Oscillation criteria for difference equations with non-monotone arguments

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1 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 DOI 0.86/s R E S E A R C H Open Access Oscillation criteria for difference equations with non-monotone arguments George E Chatzarakis * and Leonid Shaikhet * Correspondence: geaxatz@otenet.gr Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklio, Athens, 4, Greece Full list of author information is available at the end of the article Abstract This paper is concerned with the oscillatory behavior of first-order retarded [advanced] difference equation of the form x(n)+p(n)x(τ (n)) =0, n N 0 [ x(n) q(n)x(σ (n)) =0, n N ], where (p(n)) n 0 [(q(n)) n ] is a sequence of nonnegative real numbers and τ (n)[σ(n)] is a non-monotone sequence of integers such that τ (n) n,forn N 0 and lim τ (n)= [σ(n) n +, forn N]. Sufficient conditions, involving, which guarantee the oscillation of all solutions are established. These conditions improve all previous well-known results in the literature. Also, using algorithms on MATLAB software, examples illustrating the significance of the results are given. MSC: 39A0; 39A Keywords: difference equations; non-monotone arguments; retarded arguments; advanced arguments; oscillation; Grönwall inequality Introduction The paper deals with the difference equation with a single variable retarded argument of the form x(n)+p(n)x ( τ(n) ) =0, n N 0, (E) and the (dual) difference equation with a single variable advanced argument of the form x(n) q(n)x ( σ (n) ) =0, n N, (E ) where N 0 and N are the sets of nonnegative integers and positive integers, respectively. Equations (E) and (E ) are studied under the following assumptions: everywhere (p(n)) n 0 and (q(n)) n are sequences of nonnegative real numbers, (τ(n)) n 0 is a sequence of integers such that τ(n) n, n N 0 and lim τ(n)=, (.) The Author(s) 07. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page of 6 and (σ (n)) n is a sequence of integers such that σ (n) n +, n N. (. ) Here, denotes the forward difference operator x(n)=x(n+) x(n)and corresponds to the backward difference operator x(n)=x(n) x(n ). Set w = min n 0 τ(n). Clearly, w is a finite positive integer if (.)holds. By a solution of (E), we mean a sequence of real numbers (x(n)) n w which satisfies (E) for all n 0. It is clear that, for each choice of real numbers c w, c w+,...,c, c 0,thereexistsa unique solution (x(n)) n w of (E) which satisfies the initial conditions x( w)=c w, x( w + ) = c w+,...,x( ) = c, x(0) = c 0. By a solution of (E ), we mean a sequence of real numbers (x(n)) n 0 which satisfies (E ) for all n. Asolution(x(n)) n w (or (x(n)) n 0 )of(e)(or(e )) is called oscillatory, ifthetermsx(n) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.anequationisoscillatory if all its solutions oscillate. In the last few decades, the oscillatory behavior and the existence of positive solutions of difference equations with deviating arguments have been extensively studied; see, for example, papers [ 9] and the references cited therein. Most of these papers concern the special case where the arguments are nondecreasing, while a small number of these papers are dealing with the general case where the arguments are non-monotone. See, for example, [ 3,, 5] and the references cited therein. By the consideration of nonmonotone arguments of other than the pure mathematical interest, it approximates the natural phenomena described by equation of the type (E) or (E ). That is because there are always natural disturbances (e.g. noise in communication systems) that affect all the parameters of the equation and therefore the fair (from a mathematical point of view) monotone arguments become non-monotone almost always.. Retarded difference equations In 008 Chatzarakis, Koplatadze and Stavroulakis [4, 5]provedthat,if p(j) >, (.) where h(n)=max 0 s n τ(s), or lim inf n p(j)> e, (.3) then all solutions of (E) oscillate.

3 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 3 of 6 It is obvious that there is a gap between the conditions (.)and(.3) when the limit lim n p(j) does not exist. How to fill this gap is an interesting problem which has been investigated by several authors. For example, in 009, Chatzarakis, Philos and Stavroulakis [6]proved that if p(j)> a a a, (.4) where a = lim inf n p(j), then all solutions of (E) oscillate. In 0, Braverman and Karpuz [3]provedthatif h(n) p(j) >, (.5) i=τ(j) then allsolutions of (E)oscillate,while, in 04,Stavroulakis [5], improved (.5)to h(n) p(j) > a a a. (.6) i=τ(j) In 05, Braverman, Chatzarakis, and Stavroulakis []provedthatifforsomer N or p(j)a ( ) r h(n), τ(j) >, (.7) p(j)a r ( ) a a a h(n), τ(j) >, (.8) where n [ ] n [ ( )] a (n, k)=, ar+ (n, k)= a r i, τ(i), (.9) i=k then all solutions of (E) oscillate. Recently, Asteris and Chatzarakis []provedthatifforsomel N i=k h(n) >, (.0) p l (j) where [ p l (n)=p(n) + n h(n) ] p l (j) (.) with p 0 (n)=p(n), then all solutions of (E) oscillate.

4 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 4 of 6. Advanced difference equations In 0, Chatzarakis and Stavroulakis [7]provedthat,if or ρ(n) q(i)>, (.) ρ(n) q(i)> ( b), (.3) where ρ(n)=min s n σ (s)andb = lim inf σ (n) +, then all solutions of (E )oscillate. In 05, Braverman, Chatzarakis, and Stavroulakis []provedthatifforsomer N or where ρ(n) q(j)b ( ) r ρ(n), σ (j) >, (.4) j=n ρ(n) q(j)b ( ) b b b r ρ(n), σ (j) >, (.5) j=n k [ ] b (n, k)= q(i), b r+ (n, k)= + k [ ( )] q(i)b i, σ (i) + r (.6) and b = lim inf σ (n) +, then all solutions of (E )oscillate. Recently, Asteris and Chatzarakis []provedthatifforsomel N where ρ(n) q(i) σ (i) j=ρ(n)+ >, (.7) q l (j) [ q l (n)=q(n) + ρ(n) + q(i) σ (i) j=ρ(n)+ ] q l (j) (.8) with q 0 (n)=q(n), then all solutions of (E )oscillate. In this paper we study further (E) and (E ) and derive new sufficient oscillation conditions. Examples illustrate cases when the results of the present paper imply oscillation while previously known results fail.

5 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 5 of 6 Main results. Retarded difference equations We study further (E) and derive a new sufficient oscillation condition, involving, which essentially improves all the previous results. Let h(n)= max τ(s), n 0. (.) 0 s n Clearly, the sequence h(n) is nondecreasing and τ(n) h(n) n foralln 0. The proof of our main result is essentially based on the following lemmas. Lemma Assume that (.) holds and 0<a := lim inf Then we have lim inf n n p(j). p(j)=lim inf where h(n) is defined by (.). n p(j)=a, (.) Proof Since h(n) is nondecreasing and τ(n) h(n) n foralln 0, we have n p(j) n p(j). Therefore lim inf n p(j) lim inf n p(j). If (.) does not hold, then there exist a >0andasubsequence(θ(n)) such that θ(n) as n and lim θ(n) j=h(θ(n)) p(j) a < a. But h(θ(n)) = max 0 s θ(n) τ(s), hence there exists θ (n) θ(n), θ (n) N 0 h(θ(n)) = τ(θ (n)), and consequently such that θ(n) j=h(θ(n)) p(j)= θ(n) j=τ(θ (n)) p(j) θ (n) j=τ(θ (n)) p(j).

6 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 6 of 6 It follows that ( θ (n) j=τ(θ (n)) p(j)) n= is a bounded sequence having a convergent subsequence, say θ (n k ) j=τ(θ (n k )) p(j) c a, ask which implies that lim inf n p(j) a < a. This contradicts (.). The proof of the lemma is complete. Lemma ([6], Lemma.) In addition to hypothesis (.), assume that h(n) is defined by (.), 0<a := lim inf n p(j) e (.3) and x(n) is an eventually positive solution of (E). Then lim inf x(n +) x(h(n)) a a a. (.4) Theorem Assume that (.) holds, h(n) is defined by (.) and a by (.3). If for some l N h(n) p l (j) > a a a, (.5) where p l (n) is defined by (.), then all solutions of (E) oscillate. Proof Assume, for the sake of contradiction, that (x(n)) n w is a nonoscillatory solution of (E). Then it is either eventually positive or eventually negative. As ( x(n)) n w is also a solution of (E), we may restrict ourselves only to the case where x(n) > 0 for all large n.let n w be an integer such that x(n) > 0 for all n n.then,thereexistsn n such that x(τ(n)) > 0, n n. In view of this, equation (E) becomes x(n)= p(n)x ( τ(n) ) 0, n n, which means that the sequence (x(n)) is eventually decreasing. Therefore, since τ(n)<n, (E) implies x(n)+p(n)x(n) 0. (.6)

7 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 7 of 6 Applying the discrete Grönwall inequality, we obtain n x(k) x(n), for all n k n. (.7) i=k Summing up (E) from τ(n)ton,wehave x(n) x ( τ(n) ) + n Since τ(i) h(i) h(n), (.7)and(.8)give x ( τ(i) ) =0. (.8) x(n) x ( τ(n) ) + x ( h(n) ) n h(n) Multiplying the last inequality by p(n), we get p(j) 0. p(n)x(n) p(n)x ( τ(n) ) + p(n)x ( h(n) ) n which, in view of (E), becomes h(n) p(j) 0, x(n)+p(n)x(n)+p(n)x ( h(n) ) n Since h(n)<n,thelastinequalitygives h(n) p(j) 0. or x(n)+p(n)x(n)+p(n)x(n) [ x(n)+p(n) + Therefore n n h(n) h(n) p(j) 0, ] x(n) 0. p(j) x(n)+p (n)x(n) 0, where [ p (n)=p(n) + n h(n) ]. p(j)

8 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 8 of 6 Repeating the above argument leads to a new estimate, x(n)+p (n)x(n) 0, where [ p (n)=p(n) + n h(n) ]. p (j) Continuing by induction, for sufficiently large n we get x(n)+p l (n)x(n) 0, where [ p l (n)=p(n) + n h(n) ]. p l (j) Clearly, by the Grönwall inequality, we have x ( τ(i) ) x ( h(n) ) h(n) p l (j). (.9) Summing up (E) from h(n)ton,wehave x(n +) x ( h(n) ) + x ( τ(i) ) =0. (.0) Combining (.9) and(.0), we have or x(n +) x ( h(n) ) + x ( h(n) ) h(n) p l (j) 0, h(n) +) x(n p l (j) x(h(n)). Therefore h(n) x(n +) lim inf p l (j) x(h(n)). (.)

9 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 9 of 6 By combining Lemmas and, it becomes obvious that inequality (.4) is fulfilled. So, (.)leadsto h(n) p l (j) a a a, which contradicts (.5). The proofof the theoremis complete. Example Consider the retarded difference equation x(n)+ 79,000 x( τ(n) ) =0, n N 0, (.) with (see Figure (a)) n 3, ifn =5μ, n, ifn =5μ +, τ(n)= n, ifn =5μ +, n 4, ifn =5μ +3, n 3, ifn =5μ +4, μ N 0. By (.), we see (Figure (b)) that h(n)= max 0 s n τ(s)= n 3, ifn =5μ, n, ifn =5μ +, n, ifn =5μ +, n, ifn =5μ +3, n 3, ifn =5μ +4, μ N 0. Figure The graphs of τ (n)andh(n).

10 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 0 of 6 Observe that the function F : N 0 R + defined as F l (n)= h(n) p l (j) attains its maximum at n =5μ +4,μ N 0,foreveryl N. Specifically, by using an algorithm of Matlab software, we obtain F (n) = = = 5μ+4 i=5μ h(n) p (j) h(n) 3μ 79,000 p(j)[ + j k=τ(j) p(k) h(j) m=τ(k) 79,000 [ + j k=τ(j) 79,000 p(m) ] h(j) m=τ(k) ], Thus F (n)= h(n) p (j) Since a = lim inf n p(j)=lim inf 5μ j=5μ p(j)=0.79, we have > a a a , that is, condition (.5) oftheorem is satisfied for l =. Therefore, all solutions of (.) oscillate. Observe, however, that a =0.79< e, p(j)= μ 5μ+4 i=5μ < a a a p(j)= = 0.76 <, ,

11 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page of 6 = μ h(n) p(j) 5μ+4 j=5μ+ i=τ(j) =0.79 μ + 5μ i=τ(5μ+3) =0.79 μ + 5μ i=5μ μ i=τ(j) { 5μ i=τ(5μ+) { 5μ i=5μ } μ i=τ(5μ+4) { ( = < a a a <. 5μ i=τ(5μ+) } μ i=5μ , μ i=5μ ) +} <, 0.79 That is, conditions (.), (.3), (.4), (.5) (.7) (forr =),(.6) (.8) (forr =)and (.0)arenotsatisfied. Figure, illustrates that the three different solutions of (.), as predicted, appear to be oscillatory. Figure Three solutions of (.).

12 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page of 6 Notation It is worth noting that the improvement of condition (.5) to the corresponding condition (.4) is significant, approximately 38.45%, if we compare the values on the lefthand side of these conditions. Also, the improvement compared to condition (.5)(or(.6) or (.7)or(.8)) is very satisfactory, around 7.8%. Also, observe that conditions (.7), (.8), and (.0) do not lead to oscillation for the first iteration. On the contrary, condition (.5) is satisfied from the first iteration. This means that our condition is better and much faster than (.7), (.8), and (.0).. Advanced difference equations A similar oscillation theorem for the (dual) advanced difference equation (E )canbederived easily. The proof of this theorem is omitted, since it is quite similar to the proof for a retarded equation. Let ρ(n)=min σ (s), n. (.3) s n Clearly, the sequence ρ(n) is nondecreasing and σ (n) ρ(n) n +foralln. Theorem Assume that (. ) holds and ρ(n) is defined by (.3). If for some l N ρ(n) q(i) σ (i) j=ρ(n)+ q l (j) > b b b, (.4) where q l (n) is defined by (.8) and 0<b = lim inf σ (n) + /e, then all solutions of (E ) oscillate. Example Consider the advanced difference equation x(n) 34,000 x( σ (n) ) =0, n N 0, (.5) with (see Figure 3(a)) n +, ifn =5μ, n +5, ifn =5μ +, σ (n)= n +, ifn =5μ +, n +, ifn =5μ +3, n +, ifn =5μ +4, μ N 0. By (.3), we see (Figure 3(b)) that ρ(n)=min s n σ (s)= n +, ifn =5μ, n +, ifn =5μ +, n +, ifn =5μ +, n +, ifn =5μ +3, n +, ifn =5μ +4, μ N 0.

13 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 3 of 6 Figure 3 The graphs of σ (n)andρ(n). Observe that the function F : N 0 R + defined as ρ(n) F l (n)= q(i) σ (i) j=ρ(n)+ q l (j) attains its maximum at n =5μ +,μ N 0,foreveryl N. Specifically, by using an algorithm of Matlab software, we obtain Thus Since we have ρ(n) F (n)= q(i) ρ(n) = q(i) = b = lim inf 5μ+3 i=5μ+ σ (i) j=ρ(n)+ σ (i) j=ρ(n)+ 34,000 q (j) q(j)[ + σ (j) k=j+ q(k) σ (k) m=ρ(j)+ σ (i) j=5μ+4 ρ(n) F (n)= σ (n) + q(i)=lim inf 34,000 [ + σ (j) k=j+ q(i) σ (i) j=ρ(n)+ 5μ+ i=5μ > b b b 34,000 q (j) q(i)= 34,000 =0.705, 0.98 q(m) ] σ (k) m=ρ(j)+ ] ,000 34

14 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 4 of 6 that is, condition (.4)ofTheorem is satisfied for l =. Therefore, all solutions of (.5) oscillate. Observe, however, that ρ(n) q(i)= μ 5μ+3 i=5μ+ q(i)= < ( b) 0.990, ρ(n) q(j)b ( ) ρ(n), σ (j) j=n = μ 5μ+3 j=5μ+ 34,000 σ (j) i=5μ+4 { = 34 σ (5μ+),000 μ i=5μ+4 { = 34 5μ+6,000 μ i=5μ+4 = 34 {,000 ( 34 34,000 σ (5μ+) 34 +,000 i=5μ+4 34,000,000 ) < b b b <. + 34,000 34,000 =0.55<, 5μ+3 i=5μ , σ (5μ+3) 34 +,000 i=5μ+4 34,000 + } <, 5μ+4 i=5μ+4 } 34,000 } 34,000 That is, conditions (.), (.3), (.4), (.5),and (.7)arenot satisfied. Figure 4, illustrates three different solutions of (.5), as predicted, appeared to be oscillatory. Notation It is worth noting that the improvement of condition (.4)to the corresponding condition (.) issignificant, approximately93%, if wecomparethevaluesontheleftside of these conditions. Also, the improvement compared to condition (.4)(or(.5)) is very satisfactory, around 46.3%. Also, observe that conditions (.4)and(.5) do not lead to oscillation for first iteration. On the contrary, condition (.4) is satisfied from the first iteration. This means that our condition is better and much faster than (.4)and(.5)..3 Deviating difference inequalities A slight modification in the proofs of Theorems and leads to the following results about deviating difference inequalities. Theorem 3 Assume that all conditions of Theorem hold. Then (i) the retarded difference inequality x(n)+p(n)x ( τ(n) ) 0, n N 0, has no eventually positive solutions;

15 Chatzarakis and Shaikhet Advances in Difference Equations (07) 07:6 Page 5 of 6 Figure 4 Three solutions of (.5). (ii) the retarded difference inequality x(n)+p(n)x ( τ(n) ) 0, n N 0, has no eventually negative solutions. Theorem 4 Assume that all conditions of Theorem hold. Then (i) the advanced difference inequality x(n) q(n)x ( σ (n) ) 0, n N, has no eventually positive solutions; (ii) the advanced difference inequality x(n) q(n)x ( σ (n) ) 0, n N, has no eventually negative solutions. Competing interests The authors declare that they have no competing interests. Authors contributions The authors declare that they have made equal contributions to the paper. Author details Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklio, Athens, 4, Greece. School of Electrical Engineering, Tel-Aviv University, Tel-Aviv, 69978, Israel. Acknowledgements The first author was supported by the Special Account for Research of ASPETE through the funding program Strengthening research of ASPETE faculty members. Received: 0 September 06 Accepted: 5 February 07

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