Existence results and monotone iterative technique for impulsive hybrid functional differential systems with

Existence results and monotone iterative technique for impulsive hybrid functional differential systems with

anticipation and retardation

Bashir Ahmad

^a

, S. Sivasundaram

^b,*

aDepartment of Mathematics, Faculty of Science, King Abdul Aziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

bDepartment of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA

Abstract

We study some existence results for impulsive hybrid functional differential equations with anticipation and retardation by using the existence theory of impulsive hybrid delay differential equations. The monotone iterative technique relative to the coupled lower and upper solutions has also been developed for impulsive hybrid functional differential equations with both anticipation and retardation.

2007 Elsevier Inc. All rights reserved.

Keywords: Impulsive hybrid differential equations with anticipation and retardation; Existence theory; Upper and lower solutions;

Monotone method

1. Introduction

A number of research papers has recently dealt with the theoretical developments and applications in the modeling and computing of anticipatory systems in certain fields of natural and artificial systems. Computing anticipatory systems involve differential delayed-advanced equations. Delayed systems are based on a memory of past states and advanced systems are systems which depend explicitly on their anticipatory future potential states. With the help of the laws of evolution considered at the current time for any physical system, the past and future states need to be defined by the new variables at the current time, taking into account some hidden mechanisms for their existence and knowledge at the current time because the past states no more exist at the current time and the future states are not yet actualized. Some delayed-advanced systems can be transformed to functional differential systems with FPP (future, present and past) dependence. Mathematically, new variables defined by equations at the current time, are introduced in view of computing, by synchronization, past and/or future states. However, the investigation of general functional differential systems with FPP

0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2007.07.065

* Corresponding author.

E-mail addresses:seenithi@gmail.com,siva@erau.edu(S. Sivasundaram).

Applied Mathematics and Computation 197 (2008) 515–524

www.elsevier.com/locate/amc

dependence has its own challenges and there do exist several approaches to deal with such problems, for instance, see[1]. The potential applications[2,3]of anticipatory systems can be found in decision theory, cha- otic epidemics, wavelet theory, etc. Recently, Lakshmikantham [4]developed the existence theory for func- tional differential equations with both retardation and anticipation.

In this paper, we discuss some existence results for impulsive hybrid functional differential equations with retardation and anticipation. We also develop the monotone iterative technique to obtain the extremal solu- tions of the impulsive hybrid functional differential equations involving both anticipation and retardation. It is well known that the method of upper and lower solutions coupled with its associated monotone iteration scheme is an interesting and powerful mechanism that offers the theoretical as well constructive existence results for nonlinear problems in a closed set, generated by the lower and upper solutions, for instance, see [5–9]. To the best of our knowledge, this technique has not been yet applied to the retardation-anticipatory functional differential systems with impulse. Impulsive hybrid dynamical systems are characterized by the occurrence of abrupt change in the state of the system which occur at certain time instants over a period of negligible duration. The presence of impulse means that the state trajectory does not preserve the basic properties which are associated with nonimpulsive hybrid dynamical systems. Thus, our consideration is quite interesting and help understand further the retardation-anticipatory functional differential systems with impulse. In Section2, we present some existence results for impulsive hybrid functional differential equations with retardation and anticipation. In Section3, the monotone iterative technique together with the coupled lower and upper solutions[5]is employed to obtain the extremal solutions of the impulsive hybrid functional differential equations involving both anticipation and retardation. Under suitable conditions, it has also been shown that the minimal and maximal solutions yield the unique solution of the problem at hand.

2. Some existence results

Consider the impulsive hybrid functional differential system involving both anticipation and retardation x⁰ðtÞ ¼fðt;x_t;x^tÞ; t6¼t_k; t2 ½t0;T; t₀P0;

x_t^þ

k ¼I_kðxtkÞ; t¼t_k; k2Z^þ;

x_t0¼/₀2C₁; x^T¼w₀2C₂; 8>

<

>: ð2:1Þ

where C₁¼Cð½h1;0;RⁿÞ, C₂¼Cð½0;h₂;RⁿÞ, h₁, h₂P0; f 2PC½½t0;T D;Rⁿ, D being an open set in C1·C2andxt(s) =x(t+s),h16s60,x^T(r) =x(T+r), 06r6h2. I_k2CðRⁿ;RⁿÞfor each k2Z⁺ and {tk} is a sequence of points such that 06t0<t1< tk<tk+1=T.

The function/0is generally known as delay or retardation or the past information. We may call/0(s) as the past information defined ont0h16s6t0,/0(t0) as the present andw0as the desired future potential. With this in mind,(2.1)may be regarded as the functional differential system with FPP (future, present and past) dependence. Whenever, a desired anticipation is involved, one needs to make appropriate decisions to predict the desired future event. The decisions made continuously from the present and past history, will effect the future prospectus of an event but it does not guarantee the occurrence of the desired results. Since the past and present cannot be changed, one is forced to relate the other factors in some form to achieve the anticipated outcome.

In order to prove the existence theorem for the posed problem(2.1), one may choose a decision function z2PC½½t₀;T;Rⁿ such that z(t0) =/0(t0) and z(T) =w0(T), that is, z^t=z(t+r), 06r6h2 is defined for t2[t0,T] withz^T=w0as the tail end. With this chosen decision function, the system(2.1)takes the form

x⁰ðtÞ ¼fðt;x_t;z^tÞ ¼Fðt;x_tÞ; t6¼t_k; t2 ½t0;T; x_t^þ

k ¼I_kðxtkÞ; t¼t_k; k2Z^þ;

x_t0¼/₀; 8>

<

>: ð2:2Þ

which is an impulsive hybrid functional differential system with only delay and employs the future information from [t0,T]. Here, F 2PC½½t0;T C₁;Rⁿ means that F :ðt_k1;t_k C₁!Rⁿ is continuous and for each

xt2C1, lim_ðt;ptÞ!ðt^þ

k;x_tÞFðt;p_tÞ ¼Fðt^þ_k;x_tÞexists. By a solution of(2.2), we mean a piecewise continuous function x(t0,/0)(t) on [t0,T] which is left continuous on (tk,tk+1] and is defined by

xðt0;/₀ÞðtÞ ¼

/₀; t0h16t6t0; x₀ðt0;/₀ÞðtÞ; t₀6t6t₁; x₁ðt₁;/₁ÞðtÞ; t₁<t6t₂;

x_kðtk;/_kÞðtÞ; t_k<t6t_kþ1; 8>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>:

where xk(tk,/k)(t) is the solution of the functional differential equation with retardation x⁰ðtÞ ¼Fðt;xtÞ;

x_t^þ

k ¼/_k;k2Z^þ.

We need the following results to establish the existence theory for the impulsive hybrid functional differen- tial system involving both anticipation and retardation (for the proof of these results, see[6,7]).

Theorem 2.1. Let g2PC½RþR²;R be such that g(t, u, v) is nondecreasing in u, v for each t2Rþ. v_kðu;vÞ 2C½R²;Ris nondecreasing in u, v for each k2Z⁺. Further, assume that the maximal solutionrðtÞP0of the problem

u⁰ðtÞ ¼gðt;u;vÞ; t6¼t_k; uðt^þ_kÞ ¼v_kðuðt_kÞ;vðt_kÞÞ; t¼t_k; uðt0Þ ¼u0;

8>

<

>: ð2:3Þ

exists on [t0,1).Then the maximal solution r(t)of u⁰ðtÞ ¼gðt;u;rðtÞÞ; t6¼t_k;

uðt^þ_kÞ ¼v_kðuðtkÞ;rðtkÞÞ; t¼t_k; uðt0Þ ¼u₀;

8>

<

>: ð2:4Þ

exists for tPt0andrðtÞ ¼rðtÞ; tPt0.

Theorem 2.2. Suppose that the assumptions ofTheorem2.1hold. Further, letm2PC¹½R;Rbe such that m(t) is left continuous at tkfor each k2Z⁺and for every v6rðtÞ;tPt₀;we have

DmðtÞ6gðt;mðtÞ;vðtÞÞ; t6¼t_k; mðt^þ_kÞ6v_kðmðtkÞ;vðtkÞÞ; t¼t_k; uðt0Þ ¼u₀;

8>

<

>:

where DmðtÞ ¼lim inf_h!0¹_h½mðtþhÞ mðtÞ. Then m(t)6r(t), tPt0, where r(t) is the maximal solution of (2.4).

We now state an existence result for(2.2)for a chosenz(t). We omit the proof as it is based on standard arguments[6,7].

Theorem 2.3. LetF 2PC½½t0;T C_q;Rⁿandvk: Cq!Cq, where Cq={/2C1:j/j0<q}. Then there exists an a>0 such that there is a solution x(t0,/0)(t) of (2.2) existing on [t0, t0+a] for every given initial value /02Cqat t = t0.

Theorem 2.4 (Global existence result). Suppose that the solutions of(2.2)exist locally for any choice of z(t) satisfying

jz^tj₀6rðtÞ; t2 ½t₀;T; ð2:5Þ

whererðtÞ ¼rðt;t0;u0Þis the maximal solution of u⁰ðtÞ ¼gðt;u;uÞ; t6¼t_k;

uðt^þ_kÞ ¼v_kðuðtkÞ;uðtkÞÞ; t¼t_k; uðt0Þ ¼u0;

8>

<

>: ð2:6Þ

existing on [t0, T]. Hereg2PC½½t0;T R²;Ris such that g(t, u, v) is nondecreasing in u, v for each t2[t0, T]

andv_kðu;vÞ 2C½R²;Ris nondecreasing in u, v for each k2Z⁺. Further, for/2C1withj/j0=j/(0)jand for any z satisfying(2.5), the function f in(2.2)has the estimate

jfðt;/;z^tÞj6gðt;j/ð0Þj;jz^tj₀Þ: ð2:7Þ

Moreover, we require that

jIkðxtkÞj6v_kðjxtkjÞ; t¼t_k: ð2:8Þ

Let r(t) = r(t, t0, u0) be the maximal solution of u⁰ðtÞ ¼gðt;u;rðtÞÞ; t6¼t_k; uðt^þ_kÞ ¼v_kðuðt_kÞ;rðt_kÞÞ; t¼t_k; uðt0Þ ¼u0;

8>

<

>:

existing on [t0, T]. Then the largest interval of existence of any solution x(t0,/0)(t) of(2.2)is [t0, T].

Proof. Letx(t0,/0)(t) be any solution of(2.2)existing on some interval [t0,t1] witht1<T. Fort2[t0,t1], we set m(t) =jx(t0,/0)(t)j so that mt=jxt(t0,/0)j. In view of (2.7), we obtain the differential inequality Dm(t)6g(t,m(t),jz^tj0), which, on using(2.5)together with the fact thatg(t,u,v) is nondecreasing inv, yields

DmðtÞ6gðt;mðtÞ;rðtÞÞ; t2 ½t0;t₁:

We choosejmt₀j ¼ j/₀j₀6u0. ByTheorem 2.2, it follows that:

mðtÞ6rðtÞ; t2 ½t₀;t₁: ð2:9Þ

For anyt2;t3 satisfyingt0<t2<t3<t1;using the monotonicity ofg(t,u,v) inu and(2.9), we obtain jxðt0;/₀Þðt3Þ xðt0;/₀Þðt2Þj6

Z t3 t₂

gðs;xsðt0;/₀Þ;rðsÞÞds6 Z t3

t₂

gðs;rðsÞ;rðsÞÞds¼rðt3Þ rðt2Þ;

where we have used the fact thatr(t) is nondecreasing int(because ofg(t,u,v)P0). Hence m_t6rðtÞ; t2 ½t0;t1:

Now, utilizing the monotone character ofv(u,v) in (u,v) andm(t1)6r(t1), we get mðt^þ₁Þ6v₁ðmðt1Þ;jz^t1j₀Þ6v₁ðmðt1Þ;rðt1ÞÞ6v₁ðrðt1Þ;rðt1ÞÞ ¼r^þ₁:

Next, we define lim_t!t^þ

1xðt0;/₀ÞðtÞ ¼xðt0;/₀Þðt1Þand setx_t1ðt0;/₀Þ ¼/₁as the new function att=t1. Now, for t1<t6t2, it follows by the earlier arguments thatmt6r(t),t2(t1,t2] andmðt^þ₂Þ6r^þ₂. Continuing this process successively, we get the existence of a solution of(2.2)on [t0,T]. This completes the proof. h

Corollary 2.5. The proof ofTheorem2.4for the existence of a solution of (2.2)remains valid on [t0,1] if the functions involved are assumed to hold on [t0,1], instead of [t0, T]. In this case, the sequence of points {tk}sat- isfies 06t0<t1< tk<tk+1 withlim_k!1tk=1.

In order to claim that the solution x(t0,/0)(t) of(2.2)existing on [t0, T], is the solution of(2.1)for a chosen z(t) on [t0, T], we need to show that

xðt0;/₀ÞðTÞ ¼w₀ðTÞ ¼zðTÞ: ð2:10Þ

In general, it is not true that(2.10)holds for an arbitrary choice of z(t). However, if we find at least one z(t) sat- isfying(2.10), then we have an existence result for(2.1). Lakshmikantham[4]has argued that there exists a func- tion z(t) such that(2.10)holds if the functional f in(2.2)satisfies the condition

2e^at</₁ð0Þ /₂ð0Þ; fðt;/₁;z^t₁Þ fðt;/₂;z^t₂Þ>þe^atj/₁ð0Þ /₂ð0Þj²6c₁e^atjz1z₂j²₀;

where jz1z₂j₀ ¼maxt06t6Tþh2jz1ðtÞ z₂ðtÞj;ðc1=aÞ¹⁼²<1;c₁>0;a>0; and /₁;/₂2X ¼ ½/₁;/₂2C₁: maxh16s60j/₁ðsÞ /₂ðsÞje^aðtþsÞ¼ j/₁ð0Þ /₂ð0Þje^at.

Let us consider another type of impulsive hybrid functional differential system with anticipation and retardation given by

x⁰ðtÞ ¼fðt;xt;x^tÞ; t6¼tk; x_t^þ

k ¼I_kðxtkÞ; t¼t_k;

x_t0¼/₀; x^t0¼w₀; 8>

<

>: ð2:11Þ

where w₀2PCð½t0;T;RⁿÞor w₀2PCð½t0;1;RⁿÞwith limt!1w0(t) = 0. Here, the decision (anticipative) func- tion starts from the value t = t₀either on the finite interval [t₀, T] or on the semi-infinite interval [t₀,1] and there is no need to choose z(t) as before, sincew0takes the position of z(t).

Now, we present a set of sufficient conditions in terms of Lyapunov like function which ensures that the evolution process x(t0,/0,w0)(t) for the impulsive hybrid problem(2.11)withw0defined on [t0,1] possesses the property as w0, that is

t!1limxðt0;/₀;w₀ÞðtÞ ¼0¼lim

t!1w₀ðtÞ: ð2:12Þ

Theorem 2.6. Suppose that the solutions x(t0,/0,w0)(t) of(2.11)exist for tPt0. Further, the following assump- tions hold:

(i) There exists a V :RþRⁿ!R_þ such that V(t, x) is continuous in ðtk1;t_k Rⁿ and for each x2Rⁿ, lim_ðt;yÞ!ðt^þ

k;xÞVðt;yÞ ¼Vðt^þ_k;xÞexists. Moreover, V(t, x) is positive definite and V(t, 0) = 0;

(ii) For/2X0¼ ½/2C₁:max_h16s60Vðtþs;/ðsÞÞe^aðtþsÞ¼Vðt;/ð0ÞÞe^at;a>0, DVðt;/ð0ÞÞe^atþaVðt;/ð0ÞÞe^at6gðt;Vðt;/ð0ÞÞe^at; Wðt;jw^t₀j₀ÞÞ; t6¼t_k; Vðt^þ_k;I_kðxÞÞe^at6v_kðtk;Vðt;/ð0ÞÞe^at; Wðt;jw^t₀j₀ÞÞ; t¼t_k;

(

whereWðt;jw^t₀j₀Þ 2PC½Rⁿ_þ;R_þand g(t, u, v),vk(u, v) satisfy the assumptions ofTheorem 2.1;

(iii) Wðt;jw^t₀j₀Þ6rðtÞ;tPt₀ withw^t₀¼w₀ðtþrÞ;06r6h₂;

(iv) the maximal solution r(t) = r(t, t0, u0), u0P0, is bounded on [t0,1].

Then every solution x(t0,/0,w0)(t) of(2.11)is such that(2.12)is satisfied.

Proof. Since we have assumed the existence of the solutionsx(t0,/0,w0)(t) of(2.11) fortPt0, we set mðtÞ ¼Vðt;xðt0;/₀;w₀ÞðtÞÞe^at:

For t2[t0,t1], following the procedure used in the proof of Theorem 2.4, we obtain the differential inequality:

DmðtÞ6gðt;mðtÞ;Wðt;jw^t₀j₀ÞÞ:

Using (iii) and monotone character ofg(t,u,v) in v, the above inequality becomes DmðtÞ6gðt;mðtÞ;rðtÞÞ;

which, by Theorem 2.2, yieldsm(t)6r(t,t0,u0), t2[t0,t1]. Now, using the hypothesis (ii) and the monotone nature ofv(u,v) in (u,v) together withm(t1)6r(t1), we obtain

mðt^þ₁Þ6v₁ðt1;Vðt1;/ð0ÞÞe^at1;Wðt;jw^t₀¹j₀ÞÞ6v₁ðt1;mðt1Þ;rðt1ÞÞÞ6v₁ðt1;rðt1;t₀;u₀Þ;rðt1ÞÞ ¼r^þ₁:

Next, lettingt2(t1,t2] withmðt^þ₁Þ6r^þ₁;it follows by the earlier arguments thatm(t)6r(t,t0,u0),t2(t1,t2] andmðt^þ₂Þ6r^þ₂. Repeating this process, we conclude that

mðtÞ6rðt;t₀;u₀Þ; tPt₀;

providedjm_t0j₀6u₀:In view of assumption (iv), it then implies that Vðt;xðt0;/₀;w₀ÞðtÞÞe^at6rðt;t₀;u₀Þ6N_r; tPt₀;

whereNr> 0 provides the bound onr(t,t0,u0). By the positive definiteness ofV(t,x), it follows that(2.12)is true. This completes the proof. h

3. Monotone iterative technique

In this section, the method of upper and lower solutions together with its associated monotone iteration scheme will be applied to find the unique solution of the impulsive hybrid functional differential equations involving both anticipation and retardation given by

x⁰ðtÞ ¼fðt;xðtÞ;x_t;x^tÞ; t6¼t_k; t2J ¼ ½t0;T; t₀P0;

Dxj_t¼tk ¼IkðxðtkÞÞ; k¼1;2;. . .;m;

xt₀¼/₀2C1; x^T¼w₀2C2; 8>

<

>: ð3:1Þ

where C₁¼Cð½h₁;0;RÞ; C₂¼Cð½0;h₂;RÞ; h₁; h₂P0; f 2PC½JRC₁C₂;R; and xt(s) =x(t+s), h16s60, x^t(r) =x(t+r), 06r6h2. I_k 2CðR;RÞ for each k= 1, 2,. . .,m, and {tk} is a sequence of points such that 06t0<t1< tm<T. As before, by a solution of (3.1), we mean a piecewise continuous functionx(t) on [t0,T] which is left continuous on (tk,tk+1].

We need the following known results in suitable form relative to linear impulsive hybrid functional differ- ential inequalities[8].

Lemma 3.1. Assume that

(i) pðtÞ 2PCð½t0h1;Tþh2;RÞ \C¹ðJ1;RÞsatisfies p⁰ðtÞ6MpðtÞ NR0

h1p_tðsÞds; t6¼t_k; t2J ¼ ½t0;T; Dpj_t¼tk 6LkpðtkÞ; k¼1;2;. . .;m;

(

where MP0, 06Lk61, k = 1, 2,. . ., m andJ1¼J ftkg^k¼m_k¼1;

(ii) p_t0ðsÞ60;h16s60,pðtÞ 2PCð½t0h1;t0;RÞ \C¹ðJ2;RÞ,whereJ2¼ ½t0h1;t0 ftlg¹_l¼r;ftlg¹_l¼ris the set of points of discontinuity of p(t), p⁰(s)6k/(T + h1) where min_½t0h1;t₀pðsÞ ¼ k;kP0;

½MþNh₁ðTþh₁Þ61; pðt^þ_iÞ pðt_iÞ6L_ipðt_iÞ and D1=max{t_r+ h1, t_r+1t_r, . . .,t₁, t1, t2t1,. . ., Ttm}is such that

D16

Qm

k¼rð1LkÞ 1þPm

j¼r

Qm

k¼jð1L_kÞðTþh1Þ:

Then p(t)60 on J.

Lemma 3.2. Let pðtÞ 2PCð½t₀h₁;Tþh₂;RÞ \C¹ðJ₁;RÞsatisfies p⁰ðtÞ6M1pðtÞ þN₁

Z 0 h1

p_tðsÞdsþN₂ Z h₂

0

p^tðuÞdu; t6¼t_k; t2J;

where M1, N1, N2P0 with [h1N1+ h2N2]<M1, and suppose that all other assumptions of Lemma 3.1 hold.

Thenp_t060; p^T60implies p(t)60 on J.

For convenience, let us set the following assumptions relative to(3.1):

(A1) There exista0;b₀2PCðJ;RÞsatisfying a⁰₀ðtÞ6fðt;a0ðtÞ;a0t;b^t₀Þ; t6¼t_k; t2J;

Da0j_t¼tk 6I_kða0ðtkÞÞ; k¼1;2;. . .;m;

a0t₀¼/₁; a^T₀ ¼w₁; 8>

<

>:

b⁰₀ðtÞPfðt;b₀ðtÞ;b_0t;a^t₀Þ; t6¼t_k; t2J;

Db₀j_t¼tk PI_kðb0ðtkÞÞ; k¼1;2;. . .;m;

b_0t0 ¼/₂; b^T₀ ¼w₂; 8>

<

>:

such that/16/06/2,w16w06w2,a0(t)6b0(t), t2J and/1,/22C1,w1,w22C2. (A₂) f(t, x,/,w) is nonincreasing in wfor each (t, x,/).

(A3) There exist M, NP0 and an arbitraryf2C2such that fðt;x;/;fÞ fðt;y;w;fÞPMðxyÞ N

Z 0 h1

ð/wÞðsÞds; t2J;

whenevera0(t)6y6x6b0(t), a0t6w6/6b0t.

(A4) a0t/0,/0b0t satisfy the assumption (ii) ofLemma3.1.

(A5) There exist constants 06Lk61, k = 1, 2,. . ., m such that I_kðxÞ I_kðyÞPLkðxyÞ;

whenevera₀(t_k)6y6x6b₀(t_k).

Theorem 3.1. Let the assumptions (A1)–(A5) be satisfied. Then there exist monotone sequences {an(t)}, {bn(t)}, which converge uniformly to the minimal solutionq(t) and the maximal solution r(t) of(3.1)on [t0h1, T + h2].

If, in addition, we require

(A6) fðt;x;/₁;w₂Þ fðt;x;/₂;w₁Þ6 M₁ðxyÞ þN₁R0

h1ð/1 /₂ÞðsÞdsþN₂Rh2

0 ðw1w₂ÞðuÞdu; where a0ðtÞ6/₂6/₁6b₀ðtÞ;a^T₀ 6w₂6w₁6b^T₀ and [h1N1+ h2N2]<M1,

then q(t) = r(t) = x(t) is the unique solution of(3.1)on J.

Proof. For eachn= 1, 2, 3,. . ., we consider the following linear problems:

a⁰_nþ1ðtÞ ¼fðt;an;ant;b^t_nÞ Mðanþ1anÞ NR0

h1ðaðnþ1Þta_ntÞðsÞds; t6¼t_k; t2J;

Da_nþ1j_t¼tk ¼I_kðanðtkÞÞ L_kðanþ1ðtkÞ a_nðtkÞÞ; k¼1;2;. . .;m;

8>

<

>: ð3:2Þ

b⁰_nþ1ðtÞ ¼fðt;b_n;b_nt;a^t_nÞ Mðb_nþ1b_nÞ NR0

h1ðb_ðnþ1Þtb_ntÞðsÞds; t6¼t_k; t2J;

Db_nþ1j_t¼tk ¼I_kðb_nðtkÞÞ L_kðb_nþ1ðtkÞ b_nðtkÞÞ; k¼1;2;. . .;m;

8>

<

>: ð3:3Þ

witha_ðnþ1Þt0 ¼/₀;b_ðnþ1Þt0¼/₀anda^T_nþ1;b^T_nþ1are chosen so that

a^T₀ 6a^T_n 6a^T_nþ16w₀6b^T_nþ16b^T_n 6b^T₀; ð3:4Þ anda^T_n;b^T_n converge uniformly tow0on [0,h₂]. In passing, we remark that we can fixa^T_n;b^T_n in(3.4)according to our choice, for instance, a^T_n;b^T_n may be taken to be the sequences translating w0 appropriately, that is, a^T_n ¼w₀n_n;b^T_n ¼w₀þg_nwithan(T) =w0(T)nn,bn(T) =w0(T) +gnfor eachn, wherenn,gn> 0 are mono- tone sequences tending to zero asn! 1. Another simple choice for(3.4)isa^T₀ ¼w₀;b^T₀ ¼w₀:In case we take /0satisfyinga0t₀ 6/₀6b_0t0;w0must satisfya1(T)6w0(T)6b1(T) for a suitable choice of(3.4).

We note that each of the linear problems(3.2) and (3.3)has a unique solution on [t0h1,T+h2]. We want to show that

a06a16a26 6an6b_n6 6b₂6b₁6b₀; t2J: ð3:5Þ For this purpose, we setp=a0a1and first prove thata06a1on J.

p⁰¼a⁰₀a⁰₁6fðt;a0ðtÞ;a0t;b^t₀Þ fðt;a0;a0t;b^t₀Þ þMða1a0Þ þN Z 0

h1

ða1ta0tÞðsÞds

6MpN Z 0

h1

p_tðsÞds; t6¼t_k; t2J;

Dpj_t¼tk ¼Da0j_t¼tkDa1j_t¼tk

6I_kða0ðt_kÞÞ I_kða0ðt_kÞÞ þL_kða1ðt_kÞ a0ðt_kÞÞ 6LkpðtkÞ; k¼1;2; ;. . .;m;

and

p_t0¼a0t₀a1t₀ 60:

Hence, byLemma 3.1, it follows thata06a1onJ. In a similar manner, one can show thatb16b0onJ. Next we prove thata16b1onJ. Lettingp=a1b1and using(3.2) and (3.3)together with the assumptions (A2), (A3) and (A5), we have

p⁰¼a⁰₁b⁰₁¼fðt;a0;a0t;b^t₀Þ Mða1a0Þ N Z 0

h1

ða1ta0tÞðsÞds

fðt;b₀;b_0t;a^t₀Þ þMðb₁b₀Þ þN Z 0

h1

ðb_1tb_0tÞðsÞds

6Mðb₀a0Þ þN Z 0

h1

ðb_0ta0tÞðsÞdsMða1a0Þ N Z 0

h1

ða1ta0tÞðsÞds

þMðb₁b₀Þ þN Z 0

h1

ðb_1tb_0tÞðsÞds6MpN Z 0

h1

p_tðsÞds; t6¼t_k; t2J;

Dpj_t¼tk ¼Da1j_t¼tkDb₁j_t¼tk ¼I_kða0ðtkÞÞ L_kða1ðtkÞ a0ðtkÞÞ I_kðb₀ðtkÞÞ þL_kðb₁ðtkÞ b₀ðtkÞÞ 6L_kðb₀ðtkÞ a0ðtkÞÞ L_kða1ðtkÞ a0ðtkÞÞ þL_kðb₁ðtkÞ b₀ðtkÞÞ

¼ LkpðtkÞ; k¼1;2;. . .;m;

andp_t0 ¼0:Again, byLemma 3.1, we find thatp(t)60 onJ, that is,a16b1onJ. Thus, we get

a06a16b₁6b₀ onJ: ð3:6Þ

Now, forp> 1, we assume that

ap16ap6b_p6b_p1 on J; ð3:7Þ

and prove that

ap6apþ16b_pþ16b_p on J: ð3:8Þ

Settingp=apap+1and making use of the assumptions (A2), (A3) and (A5) together with(3.7), we obtain p⁰¼a⁰_pa⁰_pþ1¼fðt;a_p1;a_ðp1Þt;b^t_p1Þ

Mðapap1Þ N Z 0

h1

ðaptaðp1ÞtÞðsÞdsfðt;ap;apt;b^t_pÞ

þMðapþ1a_pÞ þN Z 0

h1

ðaðpþ1Þta_ptÞðsÞds

6Mðapa_p1Þ þN Z 0

h1

ðapta_ðp1ÞtÞðsÞds

Mðapa_p1Þ N Z 0

h1

ðapta_ðp1ÞtÞðsÞdsþMðapþ1a_pÞ þN Z 0

h1

ðaðpþ1Þta_ptÞðsÞds

6MpN Z 0

h1

p_tðsÞds; t6¼t_k; t2J;

Dpj_t¼tk ¼Dapj_t¼tkDapþ1j_t¼tk ¼I_kðap1ðtkÞÞ L_kðapðtkÞ ap1ðtkÞÞ I_kðapðtkÞÞ þL_kðapþ1ðtkÞ a_pðtkÞÞ

6L_kðapðtkÞ ap1ðtkÞÞ L_kðapðtkÞ ap1ðtkÞÞ þL_kðapþ1ðtkÞ apðtkÞÞ

¼ L_kpðt_kÞ; k¼1;2;. . .;m:

Sincep_t0¼0;it follows byLemma 3.1thatp(t)60 onJ, that is,ap6ap+1onJ. Using a similar procedure, it can be shown thatbp+16bponJ. Now, we definep=ap+1bp+1and employ the earlier arguments to arrive at

p⁰6MpN Z 0

h1

p_tðsÞds; t6¼tk; t2J; Dpj_t¼tk 6L_kpðt_kÞ; k¼1;2;. . .;m;

and p_t0 ¼0; which imply that ap+16bp+1 on J. Combining these conclusions proves (3.8). Therefore, by induction, (3.5) is valid on J, which together with (3.4) implies that (3.5)also holds on [t0,T+h2]. As the sequences {an}, {bn} are bounded by (3.5), one can apply the standard arguments [7–9] to conclude that sequences {an}, {bn} converge uniformly onJ and in factan!q,bn!ron J. Also, we have

q⁰ðtÞ ¼fðt;q;q_t;r^tÞ; t6¼t_k; Dqj_t¼tk ¼I_kðqðtkÞÞ; k¼1;2;. . .;m; q_t0¼/₀; r⁰ðtÞ ¼fðt;r;r_t;q^tÞ; t6¼t_k; Drj_t¼tk ¼I_kðrðt_kÞÞ; k¼1;2;. . .m; r_t0 ¼/₀; withq6ronJandq^T=r^T.

In order to show thatq,rare respectively coupled minimal and maximal solutions of(3.1), letx(t) be any solution of(3.1)withxt0 ¼/₀;x^T¼w₀satisfyinga06x6b0onJ. By definition ofq,r, we haveq^T=x^T=r^T. Therefore, it is enough to show that q6x6ronJ. Lettingp=a1xso thatp_t

0¼0 and using the earlier arguments, we obtain

p⁰¼a⁰₁x⁰¼fðt;a0;a0t;b^t₀Þ Mða1a0Þ N Z 0

h1

ða1ta0tÞðsÞdsfðt;x;x_0t;x^tÞ

6Mðxa0Þ þN Z 0

h1

ðxta0tÞðsÞdsMða1a0Þ N Z 0

h1

ða1ta0tÞðsÞds

6MpN Z 0

h1

p_tðsÞds; t6¼t_k; t2J;

Dpj_t¼tk ¼Da1j_t¼tkDxj_t¼tk ¼I_kða0ðtkÞÞ L_kða1ðtkÞ a0ðtkÞÞ I_kðxtkÞ 6L_kðxðtkÞ a0ðtkÞÞ L_kða1ðtkÞ a0ðtkÞÞ

¼ LkpðtkÞ; k¼1;2;. . .;m:

Hence, byLemma 3.1, we geta16xon J. A similar procedure can be employed to show thatx6b1on J.

Proceeding as before, it can be shown thatan+16x6bn+1onJ. Thusq,rare respectively coupled minimal and maximal solutions of(3.1).

Next, we show the uniqueness of the solution of (3.1). Setting p=rq and using q^t6r^t together with assumption (A6), we obtain

p⁰¼r⁰ðtÞ q⁰ðtÞ ¼fðt;r;rt;q^tÞ fðt;q;q_t;r^tÞ 6M1pðtÞ þN₁

Z 0 h1

ðsÞp_tðsÞdsþN₂ Z h2

0

p^tðsÞds; t6¼t_k; Dpj_t¼tk ¼I_kðrðtkÞÞ I_kðqðtkÞÞ6LkpðtkÞ; k¼1;2;. . .;m;

andp_t

0¼0; p^T¼0. ByLemma 3.2, we find thatr6q, which yieldsq=ronJ. Consequently,x=q=ris the unique solution of(3.1)onJ withx_t0 ¼/₀; x^T ¼w₀. This completes the proof. h

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