n zn−s2, γˆ_n⁽¹⁾w_n≥w_n−1+ ˆa⁽¹⁾_n zn+ ˆb⁽¹⁾_n w_n−s1+ ˆc⁽¹⁾_n zn−s2, γˆ_n⁽²⁾z_n≥z_n−1+ ˆa⁽²⁾_n w_n+ ˆb⁽²⁾_n w_n−s1+ ˆc⁽²⁾_n z_n−s2 (n=1, 2,...), w_n≥0, w_n≥0 n∈I1

, z_n≥0, z_n≥0 n∈I2

,

(3.13)

whereγ⁽n^l),a⁽n^l),bn^(l), andc⁽n^l),l=1, 2, are the same as that in (3.7) and ˆγ⁽n^l), ˆa⁽n^l), ˆb⁽n^l), and ˆc⁽n^l)

are nonnegative with ˆγ⁽n^l)>0,n=1, 2,....

Lemma3.2. Let(w_n,z_n,w_n,z_n)satisfy (3.13), and let

a⁽¹⁾_n a⁽²⁾_n aˆ⁽¹⁾_n aˆ⁽²⁾_n <γ_n⁽¹⁾γ⁽²⁾_n γˆ⁽¹⁾_n γˆ_n⁽²⁾ (n=1, 2,...). (3.14) Then(wn,zn,w_n,z_n)≥(0, 0, 0, 0)for everyn.

Proof. By (3.13) withn=1, we have

γ⁽¹⁾1 w1≥a⁽¹⁾1 z1, γ⁽²⁾1 z1≥a⁽²⁾1 w1, γˆ⁽¹⁾1 w1≥aˆ⁽¹⁾1 z1, γˆ⁽²⁾1 z1≥aˆ⁽²⁾1 w1. (3.15)

This implies that w1≥

a⁽¹⁾1

γ⁽¹⁾1

z1≥ a⁽¹⁾1

γ1⁽¹⁾

aˆ⁽²⁾1

γˆ⁽²⁾1

w1≥ a⁽¹⁾1

γ1⁽¹⁾

aˆ⁽²⁾1

γˆ1⁽²⁾

aˆ⁽¹⁾1

γˆ1⁽¹⁾

z1

≥

a⁽¹⁾1 a⁽²⁾1

γ⁽¹⁾1 γ2⁽²⁾

aˆ⁽¹⁾1 aˆ⁽²⁾2

γˆ1⁽¹⁾γˆ⁽²⁾1

w1.

(3.16)

In view of condition (3.14), we havew1≥0. This implies thatz1≥0,w1≥0 andz1≥0 which proves the case forn=1. Assume, by induction, that (w_n,z_n,w_n,z_n)≥(0, 0, 0, 0) forn=1, 2,...,m−1 for somem >1. Then by (3.13),

γ_m⁽¹⁾w_m≥a⁽¹⁾_m z_m, γ⁽²⁾_m z_m≥a⁽²⁾_mw_m, γˆ_m⁽¹⁾w_m≥aˆ⁽¹⁾_m z_m, γˆ⁽²⁾_m z_m≥aˆ⁽²⁾_mw_m. (3.17) This leads to

wm≥

a⁽¹⁾ma⁽²⁾m

γm⁽¹⁾γ⁽²⁾m

aˆ⁽¹⁾m aˆ⁽²⁾m

γˆ⁽¹⁾m γˆm⁽²⁾

wm. (3.18)

It follows again from (3.14) thatw_m≥0 from which we obtainz_m≥0,w_m≥0 andz_m≥0.

The conclusion of the lemma follows from the principle of induction.

To obtain comparison results among solutions, we need to impose a condition on the time incrementk. Define

σ1⁽¹⁾≡max ∂ f⁽¹⁾

∂u

u,v,us,vs

; (u,v),us,vs

∈᏿

, σ2⁽¹⁾≡max∂ f⁽¹⁾

∂v

u,v,us,vs

; (u,v),us,vs

∈᏿

, σ1⁽²⁾≡max∂ f⁽²⁾

∂u

u,v,us,vs

; (u,v),us,vs

∈᏿

, σ2⁽²⁾≡max

∂ f⁽²⁾

∂v

u,v,u_s,v_s; (u,v),u_s,v_s∈᏿

.

(3.19)

Our condition onkis given by

kK⁽¹⁾+K⁽²⁾<1, kσ2⁽¹⁾

kσ1⁽²⁾

<1−kσ1⁽¹⁾

1−kσ2⁽²⁾

, (3.20)

whereK^(l),l=1, 2, are the Lipschitz constants in (2.1). Since σ1⁽¹⁾≤K⁽¹⁾,σ2⁽²⁾≤K⁽²⁾, it follows that kσ1⁽¹⁾<1 and kσ2⁽²⁾<1. Notice that σ2⁽¹⁾ andσ1⁽²⁾ are nonnegative while σ1⁽¹⁾andσ2⁽²⁾are not necessarily nonnegative. The following comparison theorem is for quasimonotone nondecreasing functions.