Open access peer-reviewed chapter

Positive Periodic Solutions for First-Order Difference Equations with Impulses

By Mesliza Mohamed, Gafurjan Ibragimov and Seripah Awang Kechil

Submitted: September 9th 2020Reviewed: December 11th 2020Published: January 8th 2021

DOI: 10.5772/intechopen.95462

Downloaded: 147

Abstract

This paper investigates the first-order impulsive difference equations with periodic boundary conditions

Keywords

  • Impulse
  • difference equation
  • boundary value problem
  • Green’s function
  • fixed point theorem

1. Introduction

Let Rdenote the real numbers and R+the positive real numbers. Let J=0T=01T. In this paper we investigate the existence of positive periodic solutions for nonlinear impulsive difference equations

Δxn=fnxn,nJ,nnk,Δxnk=Ikxnk,k=1,2,,m,x0=xT,E1

where Δdenotes the forward difference operators, i.e., Δxn=xn+1xn, fCJ×R+R+, IkCR+R+.

Boundary value problems for difference equations and impulsive differential equation have been widely received attentions from many authors (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]) and reference therein. However, as far as we know, the theory of difference equation for boundary value problems (BVPs) with impulses is rather less, there are still lots of work and research that should be done. In [3], He and Zhang obtained the criteria on the existence of minimal and maximal solutions of (1) by using the method of upper and lower solutions and monotone iterative technique. The similar techniques were applied in [11] for the problem

Δxn=fnxn,nJ,nnk,Δxnk=Ikxnk,k=1,2,,p,Mx0NxT=C,E2

where fCJ×R+R+, IkCR+R+.Motivated by the work of [3, 13] we establish the existence of positive periodic solutions for (1) by using the fixed point-theorem in cones following the ideas of [13]. The results herein improve some of the results in [3, 11].

Throughout the paper, we make the following assumptions:

  1. 0<M<1, 0<Lk<M,fnxn+Mxn0.

  2. MLkxnk+Ikxnk+Lkxnk0.

This paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, by applying the fixed point theorem in cones, we obtain some new existence theorems for the impulsive difference equations with periodic boundary conditions.

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2. Preliminaries

Let

E=x:JR:x0=xT

with the norm x=maxnJxn.Then Eis a Banach space.

Consider the following linear impulsive difference equations with periodic boundary condition

Δxn+Mxn=σn,nJ,nnk,Δxnk=Lkxnk+Ikxnk+Lkxnk,k=1,2,,m,x(0=xT,E3

where Mis a constant, IkCR+R+and σCJR+.

The following lemma transforms the analysis of PBVP (3) to the analysis of summation equation. We denote Gnj, the Green’s function of the problem (3).

Lemma 1.LetA1andA2hold,σCJR+. ThenxEis a solution of PBVP(3)if and only ifxis a solution of the following impulsive summation equation

xn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk++Lkxnk,E4

where

Gnj=111MT1Mn1Mj+1,0jn1,1MT+n1Mj+1,njT1.E5

ProofFor convenience, we give the proof for the corresponding linear case (3). Consider first the homogeneous equation

Δun+Mun=0,nJ,nnk,

which is easily solved by iteration. We have

un=1Mn,

with u0=1. Now the first Eq. (3) can be solved by substituting xn=unyninto Eq. (3), where yis to be determined:

un+1yn+1unyn1M=σn

or

Δyn=σnEun,nnk.

So from (3), we see that ynsatisfies

Δyn=σn1Mn+1,nnk,Δynk=MLk1Mynk+Ikxnk+Lkxnk1Mnk+1,k=1,2,,m,y0=yT1MT.E6

From (6), we have

yn=y0+j=0,jnkn1σj1Mj+1+0<nkn1MLk1Mynk+Ikxnk+Lkxnk1Mnk+1.E7

Letting n=Tin (7), we have

yT=y0+j=0,jnkT1σj1Mj+1+0<nkT1MLk1Mynk+Ikxnk+Lkxnk1Mnk+1.E8

Applying (8) and the boundary condition y0=yT1MT, we get

y0=1MT11MTj=0,jnkT1σj1Mj+1+0<nkT1(MLk1Mynk+Ikxnk+Lkxnk1Mnk+1.E9

Substituting (9) into (7) and using yn=xn1Mn,nJ, we get the unique solution of (3)

xn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk),

where Gnjis given in (5). The proof is complete. .

Consider the PBVP (1). To define a cone, we observe that

1Mn11MTGnnj111MT.

Define

A1Mn11MT,
B111MT.

We denote the cone in Eby

K=xEn0Tandxnσx,

where σ=AB01. Define a mapping T:EEby

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk).E10

Since fis continuous, Tis also a continuous map. Before starting the main results, we shall give some important lemmas. The next Lemma is essential in obtaining our results.

Lemma 2.The mappingTmapsKintoK, i.eTKK.

ProofFor any xK, it is easy to see that TxE. From (10) we have

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)j=0,jnkT1j+0<nkT1BMLkxnk+Ikxnk+Lkxnk).

Noting that

Gnjσj0,GnnkMLkxnk+Ikxnk+Lkxnk0.

We can also obtain

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)j=0,jnkT1j+0<nkT1AMLkxnk+Ikxnk+Lkxnk)ABTxnσTx.

Hence TKK. The proof is complete. .

The following lemma is crucial to prove our main results.

Lemma 3.(GuoKrasnoselskii's fixed point theorem [14]) Let Ebe a Banach space and let KEbe a cone E. Assume Ω1,Ω2are open subsets of Ewith 0Ω1, Ω¯1Ω2, and let

T:KΩ¯2\Ω1K

be a completely continuous operator such that either.

  1. Txxfor xKΩ1and Txxfor xKΩ2; or

  2. Txxfor xKΩ1and Txxfor xKΩ2.

Then Thas a fixed point in KΩ¯2\Ω1.

We are now in a position to apply the preceding results to obtain the existence of positive periodic solutions to (1).

3. Main results

In this section, we state and prove our main findings.

Theorem 1.Suppose that the following conditions hold

limx0+fnxx=M,limx0+k=1mIkxx=0,E11
limx+fnxx=,limx+k=1mIkxx=.E12

Then the problem (1) has at least one positive periodic solution.

Proof0<r<R<, setting

Ω1=xE:x<r,Ω2=xE:x<R.

We have 0Ω1, Ω1Ω2. It follows from (11) that there exists r>0so that for any 0<xr,

fnxnc1xMx,k=1mIkc2x,

where c1,c2are positive constants satisfying

σBTc1+TmM+c2<1.

Therefore for xK,with x=r,

fnxn+Mxc1x,k=1mIkc2x.

Moreover 0<σxxnx=r.Thus

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)n=0,nnkT1Bfnxn+Mxn+0<nkT1BMLkxnk+Ikxnk+Lkxnk)TBc1σx+TmMLk+c2+LkxσBTc1+TmM+c2xx,

which implies Txxfor xKΩ1.

On the other hand (12) yields the existence of R̂>0such that for any xR̂

fnxnη1x,k=1mIkη2x

where η1,η2>0are constants large enough such that

Tη1+M+TmM+η2>1.

Fixing RmaxrR̂σ. and letting xKwith x=R, we get xnσx=σR>R̂and fnxn+Mxη1x+Mxση1+Mx.

Thus

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)n=0,nnkT1Afnxn+Mxn
+0<nkT1AMLkxnk+Ikxnk+Lkxnk)TAση1+Mx+ATmσMLk+η2+LkxTη1+M+TmM+η2xx.

In particular Txx, xKΩ2.

Consequently by Lemma 3(i), T has a fixed point in

KΩ¯2\Ω1,

which is a positive periodic solution of (1). The proof is complete. .

Theorem 2.Suppose that the following conditions hold

limx0+fnxx=,limx0+k=1mIkxx=,E13
limx+fnxx=M,limx+k=1mIkxx=0.E14

Then the problem (1) has at least one positive periodic solution.

ProofWe follow the same strategy and notations as in the proof of Theorem 1. Firstly, we show that for r>0sufficiently large

Txx,xKΩ1.

From (13) it follows that there exists 0<x<r̂, where β1,β2are constants large enough such that σATβ1+M+TmM+β2>1.Therefore, for 0<x<r̂, if xKand x=r,then from (12),

fnxn+Mxβ1x+Mxσβ1+Mx,
k=1mIkβ2xσβ2x.

Furthermore, we obtain

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)n=0,nnkT1Afnxn+Mxn+0<nkT1AMLkxnk+Ikxnk+Lkxnk)TAσβ1+Mx+TmMLk+β2+LkxσATβ1+M+TmM+β2xx,

which implies Txx, for each xKΩ1.

Next we show that for R>0sufficiently large, Txx, xKΩ2. On the other hand (14) yields the existence of R̂>0such that for any xR̂

fnxnη1xMx,k=1mIkη2x,

where η1,η2>0are constants such that

Tη1+TmM+η2<1.

Fixing RmaxrR̂σ. and letting xKwith x=R, we get xnσx=σR>R̂and fnxn+Mxη1xη1σxand

Txn=j=0,jnkT1Gnjσj+0<nkT1GnnkMLkxnk+Ikxnk+Lkxnk)n=0,nnkT1Bfnxn+Mxn+0<nkT1BMLkxnk+Ikxnk+Lkxnk)TBση1x+BTmσMLkx+η2x+LkxTη1+TmM+η2xx.

which implies Txx, xKΩ2.

Finally, it follows from Lemma 3(ii) that T has a fixed point in

KΩ¯2\Ω1,

which is a positive periodic solution of (1). The proof is complete.

4. Conclusion

By applying the fixed point theorem in cones, we establish new existence theorems on positive periodic solutions for impulsive difference equations with periodic boundary conditions. Our main findings enrich and complement those available in the literature.

Additional classifications

AMS subject classification:39A10, 34B37

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mesliza Mohamed, Gafurjan Ibragimov and Seripah Awang Kechil (January 8th 2021). Positive Periodic Solutions for First-Order Difference Equations with Impulses, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.95462. Available from:

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