Asymptotic convergence of the solutions of a dynamic equation on discrete time scales

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 580750, 20 pages doi:10.1155/2012/580750 Research Article Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales J. Dibl´ ık,1, 2 M. R ˚ uˇ ziˇ ckov´ a,3Z. ˇ Smarda,1and Z. ˇ Sut´ a3 1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic 2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic 3Department of Mathematics, University of ˇ Zilina, 01026 ˇ Zilina, Slovakia Correspondence should be addressed to J. Dibl´ ık, [email protected].cz Received 10 September 2011; Accepted 12 November 2011 Academic Editor: Toka Diagana Copyright q2012 J. Dibl´ ık 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. The paper investigates a dynamic equation Δytnβtnytn−j−ytn−k for n→∞,wherek and jare integers such that k>j≥0, on an arbitrary discrete time scale T:{tn}with tn∈R, n∈Z∞ n0−k{n0−k,n0−k1,...},n0∈N,tn<t n1,Δytnytn1−ytn, and limn→∞tn∞. We assume β:T→0,∞. It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for n→∞. The results are presented as inequalities for the function β. Examples demonstrate that the criteria obtained are sharp in a sense. 1. Introduction We use the following notation: for an integer s, we define that Z∞ s:{s, s 1,...},andifan integer q≥s, we define Zq s:{s, s 1,...,q}. Hilger initiated in 1,2the calculus of time scales in order to create a theory that unifies discrete and continuous analyses. He defined a time scale Tas an arbitrary nonempty closed subset of real numbers. The theoretical background for time scales can be found in 3. In this paper, we use discrete time scales. To be exact, we define a discrete time scale TTtas an arbitrary unbounded increasing sequence of real numbers, that is, Tt:{tn}, where tn∈R,n∈Z∞ n0−k,n0∈N,k>0 is an integer, tn<t n1, and limn→∞tn∞. For a fixed v∈Z∞ n0−k, we define a time scale TvTvt:{tn}, where n∈Z∞ v. Obviously, Tn0−ktTt. In addition, for integers s,q,q≥s≥n0−k, we define the set Tq sTq st:{ts,t s1,...,t q}. 2 Abstract and Applied Analysis In the paper we study a dynamic equation Δytnβtnytn−j−ytn−k1.1 as n→∞.Thedifference is defined as usual: Δytn:ytn1−ytn, integers kand jin 1.1satisfy the inequality k>j≥0, and β:T→R:0,∞. Without loss of generality, we assume that tn0−k>0this is a technical detail, necessary for some expressions to be well defined. Throughout the paper, we adopt the notation k ik1Bti0 where kis an integer and Bdenotes the function under consideration. The results concern the asymptotic convergence of all solutions of 1.1. First we prove that, in the general case, the asymptotic convergence of all solutions is determined only by the existence of an increasing and bounded solution. Therefore, our effort is focused on developing criteria guaranteeing the existence of such solutions. The proofs of the results are based on comparing the solutions of 1.1with those of an auxiliary inequality with the same left-hand and right-hand sides as in 1.1. We also illustrate general results using examples with particular time scales. The problem concerning the asymptotic convergence of solutions in the continuous case, that is, in the case of delayed differential equations or other classes of equations, is a classical one and has attracted much attention recently we refer, e.g., to the papers 4–11. The problem of the asymptotic convergence of solutions of discrete and difference equations with delay has not yet received much attention. Some recent results can be found, for example, in 12–19. Comparing the known investigations with the results presented, we can see that our results give sharp sufficient conditions of the asymptotic convergence of solutions. This is illustrated by examples. Nevertheless, we are not concerned with computing the limits of the solutions as n→∞. The paper is organized as follows. In Section 2, auxiliary definitions and results are collected. An auxiliary inequality is studied, and the relationship of its solutions with the solutions of 1.1is derived. Section 3 contains results concerning the convergence of all solutions of 1.1. The criteria of existence of an increasing and convergent solution of 1.1are established in Section 4. Examples illustrating the sharpness of the results derived are discussed as well. 2. Auxiliary Definitions and Results Let C:CTn0 n0−k,Rbe the space of discrete functions mapping the discrete interval Tn0 n0−kinto R.Letv∈Z∞ n0be given. The function y:Tv−k→Ris said to be a solution of 1.1on Tv−kif it satisfies 1.1for every n∈Z∞ v.Asolutionyof 1.1on Tv−kis asymptotically convergent if the limit limn→∞ytnexists and is finite. For a given v∈Z∞ n0and ϕ∈C, we say that yytv,ϕis asolution of 1.1defined by the initial conditions tv,ϕif ytv,ϕis a solution of 1.1on Tv−k and ytv,ϕtvmϕtmfor m∈Z0 −k. 2.1. Auxiliary Inequality The inequality Δωtn≥βtnωtn−j−ωtn−k2.1 Abstract and Applied Analysis 3 is a helpful tool in the analysis of solutions of 1.1.Letv∈Z∞ n0. The function ω:Tv−k→R is said to be a solution of 2.1on Tv−kif ωsatisfies 2.1for n∈Z∞ v.Asolutionωof 2.1on Tv−kis asymptotically convergent if the limit limn→∞ωtnexists and is finite. We give some properties of solutions of inequalities of type 2.1to be used later on. We will also compare the solutions of 1.1with those of 2.1. Lemma 2.1. Let ϕ∈Cbe increasing (nondecreasing, decreasing, nonincreasing) on Tn0 n0−k. Then the solution yn0,ϕtnof 1.1,wheren∈Z∞ n0is increasing (nondecreasing, decreasing, nonincreasing) on Tn0, too. Lemma 2.2. Let ϕ∈Cbe increasing (nondecreasing) and ω:T→Rbe a solution of inequality 2.1with ωtmϕtm,m∈Zn0 n0−k. Then, ωtn,wheren∈Z∞ n0is increasing (nondecreasing). The proofs of both lemmas above follow directly from the form of 1.1,2.1,andfrom the properties βtn>0, n∈Z∞ n0−k,k>j≥0. Theorem 2.3. Let ω:T→Rbe a solution of 2.1on T. Then there exists a solution y:T→Rof 1.1on Tsuch that ytn≤ωtn2.2 holds for every n∈Z∞ n0−k. In particular, a solution yn0,φof 1.1with φ∈C, defined by φtn:ωtn,n∈Zn0 n0−k,2.3 is such a solution. Proof. Let ωtnbe a solution of 2.1defined on T. We will show that the solution ytn: yn0,φtnof 1.1with φdefined by 2.3satisfies 2.2,thatis, yn0,φtn≤ωtn2.4 for every n∈Z∞ n0−k.LetW:T→Rbe defined by Wtn:ωtn−ytn.2.5 Then Wtn0ifn∈Zn0 n0−kand, in addition, Wis a solution of 2.1on T.Lemma 2.2 implies that Wis nondecreasing. Consequently, Wtnωtn−ytn≥Wtn0ωtn0−ytn00,2.6 and ytn≤ωtnfor all n≥n0. 4 Abstract and Applied Analysis 2.2. A Solution of Inequality 2.1 Now we will construct a solution of 2.1. The result obtained will help us obtain sufficient conditions for the existence of an increasing and asymptotically convergent solution of 1.1 see Theorem 4.1 below. Lemma 2.4. Let there exists a function ε:T→Rsuch that εtn1≥ n−j  in−k1 βti−1εti2.7 for every n∈Z∞ n0. Then there exists a solution ωωεof 2.1defined on Tand having the form ωεtn: n  in0−k1 βti−1εti.2.8 Proof. Assuming that ωεdefined by 2.8is a solution of 2.1for n∈Z∞ n0, we will deduce the inequality for ε.Weget Δωεtnωεtn1−ωεtnn1  in0−k1 βti−1εti−n  in0−k1 βti−1εtiβtnεtn1, ωεtn−j−ωεtn−k n−j  in0−k1 βti−1εti− n−k  in0−k1 βti−1εti n−j  in−k1 βti−1εti. 2.9 We substitute ωεfor ωin 2.1. Then, using 2.9,2.1turns into βtnεtn1≥βtn n−j  n−k1 βti−1εti.2.10 Reducing the last inequality by βtn, we obtain the desired inequality. 2.3. Decomposition of a Function into the Difference of Two Increasing Functions It is well-known that every absolutely continuous function is representable as the difference of two increasing absolutely continuous functions 20, page 318. We will need a simple analogue of this result on discrete time scales under consideration. Lemma 2.5. Every function ϕ∈Ccan be decomposed into the difference of two increasing functions ϕj∈C,j1,2, that is, ϕtnϕ1tn−ϕ2tn,n∈Zn0 n0−k.2.11 Abstract and Applied Analysis 5 Proof. Let constants Mn>0, n∈Zn0 n0−kbe such that Mn1>M nmax0,ϕ tn−ϕtn12.12 is valid for each n∈Zn0−1 n0−k.Weset ϕ1tn:ϕtnMn,n∈Zn0 n0−k, ϕ2tn:Mn,n∈Zn0 n0−k. 2.13 It is obvious that 2.11holds. Now we verify that both functions ϕj,j1,2 are increasing. The first one should satisfy ϕ1tn1>ϕ 1tnfor n∈Zn0−1 n0−k, which means that ϕtn1Mn1>ϕ tnMn2.14 or Mn1>M nϕtn−ϕtn1.2.15 We conclude that the last inequality holds because, due to 2.12, we have Mn1>M nmax0,ϕ tn−ϕtn1≥Mnϕtn−ϕtn1.2.16 The inequality ϕ2tn1>ϕ 2tnobviously holds for every n∈Zn0−1 n0−kdue to 2.12as well. 2.4. Auxiliary Asymptotic Decomposition The following lemma can be proved easily by induction. The symbol Ocapital “O”stands for the Landau order symbol. Lemma 2.6. For fixed r,σ∈R\{0}, the asymptotic representation n−rσnσ1−σr nO1 n2 2.17 holds for n→∞. 3. Convergence of All Solutions The main result of this part is the statement that the existence of an increasing and asymptotically convergent solution of 1.1implies the asymptotical convergence of all solutions. Theorem 3.1. If 1.1has an increasing and asymptotically convergent solution on Z∞ n0−k,thenall the solutions of 1.1defined on Z∞ n0−kare asymptotically convergent. 6 Abstract and Applied Analysis Proof. First we prove that every solution defined by a monotone initial function is convergent. We will assume that a monotone initial function ϕ∈Cis given. For definiteness, let ϕ be increasing or nondecreasing the case when it is decreasing or nonincreasing can be considered in much the same way.ByLemma 2.1,thesolutionyn0,ϕis monotone, that is, it is either increasing or nondecreasing. We prove that yn0,ϕis convergent. Denote the assumed increasing and asymptotically convergent solution of 1.1as y Ytn,n∈Z∞ n0−k. Without loss of generality, we assume that yn0,ϕ/ ≡Yon Z∞ n0−ksince, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume ΔYtn>0,n∈Zn0−1 n0−k.3.1 Hence, there is a constant γ>0 such that ΔYtn−γΔytn>0,n∈Zn0−1 n0−k3.2 or ΔYtn−γytn>0,n∈Zn0−1 n0−k.3.3 This implies that the function Ytn−γytnis increasing on Zn0−1 n0−k,andLemma 2.1 implies that Ytn−γytnis increasing on Z∞ n0−k.Thus, Ytn−γytn>Y tn0−γytn0,n∈Z∞ n03.4 or ytn<y tn01 γYtn−Ytn0,n∈Z∞ n03.5 and, consequently, ytnis a bounded function on Z∞ n0−kbecause of the boundedness of Ytn. Obviously, in such a case, ytnis asymptotically convergent and has a finite limit. Summarizing the previous section, we state that every monotone solution is convergent. It remains to consider a class of all nonmonotone initial functions. For the behavior of a solution yn0,ϕgenerated by a nonmonotone initial function ϕ∈C, there are two possibilities: yn0,ϕis either eventually monotone and, consequently, convergent, or yn0,ϕis eventually nonmonotone. Now we use the statement of Lemma 2.5 that every discrete function ϕ∈Ccan be decomposed into the difference of two increasing discrete functions ϕj∈C,j1,2. In accordance with the previous part of the proof, every function ϕj∈C,j1,2 defines an increasing and asymptotically convergent solution yn0,ϕj. Now it is clear that the solution yn0,ϕis asymptotically convergent. From Theorem 3.1, it follows that a crucial property assuring the asymptotical convergence of all solutions of 1.1is the existence of a strictly monotone and asymptotically convergent solution. In the next part, we will focus our attention on the relevant criteria. Now, in order to finish this section, we need an obvious statement concerning the asymptotic convergence. From Lemma 2.1 and Theorem 2.3, we immediately derive the following result. Abstract and Applied Analysis 7 Theorem 3.2. Let ωbe an increasing and bounded solution of 2.1on T. Then there exists an increasing and asymptotically convergent solution yof 1.1on T. Combining the statements of Theorems 2.3,3.1,and3.2, we get a series of equivalent statements. Theorem 3.3. The following three statements are equivalent. aEquation 1.1has a strictly monotone and asymptotically convergent solution on Z∞ n0−k. bAll solutions of 1.1defined on Z∞ n0−kare asymptotically convergent. cInequality 2.1has a strictly monotone and asymptotically convergent solution on Z∞ n0−k. 4. Increasing Convergent Solutions of 1.1 This part deals with the problem of detecting the existence of asymptotically convergent increasing solutions. We provide sufficient conditions for the existence of such solutions of 1.1. The important theorem below is a consequence of Lemma 2.1,Theorem 2.3,and Lemma 2.4. Theorem 4.1. Let there exists a function ε:T→Rsatisfying ∞  in0−k1 βti−1εti<∞, εtn1≥ n−j  in−k1 βti−1εti 4.1 for every n∈Z∞ n0. Then the initial function ϕtn: n  in0−k1 βti−1εti,n∈Zn0 n0−k4.2 defines an increasing and asymptotically convergent solution ytn0,ϕtnof 1.1on Tsatisfying ytn0,ϕtn≤ n  in0−k1 βti−1εti4.3 for every n∈Z∞ n0. Although Theorem 4.1 itself can serve as a source of various concrete criteria, later we will apply its following modification which can be used easily. Namely, assuming that βin 1.1can be estimated by a suitable function, we can deduce that 1.1has an increasing asymptotically convergent solution. We consider such a case. 8 Abstract and Applied Analysis Theorem 4.2. Let there exist functions β∗:T→Rand ε:T→Rsuch that the inequalities βtn≤β∗tn,4.4 εtn1≥ n−j  in−k1 β∗ti−1εti4.5 hold for all n∈Z∞ n0−k, and moreover ∞  in0−k1 β∗ti−1εti<∞.4.6 Then there exists an increasing and asymptotically convergent solution y:T→Rof 1.1satisfying ytn≤ n  in0−k1 βti−1εti4.7 for every n∈Z∞ n0. Such a solution is defined, for example, by the initial function ϕtn: n  in0−k1 βti−1εti,n∈Zn0 n0−k.4.8 Proof. From 4.5and 4.6,weget εtn1≥ n−j  in−k1 β∗ti−1εti≥ n−j  in−k1 βti−1εti, ∞> ∞  in0−k1 β∗ti−1εti≥ ∞  in0−k1 βti−1εti. 4.9 Then all assumptions of Theorem 4.1 are true. From its conclusion now follows the statement of Theorem 4.2. 4.1. Some Special Criteria It will be demonstrated by examples that, in many applications, the function β∗mentioned in Theorem 4.2 can have the form β∗tnc−γtn,4.10 where cis a positive constant and γ:T→Ris a suitable function such that γtn<cat least for all sufficiently large nand lim n→∞ γtn0.4.11 Abstract and Applied Analysis 9 Below we carry on in this way and give sufficient conditions for the existence of increasing and asymptotically convergent solutions of 1.1for general discrete time scale under consideration. For several special time scales, we derive such criteria in subsequent sections. Theorem 4.3. Let there exist constants c>0,p>0and α>0such that βtn≤c−p tn ,4.12 1 tα n1 ≥ n−j  in−k1c−p ti−11 tα i 4.13 hold for all n∈Z∞ n0−k, and moreover ∞  in0−k1 1 tα i <∞.4.14 Then there exists an increasing and asymptotically convergent solution y:T→Rof 1.1 satisfying ytn≤ n  in0−k1c−p ti−11 tα i 4.15 for every n∈Z∞ n0. Such a solution is defined, for example, by the initial function ϕtn: n  in0−k1c−p ti−11 tα i ,n∈Zn0 n0−k.4.16 Proof. We will apply Theorem 4.2 with β∗tn:c−p tn ,ε tn:1 tα n .4.17 Inequality 4.5turns into εtn11 tα n1 ≥ n−j  in−k1 β∗ti−1εti n−j  in−k1c−p ti−11 tα i 4.18 and is true due to 4.13. Inequality 4.6holds due to assumption 4.14as well because limn→∞tn∞and ∞  in0−k1 β∗ti−1εti ∞  in0−k1c−p ti−11 tα i <∞.4.19 Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.3. 16 Abstract and Applied Analysis Finally, applying some of the computations from the proof of Theorem 4.5,weget Rtn1 ln qαnαα 2ln qαnα1kj−1−p∗kj1 2ln qαnα1O1 nα2.4.51 and, for the left-hand side Ltnof 4.20, Ltn1 ln qαn1α1 ln qαnα−α ln qαnα1O1 nα2.4.52 Comparing Ltnand Rtn, we see that, for Ltn≥Rtn, −α ln qα>αkj−1 2ln qα−p∗kj1 2ln qα4.53 is sufficient. Simplifying it, we get p∗kj1>α kj1,4.54 and, finally, p∗>α. This inequality is assumed, and therefore 4.21is valid if n0is sufficiently large. It remains to prove that 4.22holds for α>1. But it is a well-known fact that the series ∞  in0−k1 1 ln tiα ∞  in0−k1 1 iαln qα4.55 is convergent for α>1. Thus, all assumptions of Theorem 4.4 are true, and, from its conclusions, we deduce that all conclusions of Theorem 4.7 are true. Example 4.8. Consider 1.1, where βtn:1 n1n−j in−k1ln q/ln ti 1 n1n−j in−k11/i.4.56 Then it is easy to verify that 1.1has a solution ytn n  i1 ln q ln ti  n  i1 1 i,4.57 which is the nth partial sum of harmonic series and, as such, is divergent as n→∞.Now we asymptotically compare the function βwith the right-hand side of 4.49. Proceeding as in Example 4.6,weget βtn1 k−j−kj1 2k−jnO1 n2.4.58 Abstract and Applied Analysis 17 Inequality 4.49is valid if βtn1 k−j−kj1 2k−jnO1 n2≤1 k−j−p∗kj1 2k−jn,4.59 that is if p∗<1. This inequality is the opposite to p∗>1 guaranteeing the existence of an increasing and asymptotically convergent solution. Thus, the example also shows that criterion 4.49is sharp in a sense. 4.5. A General Criterion for the Existence of an Increasing and Asymptotically Convergent Solution Analysing two criteria for the existence of an increasing and asymptotically convergent solution y:T→Rof 1.1expressed by 4.12and 4.20, that is, by inequalities βtn≤c−p tn , βtn≤c−p ln tn 4.60 with suitable constants cand p, we can state the following. The first criterion 4.12can successfully be used, for example, for the time scale Tt{tn}, where tnn. In this case, as stated in Theorem 4.5,4.30,thatis, βtn≤1 k−j−p∗kj1 2k−jtn 1 k−j−p∗kj1 2k−jn 4.61 is assumed with a p∗>1. The second criterion 4.20can successfully be used, for example, for the time scale Tt{tn}where tnqnand q>1. Then, as stated in Theorem 4.7,4.49,thatis, βtn≤1 k−j−p∗kj1ln q 2k−jln tn 1 k−j−p∗kj1 2k−jn4.62 is assumed with a p∗>1. Comparing 4.61and 4.62, we see that, although their left-hand sides are different due to different meaning of tnin every case, their right-hand sides are identical. The following result gives a criterion for every discrete time scale Tt{tn}with properties described in introduction. Theorem 4.9. Let βtn≤1 k−j−p∗kj1 2k−jn4.63 holds for all n∈Z∞ n0−kand for a fixed p∗>1. Let, moreover, α∈1,p∗. Then there exists an increasing and asymptotically convergent solution y:T→Rof 1.1satisfying ytn≤ n  in0−k11 k−j−p∗kj1 2k−ji−11 iα4.64 18 Abstract and Applied Analysis for every n∈Z∞ n0. Such a solution is defined, for example, by the initial function ϕtn: n  in0−k11 k−j−p∗kj1 2k−ji−11 iα,n∈Zn0 n0−k.4.65 Proof. We will apply Theorem 4.2 with β∗tn:1 k−j−p∗kj1 2k−jn,ε tn:1 nα.4.66 Inequality 4.5turns into εtn11 n1α≥ n−j  in−k1 β∗ti−1εti n−j  in−k11 k−j−p∗kj1 2k−ji−11 iα.4.67 Asymptotic decompositions of the left-hand and right-hand sides were used in the proof of Theorem 4.5 if δn0, i.e., tnnfor every n∈Z∞ n0−kand a similar decomposition was used in the proof of Theorem 4.7. Therefore, we will not repeat it. We will only state that the above inequality holds for p∗>α.4.6holds as well because the series ∞  in0−k11 k−j−p∗kj1 2k−ji−11 iα4.68 is obviously convergent. Remark 4.10. Although Theorem 4.9 is a general result, it has a disadvantage in applications because of its implicit character. Unlike 4.61and 4.62, where the left-hand and middle parts are explicitly expressed in terms of tn, the right-hand side of the crucial inequality 4.63 cannot, in a general situation of arbitrary time scale {tn}, be explicitly expressed using only the tnterms. This is only possible if, for a given time scale, a function fis explicitly known such that ftnn. Then, 4.63can be written in the form βtn≤1 k−j−p∗kj1 2k−jftn1 k−j−p∗kj1 2k−jn.4.69 Remark 4.11. On the other hand, in a sense, Theorem 4.9 gives the best possible result. Indeed, 1.1with βtn:1 n1n−j in−k11/i 4.70 has an increasing asymptotically divergent solution ytnn i11/i. An asymptotic decomposition of the right-hand side of 4.70was performed in Example 4.6 and an increasing and asymptotically convergent solution exists if 4.63,thatis, βtn1 n1n−j in−k11/i ≤1 k−j−p∗kj1 2k−jn4.71 Abstract and Applied Analysis 19 holds, or if βtn1 k−j−kj1 2k−jnO1 n2≤1 k−j−p∗kj1 2k−jn.4.72 The last holds for p∗<1. This inequality is the opposite to p∗>1 guaranteeing the existence of an increasing and asymptotically convergent solution. Thus, the example shows that our general criterion is sharp in a sense. Acknowledgments This research was supported by the Grant P201/10/1032 of the Czech Grant Agency Prague, by the project FEKT-S-11-2921and by the Council of Czech Government MSM 00216 30503. M. R˚ uˇ ziˇ ckov´ a and Z. Suta were supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic VEGA. References 1S. 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