DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2018/2368694 2368694 Research Article Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type Migda Janusz 1 http://orcid.org/0000-0003-3188-1173 Migda Małgorzata 2 Nockowska-Rosiak Magdalena 3 Macias-Diaz Jorge E. 1 Faculty of Mathematics and Computer Science A. Mickiewicz University Umultowska 87 61-614 Poznań Poland amu.edu.pl 2 Institute of Mathematics Poznań University of Technology Piotrowo 3A 60-965 Poznań Poland put.poznan.pl 3 Institute of Mathematics Lodz University of Technology Ul. Wólczańska 215 90-924 Łódź Poland p.lodz.pl 2018 972018 2018 26 04 2018 21 06 2018 972018 2018 Copyright © 2018 Janusz Migda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the discrete Volterra type equation of the form Δ(rnΔxn)=bn+k=1nK(n,k)f(xk). We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o(ns), for given nonpositive real s, as a measure of approximation.

Ministerstwo Nauki i Szkolnictwa Wyzszego 04/43/DSPB/0095
1. Introduction

In this paper we consider the nonlinear Volterra sum-difference equation of nonconvolution type:EΔrnΔxn=bn+k=1nKn,kfxk,(1)rn,bnR,rn>0,f:RR,K:N×NR.Here N, R denote the set of positive integers and the set of real numbers, respectively. By a solution of E we mean a sequence x:NR satisfying E for large n.

Discrete Volterra equations of different types are widely used in the process of modeling of some real phenomena or by applying a numerical method to a Volterra integral equation. Let mN. The general form of a Volterra sum-difference autonomous equation is(2)Δmxn=an+k=1nKn,kfxk.Such equations can be regarded as the discrete analogue of Volterra integrodifferential equations of the form(3)xmt=ft+0tKt,sfxsds.There are relatively few works devoted to the study of equations of type (2); see, for example, . In , the asymptotic behaviors of nonoscillatory solutions of the higher-order integrodynamic equation on time scales are presented.

In most papers, the following special case of (2) is considered:(4)xn+1=anxn+bn+k=1nKn,kxk;see, e.g., , , , , or . For some recent results devoted to nonlinear Volterra equations we refer to [5, 1722] and references therein.

Note, that equation E generalizes the second-order discrete Volterra difference equation of type (2):(5)Δ2xn=bn+k=1nKn,kfxk.On the other hand, if K(n,k)=0 for kn, then denoting an=K(n,n) equation E takes the form(6)ΔrnΔxn=anfxn+bn.Hence second-order difference equation (6) is a special case of E. The results on asymptotic properties and oscillation of equations of type (6) can be found, i.e., in .

Our main goal is to present sufficient conditions for the existence of a solution x to equation E such that(7)xn=k=1n-1crk+d+ons,where c,dR and s(-,0]. We give also sufficient conditions for a given solution x of equation E to have an asymptotic property (7). Moreover, in Section 5 we show applications of the obtained results to linear Volterra equation of type E. We present also some results for the case when (rn) is a potential sequence.

2. Preliminaries

We will denote by SQ the space of all sequences x:NR. If x,y in SQ, then xy and x denote the sequences defined by xy(n)=xnyn and x(n)=xn, respectively. Moreover,(8)x=supxn:nN.If xSQ, sR, and limnn-sxn=0, then we write xn=o(ns). Analogously, xn=O(ns) denotes the boundedness of the sequence (n-sxn).

The following two lemmas will be useful in the proof of our main results.

Lemma 1.

Assume uSQ, nN, and(9)j=11rji=jui<.Then(10)j=n1rji=juij=ni=1jujri<.

Proof.

We have(11)j=n1rji=jui=1rnun+un+1+un+2++1rn+1un+1+un+2++=1rnun+1rn+1rn+1un+1+1rn+1rn+1+1rn+2un+2+j=ni=1jujrij=1i=1jujri=1r1u1+1r1+1r2u2+=1r1u1+u2++1r2u2+u3++=j=11rji=jui<.

Lemma 2 ([<xref ref-type="bibr" rid="B18">27</xref>, Lemma 4.7]).

Assume y,ρSQ, and limnρn=0. In the set(12)X=xSQ:x-yρwe define a metric by the formula(13)dx,z=x-z.Then any continuous map H:XX has a fixed point.

3. Solutions with Prescribed Asymptotic Behavior

In this section we present sufficient conditions for the existence of a solution x to equation E such that(14)xn=k=1n-1crk+d+ons,where c,dR and s(-,0].

Theorem 3.

Assume s(-,0], t[s,), c,dR, y:NR, qN, α(0,),(15)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<,yn=d+ck=1n-11rk,U=n=qyn-α,yn+α,and f is continuous and bounded on U. Then there exists a solution x of E such that(16)xn=yn+ons.

Proof.

For nN and xSQ let(17)Fxn=bn+k=1nKn,kfxk.There exists L>0 such that(18)fuLfor  any  uU.Let(19)Y=xSQ:x-yα.If xY and nq, then(20)xnyn-α,yn+αU.Choose a positive number Q such that rn-1Qnt for any n. Then(21)n=11nsrnj=nk=1jKj,kQn=1nt-sj=nk=1jKj,kSince t-s0, we have(22)n=1nt-sj=nk=1jKj,kn=1j=njt-sk=1jKj,k.For jN let(23)zj=jt-sk=1jKj,k.Then we have(24)n=1j=nzj=z1+z2++z2+z3++=z1+2z2+3z3+=n=1nzn=n=1nnt-sk=1nKn,k=n=1n1+t-sk=1nKn,k<.Hence, using (21) and (22), we get(25)n=11nsrnj=nk=1jKj,k<.Analogously, replacing k=1jKj,k by bj, we obtain(26)n=11nsrnj=nbj<.Using (25) and (26) we get(27)j=11jsrji=jbi+Lk=1iKi,k<.Since s0, we have(28)j=11rji=jbi+Lk=1iKi,k<.Define a sequence ρSQ by(29)ρn=j=n1rji=jbi+Lk=1iKi,k.Define w,gSQ by(30)wn=bn+Lk=1nKn,k,gn=j=n1jsrji=jwi.By (27), gn=o(1). We have(31)n-sρn=n-sj=n1rji=jwi=j=n1nsrji=jwij=n1jsrji=jwi=gn.Hence n-sρn=o(1) and we get(32)ρn=nso1=ons.Hence there exists an index pq such that(33)ρnαfor np. Let(34)X=xSQ:x-yρ,xn=yn  for  n<p,H:YSQ,Hxn=ynfor  n<pyn+j=n1rji=jFxifor  np.We define a metric on X by formula (13). Note that XY. Let xX. By (33) and (20) we have xiU for any ip. Hence, by (18), fxiL for ip. Using (17) and (29) we obtain(35)Hxn-yn=j=n1rji=jFxij=n1rji=jFxiρnfor np. Therefore HXX. Now we show that the map H is continuous. Using (25) and the assumption s0, we have(36)n=11rnj=nk=1jKj,k<.Hence, by Lemma 1, we get(37)n=1j=1n1rjk=1nKn,k<.Let ε>0. Choose an index mp and a positive constant γ such that(38)Ln=mj=1n1rjk=1nKn,k<ε,γn=1mj=1n1rjk=1nKn,k<ε.Let(39)C=n=1myn-α,yn+α.Choose a positive δ such that if t1,t2C and t1-t2<δ, then(40)ft1-ft2<γ.Choose x,zX such that x-z<δ. Then we have(41)Hx-Hz=supnpj=n1rji=jFxi-Fzisupnpj=n1rji=jFxi-Fzi=j=p1rji=jFxi-Fzij=p1rji=jk=1iKi,kfxk-fzk.Using Lemma 1 we obtain(42)Hx-Hzj=pi=1j1rik=1jKj,kfxk-fzk.Note that fxj-fzj2L for jp and(43)fxj-fzjγfor  jp,p+1,,m.Hence we obtain(44)Hx-Hzγn=1mj=1n1rjk=1nKn,k+2Ln=mj=1n1rjk=1nKn,k<3ε.Therefore H:XX is continuous. By Lemma 2 there exists a point xX such that x=Hx. Then, for np, we have(45)xn=yn+j=n1rji=jFxi.Note that(46)ΔrnΔyn=ΔrnΔd+ck=1n-11rk=cΔrnΔk=1n-11rk=cΔ1=0for any n. Hence, for np, we get(47)ΔrnΔxn=ΔrnΔj=n1rji=jFxi=-Δrn1rni=nFxi=Fxn=bn+k=1nKn,kfxk.Therefore x is a solution of E. Since xX we have xn=yn+o(ns).

If the function f is continuous, then from Theorem 3 we get the following two results.

Corollary 4.

Assume s(-,0], t[s,), f is continuous, and(48)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any dR there exists a solution x of E such that xn=d+o(ns).

Proof.

Taking c=0, q=1, and α=1 in Theorem 3, we obtain the result.

Corollary 5.

Assume t(-,-1), s(-,t], f is continuous, and(49)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any c,dR there exists a solution x of E such that(50)xn=d+ck=1n-11rk+ons.

Proof.

Assume c,dR and a sequence ySQ is defined by(51)yn=d+ck=1n-11rk.Since rn-1=O(nt) and t<-1, we see that y is bounded. Now, taking q=1 and α=1 in Theorem 3, we obtain the result.

Note that Corollaries 4 and 5 concern convergent solutions. However, Theorem 3 includes also divergent solutions. For example, if f(x)=x-1 for x0, s(-,0], t=0, rn-1=O1, and(52)k=1rk-1=,n=1n1-si=1nKn,i<,n=1n1-sbn<.then, by Theorem 3, for any nonzero cR and any dR there exists a solution x of E such that(53)xn=d+ck=1n-11rk+ons.Now we present an example that proves the assumption(54)n=1n1+t-si=1nKn,i<,in Theorem 3, is essential.

Example 6.

Assume rn=n, bn=0,(55)Kn,k=1n2,fx=1x+1+1,s=0, and t=0. Then equation E takes the form(56)ΔnΔxn=1n2k=1n1xk+1+1.Let c,dR and(57)yn=d+ck=1n-11rk=d+ck=1n-11k.Notice that f is continuous and bounded on R. Moreover,(58)n=1n1+t-sk=1nKn,k=n=1n1n=,and(59)Δyn=cn,ΔnΔyn=0.Assume x is a solution of (56) such that(60)xn=yn+zn,zn=ons=o1.Since Δ(nΔyn)=0, we have(61)ΔnΔxn=ΔnΔyn+ΔnΔzn=ΔnΔzn.Hence(62)ΔnΔzn=ΔnΔxn=k=1n1n21xk+1+1>0for large n. Therefore, the sequence nΔzn is eventually increasing and there exists the limit(63)λ=limnnΔzn>-.If λ<, then the sequence nΔzn is convergent in R. Hence the series(64)n=1ΔnΔxn=n=1ΔnΔznis convergent. On the other hand(65)ΔnΔxn=k=1n1n21xk+1+1>1nfor large n. Hence λ=. Therefore nΔzn>1 for large n and we get(66)n=1Δznn=11n=.But since zn0, the series n=1Δzn is convergent.

4. Asymptotic Behavior of Solutions

In this section we present sufficient conditions for a given solution x of equation E to have an asymptotic property(67)xn=k=1n-1crk+d+ons,where c,dR and s(-,0].

Theorem 7.

Assume s(-,0], t[s,),(68)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<,and x is a solution of E such that the sequence (f(xn)) is bounded. Then there exist c,dR such that(69)xn=k=1n-1crk+d+ons.

Proof.

We have(70)ΔrnΔxn=bn+k=1nKn,kfxkfor large n. Using boundedness of the sequence (f(xn)) and (68) we get(71)n=1n1+t-sΔrnΔxn<.Define w,uSQ by(72)wn=ΔrnΔxn,un=nt-swn.Choose a positive L such that rn-1Lnt for any n. Since t-s0, we have(73)n=11nsrnj=nwjLn=1nt-sj=nwj=Ln=1j=nnt-swjLn=1j=njt-swj.Moreover,(74)n=1n1+t-swn=n=1nun=u1+u2+u2+u3+u3+u3+=j=1uj+j=2uj+j=3uj+=n=1j=nuj=n=1j=njt-swj.Hence, by (73)(75)n=11nsrnj=nwj<.Since s0, we have(76)n=11rnj=nwj<n=11nsrnj=nwj<.Let(77)zn=j=n1rji=jwi.Then(78)n-szn=n-sj=n1rji=jwin-sj=n1rji=jwi=j=n1nsrji=jwij=n1jsrji=jwi=o1.Thus zn=o(ns). Let(79)yn=xn-zn.Then(80)Δyn=Δxn-Δzn=Δxn+1rni=nwi.Hence(81)rnΔyn=rnΔxn+i=nwiand we get(82)ΔrnΔyn=ΔrnΔxn-wn=ΔrnΔxn-ΔrnΔxn=0for any nN. Therefore, there exists a real constant c such that rnΔyn=c. Thus(83)yn-y1=Δy1++Δyn-1=cr1++crn-1.Hence(84)xn=yn+zn=k=1n-1crk+d+onswhere d=y1.

Corollary 8.

Assume s(-,0], t[s,), rn-1=O(nt), f is locally bounded,(85)n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<,and x is a bounded solution of E. Then there exist c,dR such that(86)xn=k=1n-1crk+d+ons.

Proof.

Since x is bounded and f is locally bounded, the sequence (f(xn)) is bounded. Hence the assertion is a consequence of Theorem 7.

Corollary 9.

Assume t[0,), rn-1=O(nt), f is locally bounded, and(87)n=1n1+ti=1nKn,i<,n=1n1+tbn<.Then any bounded solution of E is convergent.

Proof.

Assume x is a bounded solution of E. Let s=0. By Corollary 8, there exist c,dR such that(88)xn=ck=1n-11rk+d+o1.Define a sequence uSQ, by un=r1-1+r2-1++rn-1-1. Then u is increasing and bounded. Hence u is convergent. Therefore xn=cun+d+o(1) is convergent.

Corollary 10.

Assume s(-,0], t[s,), rn-1=O(nt), f is bounded,(89)n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<,and x is an arbitrary solution of E. Then there exist c,dR such that(90)xn=k=1n-1crk+d+ons.

Proof.

The assertion is an immediate consequence of Theorem 7.

In this section we present some additional results. First, we give some applications of our results to linear discrete Volterra equations of type E. From Corollary 4 we get the following result.

Corollary 11.

Assume s(-,0], t[s,),(91)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any dR there exists a solution x of equation(92)ΔrnΔxn=bn+k=1nKn,kxksuch that xn=d+o(ns).

From Corollary 5 we get the following.

Corollary 12.

Assume t(-,-1), s(-,t], and(93)rn-1=Ont,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any c,dR there exists a solution x of (92) such that(94)xn=d+ck=1n-11rk+ons.

Example 13.

Assume s=0, t=1, and(95)rn=1n-1,bn=-3n+2n+1nn-1-1n4,Kn,k=2kn6.Then (92) takes the form(96)Δ1n-1Δxn=-3n+2n+1nn-1-1n4+k=1n2kn6xk.It is easy to check that all assumptions of Corollary 11 hold. Indeed, we have(97)n=1n2k=1nKn,k=n=1n2k=1n2kn6=n=1n+1n3<,and(98)n=1n2bn<.So, for every dR, there exists a solution x of (96) such that limnxn=d. One such solution is xn=1-1/n.

In our investigations the condition(99)n=1n1+t-si=1nKn,i<plays an important role. In practice, this condition can be difficult to verify. In the following remark we present the condition, which is a little stronger but easier to check.

Remark 14.

Assume s(-,0], tR, λ(-,s-t-2), and un=O(nλ). Let ε=s-t-2-λ, L>0, unLnλ for any n. Then λ=s-t-2-ε and(100)n=1n1+t-sunLn=1n1+t-snλ=n=11n1+ε<.

Applying this remark to Corollaries 4, 5, 8, and 9, respectively, we obtain following results.

Corollary 15.

Assume s(-,0], t[s,), λ(-,s-t-2), f is continuous, and(101)rn-1=Ont,i=1nKn,i=Onλ,bn=Onλ.Then for any dR there exists a solution x of E such that xn=d+o(ns).

Corollary 16.

Assume t(-,-1), s(-,t], λ(-,s-t-2), f is continuous, and(102)rn-1=Ont,i=1nKn,i=Onλ,bn=Onλ.Then for any c,dR there exists a solution x of E such that(103)xn=d+ck=1n-11rk+ons.

Corollary 17.

Assume s(-,0], t[s,), λ(-,s-t-2), f is locally bounded,(104)rn-1=Ont,i=1nKn,i=Onλ,bn=Onλ,and x is a bounded solution of E. Then there exist c,dR such that(105)xn=k=1n-1crk+d+ons.

Corollary 18.

Assume t[0,), λ(-,-t-2), f is locally bounded, and(106)rn-1=Ont,i=1nKn,i=Onλ,bn=Onλ.Then any bounded solution of E is convergent.

Now we present some results for the case when the series(107)n=1n1+t-si=1nKn,i,n=1n1+t-sbnare strongly convergent.

Remark 19.

If u:NR and limsupnunn<1, then, by the root test,(108)n=1nλun<for any λR.

Corollary 20.

Assume tR, f is continuous, and(109)rn-1=Ont,limsupni=1nKn,in<1,limsupnbnn<1.Then for any dR and any λ(-,0] there exists a solution x of E such that(110)xn=d+onλ.

Proof.

Let dR. Choose s(-,mint,λ). By Remark 19, we have(111)n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.By Corollary 4, there exists a solution x of E such that(112)xn=d+ons=d+onλ.

Analogously, using Corollary 5, we get the following.

Corollary 21.

Assume t(-,-1), f is continuous, and(113)rn-1=Ont,limsupni=1nKn,in<1,limsupnbnn<1.Then for any c,dR and any λ(-,0] there exists a solution x of E such that(114)xn=d+k=1n-1crk+onλ.

To the end we consider the case when (rn) is a potential sequence.

Lemma 22.

If ω(1,), then(115)k=1n-11kω=n=11nω+On1-ω.

Proof.

Define uSQ and λR by(116)un=k=1n-11kω,λ=k=11kω.By [28, Theorem 2.2], we have(117)Δn1-ω=1-ωn-ω+on-ω.Since Δun=n-ω, we get(118)Δun-λΔn1-ω=n-ω1-ωn-ω+on-ω=11-ω+o111-ω.Note that n1-ω0 and (un-λ)0. Hence, by discrete L’Hospital’s Rule,(119)un-λn1-ω=11-ω+o1.Therefore(120)un=λ+11-ωn1-ω+on1-ω=k=11kω+On1-ω.

Corollary 23.

Assume t(-,-1), s(-,t], f is continuous, and(121)rn=n-t,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any μR there exists a solution x of E such that(122)xn=μ+Ont+1.

Proof.

Assume μR. Let(123)d=0,λ=n=1nt,c=μλ.By Corollary 5, there exists a solution x of E such that(124)xn=d+ck=1n-1kt+ons.By Lemma 22(125)k=1n-1kt=λ+Ont+1.Hence(126)xn=μ+Ont+1+ons=μ+Ont+1.

Corollary 24.

Assume t(-,-1), s(-,t], f is locally bounded, and(127)rn=n-t,n=1n1+t-si=1nKn,i<,n=1n1+t-sbn<.Then for any bounded solution x of E there exists a real number μ such that(128)xn=μ+Ont+1.

Proof.

By Corollary 8 there exist c,dR such that(129)xn=d+ck=1n-1kt+ons.By Lemma 22 we obtain(130)xn=d+ck=1kt+Ont+1+ons=μ+Ont+1,where  μ=d+ck=1kt.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The second author was supported by the Ministry of Science and Higher Education of Poland (04/43/DSPB/0095).

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