Positive periodic solutions for nonlinear difference equations via a continuation theorem

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DIFFERENCE EQUATIONS VIA A CONTINUATION THEOREM

GEN-QIANG WANG AND SUI SUN CHENG

Received 29 August 2003 and in revised form 4 February 2004

Based on a continuation theorem of Mawhin, positive periodic solutions are found for diﬀerence equations of the formyn+1=ynexp(f(n,yn,yn−1,...,yn−k)),n∈Z.

1. Introduction

There are several reasons for studying nonlinear diﬀerence equations of the form

yn+1=ynexp

fn,yn,yn−1,...,yn−k

, n∈Z= {0,±1,±2,...}, (1.1)

where f = f(t,u0,u1,...,uk) is a real continuous function defined onRk+2such that

ft+ω,u0,...,uk=ft,u0,...,uk, t,u0,...,uk∈Rk+2, (1.2)

andωis a positive integer. For one reason, the well-known equations

yn+1=λyn,

yn+1=µyn1−yn,

yn+1=ynexp

µ1−yn

K

, K >0,

(1.3)

are particular cases of (1.1). As another reason, (1.1) is intimately related to delay dif-ferential equations with piecewise constant independent arguments. To be more precise, let us recall that a solution of (1.1) is a real sequence of the form{yn}n∈Zwhich renders (1.1) into an identity after substitution. It is not diﬃcult to see that solutions can be found when an appropriate function f is given. However, one interesting question is whether there are any solutions which are positive andω-periodic, where a sequence{yn}n∈Z is said to beω-periodic if yn+ω=yn, forn∈Z. Positiveω-periodic solutions of (1.1) are related to those of delay diﬀerential equations involving piecewise constant independent

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arguments:

y₍_t₎₌_y₍_t₎_f_[_t_],_y_[_t_]_,_y_[_t₋_1]_,_y_[_t₋_2]_,_..._,_y_[_t₋_k_]_, _t_∈_R_, _(1.4)

where [x] is the greatest-integer function.

Such equations have been studied by several authors including Cooke and Wiener [5,6], Shah and Wiener [9], Aftabizadeh et al. [1], Busenberg and Cooke [2], and so forth. Studies of such equations were motivated by the fact that they represent a hybrid of discrete and continuous dynamical systems and combine the properties of both diff er-ential and differer-ential-difference equations. In particular, the following equation

y₍_t₎₌_ay₍_t₎₁₋_y_[_t_]_, _(1.5)

is in Carvalho and Cooke [3], whereais constant.

By a solution of (1.4), we mean a functiony(t) which is defined onRand which satis-fies the following conditions [1]: (i)y(t) is continuous onR; (ii) the derivativey(t) ex-ists at each pointt∈Rwith the possible exception of the points [t]∈R, where one-sided derivatives exist; and (iii) (1.4) is satisfied on each interval [n,n+ 1)⊂Rwith integral endpoints.

Theorem1.1. Equation (1.1) has a positiveω-periodic solution if and only if (1.4) has a positiveω-periodic solution.

Proof. Let y(t) be a positiveω-periodic solution of (1.4). It is easy to see that for any n∈Z,

y₍_t₎₌_y₍_t₎_f_n_,_y₍_n_),_y₍_n₋_1),_..._,_y₍_n₋_k₎_, _n_≤_{t < n}_{+ 1}_. _(1.6)

Integrating (1.6) fromntot, we have

y(t)=y(n) exp(t−n)fn,y(n),y(n−1),...,y(n−k). (1.7)

Since limt→(n+1)−y(t)=y(n+ 1), we see further that

y(n+ 1)=y(n) expfn,y(n),y(n−1),...,y(n−k). (1.8)

If we now letyn=y(n) forn∈Z, then{yn}n∈Zis a positiveω-periodic solution of (1.1). Conversely, let{yn}n∈Z be a positiveω-periodic solution of (1.1). Set y(n)=yn, for

n∈Z, and let the functiony(t) on each interval [n,n+ 1) be defined by (1.7). Then it is not diﬃcult to check that this function is a positiveω-periodic solution of (1.4). The

proof ofTheorem 1.1is complete.

Therefore, once the existence of a positiveω-periodic solution of (1.1) can be demon-strated, we may then make immediate statements about the existence of positive ω -periodic solutions of (1.4).

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Here we will invoke a continuation theorem of Mawhin for obtaining such solutions. More specifically, letXandY be two Banach spaces andL: DomL⊂X→Y is a linear mapping andN:X→Y a continuous mapping [7, pages 39–40]. The mappingL will be called a Fredholm mapping of index zero if dim KerL=codim ImL <+∞, and ImLis closed inY. IfLis a Fredholm mapping of index zero, there exist continuous projectors P:X→XandQ:Y→Y such that ImP=KerLand ImL=KerQ=Im(I−Q). It follows thatL|DomL∩KerP: (I−P)X→ImLhas an inverse which will be denoted byKP. IfΩis an

open and bounded subset ofX, the mappingNwill be calledL-compact on ¯ΩifQN( ¯Ω) is bounded andKP(I−Q)N: ¯Ω→Xis compact. Since ImQis isomorphic to KerLthere

exist an isomorphismJ: ImQ→KerL.

Theorem1.2 (Mawhin’s continuation theorem). LetLbe a Fredholm mapping of index zero, and letNbeL-compact onΩ¯. Suppose

(i)for eachλ∈(0, 1),x∈∂Ω,Lx=λNx;

(ii)for eachx∈∂Ω∩KerL,QNx=0anddeg(JQN,Ω∩Ker, 0)=0. Then the equationLx=Nxhas at least one solution inΩ¯ ∩domL.

As a final remark in this section, note that ifω=1, then a positiveω-periodic solution of (1.1) is a constant sequence{c}n∈Zthat satisfies (1.1). Hence

f(n,c,...,c)=0, n∈Z. (1.9)

Conversely, if c >0 such that f(n,c,...,c)=0 for n∈Z, then the constant sequence {c}n∈Zis anω-periodic solution of (1.1). For this reason, we will assume in the rest of our discussion thatωis an integer greater than or equal to 2.

2. Existence criteria

We will establish existence criteria based on combinations of the following conditions, whereDandMare positive constants:

(a1) f(t,ex0,...,exk)>0 fort∈Randx0,...,xk≥D, (a2) f(t,ex0,...,exk)<0 fort∈Randx0,...,xk≥D,

(b1) f(t,ex0,...,exk)<0 fort∈Randx0,...,xk≤ −D, (b2) f(t,ex0,...,exk)>0 fort∈Randx0,...,xk≤ −D, (c1) f(t,ex0,...,exk)≥ −Mfor (t,ex0,...,exk)∈Rk+2, (c2) f(t,ex0,...,exk)≤Mfor (t,ex0,...,exk)∈Rk+2.

Theorem2.1. Suppose either one of the following sets of conditions holds: (i) (a1),(b1), and(c1), or,

(ii)(a2),(b2), and(c1), or, (iii) (a1),(b1), and(c2), or (iv) (a2),(b2), and(c2).

Then (1.1) has a positiveω-periodic solution.

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We first need some basic tools. First of all, for any real sequence{un}n∈Z, we define a

denoted byθ1andθ2respectively.

Define the mappingsL:Xω→YωandN:Xω→Yω, respectively, by

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Lemma2.2. The mappingLdefined by (2.3)Lis a Fredholm mapping of index zero. Next we recall that a subsetSof a Banach spaceXis relatively compact if, and only if, for eachε >0, it has a finiteε-net.

Lemma2.3. A subsetSofXωis relatively compact if and only ifSis bounded.

Proof. It is easy to see that ifSis relatively compact inXω, thenSis bounded. Conversely,

if the subsetSofXωis bounded, then there is a subset

Γ:=x∈Xω| x1≤H, (2.9)

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Noting thatΩis a closed and bounded subset ofXωand f is continuous onRk+2_, rela-tion (2.14) implies thatKP(I−Q)N(Ω) is bounded inXω. In view ofLemma 2.3,KP(I−

Q)N(Ω) is relatively compact inXω. Since the closure of a relatively compact set is

rela-tively compact,KP(I−Q)N(Ω) is relatively compact inXωand henceNisL-compact on

Ω. This completes the proof.

Now, we consider the following equation

xn−x0=λ n−1

i=0

fi,exi_,exi−1_,..._,exi−k_, n_∈Z_, _(2.15)

whereλ∈(0, 1).

Lemma 2.5. Suppose (a1),(b1), and(c1)are satisfied. Then for anyω-periodic solution x= {xn}n∈Zof (2.15),

x1= max 0≤i≤ω−1

_x_i_≤_D_{+ 4}_ωM. _(2.16)

Proof. Letx= {xn}n∈Zbe aω-periodic solutionx= {xn}n∈Zof (2.15). Then

ω−1

i=0

fi,exi_,exi−1_,..._,exi−k₌₀. _(2.17)

If we write

G+

n=maxfn,exn,exn−1,...,exn−k, 0, n∈Z, (2.18)

G−

n=max−fn,exn,exn−1,...,exn−k, 0, n∈Z, (2.19)

then{G+

n}n∈Zand{G−n}n∈Zare nonnegative real sequences and

fn,exn_,exn−1_,..._,exn−k₌G+

n−G−n, n∈Z, (2.20)

as well as

_f

n,exn_,exn−1_,..._,exn−k₌G+

n+G−n, n∈Z. (2.21)

In view of (c1) and (2.19), we have

_G−

n=G−n≤M, n∈Z. (2.22)

Thus

ω−1

i=0 G−

i ≤ωM, (2.23)

and in view of (2.17), (2.20), and (2.23),

ω−1

i=0 G+

i = ω−1

i=0 G−

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By (2.21) and (2.24), we know that

ω−1

i=0

_f

i,exi_,exi−1_,..._,exi−k_≤₂ωM. _(2.25)

Letxα=max0≤i≤ω−1xiandxβ=min0≤i≤ω−1xi, where 0≤α,β≤ω−1. By (2.15), we have

xα−xβ=xα−xβ=λ

α−1

i=0

fi,exi_,exi−1_,..._,exi−k₋

β−1

i=0

fi,exi_,exi−1_,..._,exi−k

≤2

ω−1

i=0

_f

i,exi_,exi−1_,..._,exi−k_≤₄ωM.

(2.26)

If there is somexl, 0≤l≤ω−1, such that|xl|< D, then in view of (2.15) and (2.25), for

anyn∈ {0, 1,...,ω−1}, we have

_x_n₌_x_l₊_x_n₋_x_l

≤D+

n−1

i=0

fi,exi_,_exi−1_,..._,exi−k₋

l−1

i=0

fi,exi_,_exi−1_,..._,exi−k

≤D+ 2

ω−1

i=0

_f

i,exi_,exi−1_,..._,exi−k

≤D+ 4ωM.

(2.27)

Otherwise, by (a1), (b1), and (2.17),xαDandxβ≤ −D. From (2.26), we have

xα≤xβ+ 4ωM≤ −D+ 4ωM,

xβ≥xα−4ωM≥D−4ωM. (2.28)

It follows that

D−4ωM≤xβ≤xn≤xα≤ −D+ 4ωM, 0≤n≤ω−1, (2.29)

or

_x_n_≤_D_{+ 4}_ωM_, ₀_≤_n_≤_ω₋₁_. _(2.30)

This completes the proof.

We now turn to the proof ofTheorem 2.1. LetL,N,PandQbe defined by (2.3), (2.4), (2.6), and (2.7), respectively. Set

Ω=x∈Xω| x1< D, (2.31)

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byLemma 2.5, for each λ∈(0, 1) and x∈∂Ω,Lx=λNx. Next, note that a sequence x= {xn}n∈Z∈∂Ω∩KerLmust be constant:{xn}n∈Z= {D}n∈Zor{xn}n∈Z= {−D}n∈Z. Hence by (a1), (b1), and (2.11),

(QNx)n=_ωn ω−1

i=0

fi,ex0_,..._,ex0_, n∈Z_, _(2.32)

so

QNx=θ2. (2.33)

The isomorphismJ: ImQ→KerLis defined by (J(nα))n=α, forα∈R,n∈Z. Then

(JQNx)n=_ω1 ω−1

i=0

fi,ex0_,..._,ex0=_0, n∈Z. _(2.34)

In particular, we see that if{xn}n∈Z= {D}n∈Z, then

(JQNx)n=_ω1 ω−1

i=0

fi,eD,...,eD>0, n∈Z, (2.35)

and if{xn}n∈Z= {−D}n∈Z, then

(JQNx)n=_ω1 ω−1

i=0

fi,e−D,...,e−D<0, n∈Z. (2.36)

Consider the mapping

H(x,s)=sx+ (1−s)JQNx, 0≤s≤1. (2.37)

From (2.35) and (2.37), for eachs∈[0, 1] and{xn}n∈Z= {D}n∈Z, we have

H(x,s)_n=sD+ (1−s)1 ω

ω−1

i=0

fi,eD,...,eD>0, n∈Z. (2.38)

Similarly, from (2.36) and (2.37), for eachs∈[0, 1] and{xn}n∈Z= {−D}n∈Z, we have

H(x,s)_n= −sD+ (1−s)1 ω

ω−1

i=0

fi,e−D,...,e−D<0, n∈Z. (2.39)

By (2.38) and (2.39),H(x,s) is a homotopy. This shows that

degJQNx,Ω∩KerL,θ1=deg−x,Ω∩KerL,θ1=0. (2.40)

ByTheorem 1.2, we see that equationLx=Nxhas at least one solution inΩ∩DomL. In other words, (2.2) has anω-periodic solutionx= {xn}n∈Z, and hence{exn}n∈Z is a positiveω-periodic solution of (1.1).

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3. Examples

and the semi-discrete “food-limited” population model of

y₍_t₎₌_y₍_t₎_r_[_t_] andCorollary 2.6, we know that (3.1) and (3.2) have positiveω-periodic solutions.

As another example, consider the semi-discrete Michaelis-Menton model

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Since

lim

x0,...,xk→+∞0min≤t≤ω

k

i=0

ai(t)exi 1 +ci(t)exi >1,

lim

x0,...,xk→−∞0max≤t≤ω

k

i=0

ai(t)exi 1 +ci(t)exi =0,

(3.8)

we can chooseM=max0≤t≤ωr(t) and some positive numberDsuch that conditions (a2), (b2), and (c1) are satisfied. ByTheorem 2.1andCorollary 2.6, (3.4), and (3.5) have posi-tiveω-periodic solution.

References

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[2] S. Busenberg and K. Cooke,Vertically Transmitted Diseases, Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993.

[3] L. A. V. Carvalho and K. L. Cooke,A nonlinear equation with piecewise continuous argument, Diﬀerential Integral Equations1(1988), no. 3, 359–367.

[4] S. Cheng and G. Zhang,Positive periodic solutions of a discrete population model, Funct. Diﬀer. Equ.7(2000), no. 3-4, 223–230.

[5] K. L. Cooke and J. Wiener,Retarded diﬀerential equations with piecewise constant delays, J. Math. Anal. Appl.99(1984), no. 1, 265–297.

[6] ,A survey of diﬀerential equations with piecewise continuous arguments, Delay Diﬀ eren-tial Equations and Dynamical Systems (Claremont, Calif, 1990), Lecture Notes in Mathe-matics, vol. 1475, Springer, Berlin, 1991, pp. 1–15.

[7] R. E. Gaines and J. L. Mawhin,Coincidence Degree, and Nonlinear Diﬀerential Equations, Lec-ture Notes in Mathematics, vol. 568, Springer-Verlag, Berlin, 1977.

[8] M. I. Gil’ and S. S. Cheng,Periodic solutions of a perturbed diﬀerence equation, Appl. Anal.76 (2000), no. 3-4, 241–248.

[9] S. M. Shah and J. Wiener,Advanced diﬀerential equations with piecewise constant argument de-viations, Int. J. Math. Math. Sci.6(1983), no. 4, 671–703.

[10] G. Zhang and S. S. Cheng,Positive periodic solutions for discrete population models, Nonlinear Funct. Anal. Appl.8(2003), no. 3, 335–344.

Gen-Qiang Wang: Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou, Guangdong 510665, China

E-mail address:[email protected]