existence and uniqueness of asymptotically constant or periodic solutions in delayed population...

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Existence and uniqueness ofasymptotically constant or periodicsolutions in delayed population modelsMichael Radina & Youssef Raffoulba School of Mathematical Sciences, Rochester Institute ofTechnology, 85 Lomb Memorial Drive, Rochester, NY 14623, USAb Department of Mathematics, University of Dayton, Dayton, OH45269-2316, USAPublished online: 22 Jul 2013.

To cite this article: Michael Radin & Youssef Raffoul (2014) Existence and uniqueness ofasymptotically constant or periodic solutions in delayed population models, Journal of DifferenceEquations and Applications, 20:5-6, 706-716, DOI: 10.1080/10236198.2013.810734

To link to this article: http://dx.doi.org/10.1080/10236198.2013.810734

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Existence and uniqueness of asymptotically constant or periodicsolutions in delayed population models

Michael Radina* and Youssef Raffoulb1

aSchool of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive,Rochester, NY 14623, USA; bDepartment of Mathematics, University of Dayton, Dayton,

OH 45269-2316, USA

(Received 19 February 2013; final version received 26 May 2013)

By means of fixed point theory we show that the unique solution of nonlinear differenceequations of the form

DxðtÞ ¼ gðxðtÞÞ2Xt21

s¼t2L

pðs2 tÞgðxðsÞÞ

converges to a predetermined constant or a periodic solution. Then, we show thesolution is stable and that its limit function serves as a global attractor. The same theorywill be extended to two more models.

Keywords: nonlinear difference equations; constant solution; periodic solution;contraction mapping; global attractor

AMS Subject Classification: 39A10; 34A97

1. Introduction

Let Z be the set of integers and for t [ Z. In [6] the author used and we will use the

contraction mapping principle to study the convergence of solutions of nonlinear

difference equations of the form

DxðtÞ ¼ gðxðtÞÞ2 gðxðt2 LÞÞ; ð1:1Þ

where g : R! R and is continuous in x with R denoting the set of all real numbers. In

addition, motivated by the papers of [1–3] the author extended the analysis of [6] to the

following population models

DxðtÞ ¼ gðxðt2 L1ÞÞ2 gðxðt2 L1 2 L2ÞÞ; ð1:2Þ

DxðtÞ ¼ gðt; xðtÞÞ2 gðt; xðt2 LÞÞ; gðt þ L; xÞ ¼ gðt; xÞ; ð1:3Þ

where L; L1 and L2 are constants and positive integers. We note that every constant is a

solution of equations (1.1)–(1.3).

Equations (1.1)–(1.3) play major roles in population models. For example, suppose

xðtÞ is the number of individuals in a population at time t. Let the delay L be the life span

of each individual. Then the birth rate of the population is some function of xðtÞ, say

gðxðtÞÞ, and the function gðxðt2 LÞÞ can be thought of as the number of deaths per unit

q 2013 Taylor & Francis

*Corresponding author. Email: [email protected]

Journal of Difference Equations and Applications, 2014

Vol. 20, Nos. 5–6, 706–716, http://dx.doi.org/10.1080/10236198.2013.810734

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mailto:[email protected]

http://dx.doi.org/10.1080/10236198.2013.810734

time at time t. Then the difference term gðxðtÞÞ2 gðxðt2 LÞÞ represents the net change

in population per unit time. This implies that the growth of the population is governed

by (1.1).

In [4] the author used the contraction mapping principle and Schauder’s second fixed

point theorem to show the existence of periodic solution of the nonlinear system

xðnþ 1Þ ¼ AðnÞxðnÞ þ gðn; xðnÞÞ;

under periodicity conditions on AðnÞ and g. Followed by the paper [5] the author

considered the scalar delay difference equation

xðnþ 1Þ ¼ aðnÞxðnÞ þ lf ðn; xðn2 tðnÞÞ;

where tðnÞ is an integer periodic non-negative sequence. Using Krasnoselskii’s fixed point

theorem, the author was able to find intervals that are filled with the eigenvalues l on

which the above-mentioned difference equation possessed at least one positive periodic

solution.

In this paper we begin by considering the model

DxðtÞ ¼ gðxðtÞÞ2Xt21

s¼t2L

pðs2 tÞgðxðsÞÞ: ð1:4Þ

Model (1.4) assumes that the deaths of those born at time t would be distributed all along

the time period ½t; t þ L�, and certainly a few beyond t þ L. In Section 2, we will use the

contraction mapping principle to determine that constant. First, we state what it means for

xðtÞ to be a solution of (1.1). Note that since (1.1) is autonomous, we lose nothing by

starting the solution at 0.

Let c : ½2L; 0�! R be a given bounded initial function. We say xðt; 0;cÞ is a solutionof (1.1) if xðt; 0;cÞ ¼ c on ½2L; 0� and xðt; 0;cÞ satisfies (1.4) for t $ 0.

It is of importance to us to know such constants since all of our models have constant

solutions.

2. Equation (1.4)

In this section we use the notion of fixed point theory to determine the constant that all

solutions of (1.4) converge to. First we rewrite (1.4) as

DxðtÞ ¼ Dt

X21

s¼2L

pðsÞXt21

u¼tþs

gðxðuÞÞ; ð2:1Þ

where pðsÞ satisfies the condition

X21

s¼2L

pðsÞ ¼ 1: ð2:2Þ

Also, we assume that function g is globally Lipschitz. That is, there exists a constant k . 0

such that

jgðxÞ2 gðyÞj # kjx2 yj: ð2:3Þ

Journal of Difference Equations and Applications 707

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On the other hand, in order to obtain contraction, we assume there is a positive constant

j , 1 so that

kX21

s¼2L

jpðsÞjð2sÞ # j: ð2:4Þ

We note that if there is a uniform distribution of deaths over one period L; pðtÞ ¼ 1=L, then(2.2) is satisfied. Moreover, in this case condition (2.4) becomes

kX21

s¼2L

jpðsÞjð2sÞ ¼ kX21

s¼2L

1

Lð2sÞ ¼

kðLþ 1Þ

2:

Thus, condition (2.4) is satisfied for

kðLþ 1Þ

2# j:

Recall, for the constant delay case, the parallel condition of Theorem 2.1 of [6] is kL # j.

To construct a suitable mapping, we let c : ½2L; 0�! R be a given initial function. By

summing (2.1) from s ¼ 0 to s ¼ t2 1 we arrive at the expression

xðtÞ ¼ cð0Þ2X21

s¼2L

pðsÞX21

u¼s

gðcðuÞÞ þX21

s¼2L

pðsÞXt21

u¼tþs

gðxðuÞÞ: ð2:5Þ

If xðtÞ is given by (2.5) then it solves (1.4). In the next theorem we show that, given an

initial function cðtÞ : ½2L; 0�! R, the unique solution of (1.4) converges to a unique

determined constant.

Theorem 1. Assume (2.2)– (2.4) and let c : ½2L; 0�! R be a given initial function. Then,

the unique solution xðt; 0;cÞ of (1.4) satisfies xðt; 0;cÞ! r where r is unique and given by

r ¼ cð0Þ þ gðrÞX21

s¼2L

pðsÞð2sÞ2X21

s¼2L

pðsÞX21

u¼s

gðcðuÞÞ: ð2:6Þ

Proof. For j�j denoting the absolute value, the metric space ðR; j�jÞ is complete. Define a

mapping H : R! R by

Hr ¼ cð0Þ þ gðrÞX21

s¼2L

pðsÞð2sÞ2X21

s¼2L

pðsÞX21

u¼s

gðcðuÞÞ:

For a; b [ R we have

jHa2Hbj #X21

s¼2L

jpðsÞjð2sÞjgðaÞ2 gðbÞj # kX21

s¼2L

jpðsÞjð2sÞja2 bj # jja2 bj:

This shows thatH is a contraction on the complete metric space ðR; j�jÞ, and henceH has

a unique fixed point r, which implies that (2.6) has a unique solution. It remains to show

that (1.4) has a unique solution and that it converges to constant r.

M. Radin and Y. Raffoul708

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Let k�k denote the maximum norm and let M be the set bounded functions f :

½2L;1Þ! R with fðtÞ ¼ c on ½2L; 0�;fðtÞ! r as t!1. Then ðM; k�kÞ defines a

complete metric space. For f [ M, define P : M!M by

ðPfÞðtÞ ¼ c for 2 L # t # 0

and

ðPfÞðtÞ ¼ cð0Þ2X21

s¼2L

pðsÞX21

u¼s

gðcðuÞÞ þX21

s¼2L

pðsÞXt21

u¼tþs

gðfðuÞÞ for t $ 0: ð2:7Þ

For f [ M with fðtÞ! r, we have

X21

s¼2L

pðsÞXt21

u¼tþs

gðfðuÞÞ! gðrÞX21

s¼2L

pðsÞð2sÞ; as t!1:

Then, using (2.6) and (2.7), we see that

ðPfÞðtÞ! cð0Þ þ gðrÞX21

s¼2L

pðsÞð2sÞ2X21

s¼2L

pðsÞX21

u¼s

gðcðuÞÞ ¼ r:

Thus, P : M!M. It remains to show that P is a contraction.

For a; b [ M, we have

jðPaÞðtÞ2 ðPbÞðtÞj #X21

s¼2L

jpðsÞjð2sÞjgðaðsÞÞ2 gðbðsÞÞj

# kX21

s¼2L

jpðsÞjð2sÞja2 bj # jka2 bk:

Thus, P is a contraction and has a unique fixed point f [ M. Based on how the mapping

P was constructed, we conclude that the unique fixed point f satisfies (1.4). A

Remark 1. For any given initial function, Theorem 1 explicitly gives the limit to which the

solution converges to. That limit is the unique solution r of (2.6).

Remark 2. For arbitrary initial function, say h : ½2L; 0�! R, Theorem 1 shows that

xðt; 0;hÞ! r. Thus, we may think of r as being ‘global attractor’.

Remark 3. We may think of Theorem 1 as of stability results. In general, we know that

solutions depend continuously on initial functions. That is, solutions which start close

remain close on finite intervals. However, under conditions of Theorem 1 such solutions

remain close forever, and their asymptotic respective constants remain close too.

The next theorem is a verification of our claim in Remark 3.

Theorem 2. Assume the hypothesis of Theorem 1. Then every initial function is stable.

Moreover, if c1 and c2 are two initial functions with xðt; 0;c1Þ! r1 and xðt; 0;c2Þ! r2then jr1 2 r2j , 1 for positive 1.


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Proof. Let kck½2L;0� denote the supremum norm of c on the interval ½2L; 0�. Fix an initialfunction c1 and let c2 be any other initial function. Let Pi; i ¼ 1; 2 be the mapping defined

by (2.7). Then by Theorem 1 there are unique functions u1; u2 and unique constants r1 andr2 such that

P1u1 ! u1; P2u2 ! u2; u1ðtÞ! r1; u2ðtÞ! r2:

Let 1 . 0 be any given positive number and set

d ¼1 12 k

P21s¼2L jpðsÞjð2sÞ

� �1þ k


:

Then

ju1ðtÞ2 u2ðtÞj ¼ jðP1u1ÞðtÞ2 ðP2u2ÞðtÞj

# jc1ð0Þ2 c2ð0Þj þX21

s¼2L

pðsÞX21

u¼s

jgðc1ðsÞÞ2 gðc2ðsÞÞj

þX21

s¼2L

pðsÞXt21

u¼tþs

jgðu1ðsÞÞ2 gðu2ðsÞÞj

# jc1ð0Þ2 c2ð0Þj þ kX21

s¼2L

jpðsÞjð2sÞkc1 2 c2k½2L;0�

þ kX21

s¼2L

jpðsÞjð2sÞj ku1 2 u2k:

This yields

ku1 2 u2k ,1þ k


12 kP21

s¼2L jpðsÞjð2sÞ:kc1 2 c2k½2L;0� , 1;

provided that

kc1 2 c2k½2L;0� ,1 12 k


� �1þ k


:¼ d:

This shows that

jxðt; 0;c1Þ2 xðt; 0;c2Þj , 1; whenever kc1 2 c2k½2L;0� , d:

For the rest of the proof we note that juiðtÞ2 kij! 0 as t!1 implies that

jr1 2 r2j ¼ jr1 2 u1ðtÞ þ u1ðtÞ2 u2ðtÞ þ u2ðtÞ2 r2j

# jr1 2 u1ðtÞj þ ku1 2 u2k þ ju2ðtÞ2 r2j! ku1 2 u2k ðas t!1Þ , 1:

A


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3. Infinite delay model

In this section we extend the results of Section 2 to the infinite delay model

DxðtÞ ¼Xt21

s¼t2L

pðs2 tÞgðxðsÞÞ2Xt21

s¼21

qðs2 tÞgðxðsÞÞ: ð3:1Þ

The first term on the right takes into account the ideas of Section 2 while the second

term takes into account the deaths distributed over all past times. First we put (3.1) in

the form

DxðtÞ ¼ 2Dt

X21

s¼2L

pðsÞXt21

u¼tþs

gðxðuÞÞ þ Dt

Xt21

s¼21

Xs2t

u¼21

qðuÞgðxðsÞÞ; ð3:2Þ

where we have assumed (2.2) and

X21

s¼21

qðsÞ ¼ 1: ð3:3Þ

Let c : ð21; 0�! R be an initial bounded sequence. Then

xðtÞ ¼ 2X21

s¼2L

pðsÞXt21

u¼tþs

gðxðuÞÞ þXt21

s¼21

Xs2t

u¼21

qðuÞgðxðsÞÞ þ c; ð3:4Þ

where

c ¼ cð0Þ þX21

s¼2L

pðsÞX21

u¼s

gðxðuÞÞ2X21

s¼21

Xsu¼21

qðuÞgðcðsÞÞ ð3:5Þ

is a solution of (3.1). We are ready to state our next theorem.

Theorem 3. Assume (2.2), (2.3), (3.1) and there exists a constant a so that for 0 , a , 1,

we have

kX21

s¼2L

jpðsÞð2sÞj þX21

s¼21

Xsu¼21

jqðuÞj

!# a: ð3:6Þ

Then, the unique solution xðt; 0;cÞ of (3.1) satisfies xðt; 0;cÞ! r where r is unique and

given by

r ¼ c2 gðrÞX21

s¼2L

pðsÞð2sÞ þ gðrÞX21

s¼21

Xsu¼21

qðuÞ; ð3:7Þ

and c is given by (3.5).


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Proof. For j�j denoting the absolute value, the metric space ðR; j�jÞ is complete. Define a

mapping H : R! R by

Hr ¼ c2 gðrÞX21

s¼2L

pðsÞð2sÞ þ gðrÞX21

s¼21

Xsu¼21

qðuÞ:

For a; b [ R we have

jHa2Hbj #X21

s¼2L

jpðsÞð2sÞkgðaÞ2 gðbÞj þ jgðaÞ2 gðbÞjX21

s¼21

Xsu¼21

jqðuÞj

# kX21

s¼2L

jpðsÞð2sÞj þX21

s¼21

Xsu¼21

jqðuÞj

!ja2 bj

# aja2 bj:

This shows thatH is a contraction on the complete metric space ðR; j�jÞ, and henceH has

a unique fixed point r, which implies that (3.7) has a unique solution. It remains to show

that (3.1) has a unique solution and that it converges to constant r.

Let k�k denote the maximum norm and let M be the set bounded functions f :

½21;1Þ! R with fðtÞ ¼ c on ½21; 0�; fðtÞ! r as t!1. Then ðM; k�kÞ defines acomplete metric space. For f [ M, define P : M!M by

ðPfÞðtÞ ¼ c for t [ ð21; 0�;

and

ðPfÞðtÞ ¼ c2X21

s¼2L

pðsÞXt21

u¼tþs

gðfðuÞÞ þXt21

s¼21

Xs2t

u¼21

qðuÞgðfðsÞÞ; for t $ 0; ð3:8Þ

where c is given by (3.5). Due to the continuity of gwe have that for f [ Mwith fðtÞ! r,

X21

s¼2L

pðsÞXt21

u¼tþs

gðfðuÞÞ! gðrÞX21

s¼2L

pðsÞð2sÞ; as t!1:

Next we show that

Xt21

s¼21

Xs2t

u¼21

qðuÞgðfðsÞÞ! gðrÞXt21

s¼21

Xs2t

u¼21

qðuÞ; as t!1: ð3:9Þ

Again, due to the continuity of G, for f [ M with fðtÞ! r one might find positive

numbers Q and T such that for any 1 . 0 we have

jgðfðtÞÞ2 gðrÞj # Q for all t and jfðtÞ2 rj , 1 if T # t , 1:


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With this in mind, we have

Xt21

s¼21

Xs2t

u¼21

qðuÞðgðfðsÞÞ2 gðrÞÞ

�� #

XT21

s¼21

Xs2t

u¼21

jqðuÞj jðgðfðsÞÞ2 gðrÞÞj

þXt21

s¼T

Xs2t

u¼21

jqðuÞj jðgðfðsÞÞ2 gðrÞÞj

# QXT21

s¼21

Xs2t

u¼21

jqðuÞj þXt21

s¼T

Xs2t

u¼21

jqðuÞj jfðsÞ2 rj

# QXT21

s¼21

Xs2t

u¼21

jqðuÞj þ k1Xt21

s¼T

Xs2t

u¼21

jqðuÞj

# QXT2t21

s¼21

Xsu¼21

jqðuÞj þ k1Xt21

s¼21

Xs2t

u¼21

jqðuÞj:

Due to the convergence that was required by (3.6), we havePT2t21s¼21

Psu¼21jqðuÞj! 0; as t!1. Moreover, for T # t , 1 condition (3.6) implies

that k1Pt21

s¼21

Ps2tu¼21jqðuÞj # 1a. Hence (3.9) is proved. It remains to show that P is a

contraction.

For a; b [ M, we have

jðPaÞðtÞ2 ðPbÞðtÞj #X21

s¼2L

jpðsÞjð2sÞjgðaðsÞÞ2 gðbðsÞÞj

þXt21

s¼21

Xs2t

u¼21

jqðuÞj jgðaðsÞÞ2 gðbðsÞÞj

# kX21

s¼2L

jpðsÞð2sÞj þXt21

s¼21

Xs2t

u¼21

jqðuÞj

!ka2 bk

# aka2 bk:

A

4. Equation (1.4) revisited

In this section we investigate the existence of periodic solutions of

DxðtÞ ¼ gðt; xðtÞÞ2Xt21

s¼t2L

pðs2 tÞgðs; xðsÞÞ; ð4:1Þ

where

gðt þ L; xÞ ¼ gðt; xÞ: ð4:2Þ

As before, we assume that there exists a positive constant k such that for all x; y [ R

we have

jgðt; xÞ2 gðt; yÞj # kjx2 yj: ð4:3Þ


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If (2.2) holds then we may rewrite (4.1) as

DxðtÞ ¼ Dt

X21

s¼2L

pðsÞXt21

u¼tþs

gðu; xðuÞÞ: ð4:4Þ

As before, to construct a suitable mapping, we let c : ½2L; 0�! R be a given initial

function. By summing (4.1) from s ¼ 0 to s ¼ t2 1 we arrive at the expression

xðtÞ ¼X21

s¼2L

pðsÞXt21

u¼tþs

gðu; xðuÞÞ þ c; ð4:5Þ

where c is given by

c ¼ cð0Þ2X21

s¼2L

pðsÞX21

u¼s

gðu;cðuÞÞ: ð4:6Þ

Theorem 4. Assume (2.2)– (2.4), (4.2) and (4.3) and let c : ½2L; 0�! R be a given initial

function. Then, the unique solution xðt; 0;cÞ of (4.1) satisfies xðt; 0;cÞ! r as t!1 where

r is a unique L-periodic solution of (4.1).

Proof. Let k�k denote the maximum norm and let M be the set of L-periodic sequences

f : Z! Z. Then ðM; k�kÞ defines a Banach space of L-periodic sequences. For f [ M,

define P : M!M by

ðPfÞðtÞ ¼ cþX21

s¼2L

pðsÞXt21

u¼tþs

gðu;fðuÞÞ: ð4:7Þ

Next we show that

ðPfÞðt þ LÞ ¼ ðPfÞðtÞ:

To see, for f [ M we have

ðPfÞðt þ LÞ ¼ cþX21

s¼2L

pðsÞXtþL21

u¼tþsþL

gðu;fðuÞÞ

¼ cþX21

s¼2L

pðsÞXt21

l¼tþs

gðlþ L;fðlþ LÞÞ ðl ¼ u2 LÞ

¼ cþX21

s¼2L

pðsÞXt21

l¼tþs

gðl;fðlÞÞ ¼ ðPfÞðtÞ:

Hence, P maps M into M. Also, with similar argument as in the previous theorems, one

can easily show that P is a contraction. Hence, (4.7) has a unique fixed point r inM which

solves (4.1). It remains to show that ðPfÞðtÞ! rðtÞ.

Let k�k denote the maximum norm and let M be the set of bounded functions

f : ½2L;1Þ! R with fðtÞ ¼ c on ½2L; 0�; fðtÞ! rðtÞ as t!1. Then ðM; k�kÞ defines


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a complete metric space. For f [ M, define P : M!M by

ðPfÞðtÞ ¼ c; for 2 L # t # 0;

and

ðPfÞðtÞ ¼ cþX21

s¼2L

pðsÞXt21

u¼tþs

gðu;fðuÞÞ; for t $ 0:

jðPfÞðtÞ2 rðtÞj ¼X21

s¼2L

pðsÞXt21

u¼tþs

gðu;fðuÞÞ2X21

s¼2L

pðsÞXt21

u¼tþs

gðu; rðuÞÞ

��

#X21

s¼2L

jpðsÞjXt21

u¼tþs

kjfðuÞ2 rðuÞj

#X21

s¼2L

jpðsÞjXt21

u¼t2L

kjfðuÞ2 rðuÞj! 0; as t!1;

since jfðuÞ2 rðuÞj! 0; as t!1. The proof for showing P is a contraction is similar as

before and hence is omitted. Thus we have shown that P has a unique fixed point in M

which converges to r. A

We note that Remarks 1–3 and hence Theorem 2 hold for equations (3.1) and (4.1).

We end this section with the following corollary.

Corollary 1. Assume the hypothesis of Theorem 4. If there exists an r [ R such that

gðt; rÞ ¼X21

s¼2L

pðsÞgðt þ s; rÞ; ð4:8Þ

then r of Theorem 4 is constant.

Proof. Suppose (4.1) has a constant solution r. Then from (4.4) we have

0 ¼ Dr ¼ Dt

X21

s¼2L

pðsÞXt21

u¼tþs

gðu; rÞ

¼X21

s¼2L

pðsÞðgðt; rÞ2 gðt þ s; rÞÞ

¼ gðt; rÞX21

s¼2L

pðsÞ2X21

s¼2L

pðsÞgðt þ s; rÞ

¼ gðt; rÞ2X21

s¼2L

pðsÞgðt þ s; rÞ; due to ð2:2Þ:

A


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5. Future work

In the future, it would be of paramount interest to study the monotonic and periodic

behaviour of the solutions (1.1) and (1.2) as well as the coupled system of equations:

DxðtÞ ¼ g1ðxðtÞÞ2 g2ðyðtÞÞg1ðxðt2 LÞÞ

DyðtÞ ¼ g2ðyðtÞÞ2 g1ðxðtÞÞg2ðyðt2 LÞÞ

):

Note

1. Email: [email protected]

References

[1] T.A. Burton, Fixed points and differential equations with asymptotically constant or periodicsolutions, Electron. J. Qual. Theory Differ. Equ. 2004(11) (2004), pp. 1–31.

[2] L.K. Cooke, Functional-differential equations; some models and perturbation problems, inDifferential and Dynamical Systems, Academic Press, New York, 1967, pp. 167–183.

[3] L.K. Cooke, An epidemic equation with immigration, Math. Biosci. 29 (1976), pp. 135–158.[4] Y.N. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations,

Pan-American J. Math. 9 (1999), pp. 97–111.[5] Y.N. Raffoul, T-periodic solutions and a priori bound, Math. Comput. Model. 32 (2000),

pp. 643–652.[6] Y.N. Raffoul, Stability and periodicity in completely delayed equations, J. Math. Anal. Appl.

324 (2006), pp. 1356–1362.


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mailto:[email protected]

existence and uniqueness of asymptotically constant or periodic solutions in delayed population...

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