Özkan Öcalan

Iterated Oscillation Criteria for Delay Dynamic Equations of First Order

We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.

COMPARISON THEOREMS ON THE OSCILLATION AND ASYMPTOTIC BEHAVIOUR OF HIGHER-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

In this work, oscillatory and asymptotic behaviours of all solutions of higher-order neutral differential equations are compared with first-order delay differential equations, depending on two different ranges of the coefficient associated with the neutral part. Some simple examples are given to compare our results with the existing results in the literature and to illustrate the significance and applicability of our new results. Our results generalise, improve and correct some of the existing results in the literature.

Oscillation results for higher order nonlinear neutral delay difference equations

Abstract In this work, we shall consider higher order nonlinear neutral delay difference equation of the type Δ m ( y n + p n y n − l ) + q n y n − k α = 0 , where { p n } , { q n } are sequences of nonnegative real numbers, k and l are positive integers and α ∈ ( 0 , 1 ) is a ratio of odd positive integers. We obtain sufficient conditions for the oscillations of all solutions of this equation.

Further oscillation criteria for partial difference equations with variable coefficients

In this paper, some new oscillation criteria on the oscillation of first-order partial delay difference equations with nonnegative variable coefficients, which improve the recent ones under some additional conditions, are given. Some examples to illustrate the applicability of our results are also supplied of which solutions are plotted by the mathematical programming language Mathematica 7.0.

Existence of positive solutions for a neutral differential equation with positive and negative coefficients

Abstract This work is concerned with the existence of positive solutions for the neutral delay differential equation with positive and negative coefficients [ x ( t ) − R ( t ) x ( t − r ) ] ′ + P ( t ) x ( t − τ ) − Q ( t ) x ( t − σ ) = 0 , where P , Q , R ∈ C ( [ t 0 , ∞ ) , R + ) , r > 0 , τ ≥ σ ≥ 0 .

Oscillations of differential equations with several non-monotone advanced arguments

Consider the first-order advanced differential equation of the form where qi, 1 ≤ i ≤ m are functions of non-negative real numbers, and σi, 1 ≤ i ≤ m are functions of positive real numbers such that σi(t) > t for t ≥ t0. New oscillation criteria, involving lim sup and lim inf, are established, in the case of non-monotone advanced arguments. An example illustrating the results is also given.

Oscillations of difference equations with non-monotone retarded arguments

Consider the first-order retarded difference equation Δ x ( n ) + p ( n ) x ? ( n ) = 0 , n ? N 0 where ( p ( n ) ) n ? 0 is a sequence of nonnegative real numbers, and ( ? ( n ) ) n ? 0 is a sequence of integers such that ? ( n ) ≤ n - 1 , n ? 0 , and lim n ? ∞ ? ( n ) = ∞ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving lim inf , is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail.

Statement of retraction

The Editors and Publisher Taylor & Francis are retracting the above article from publication in Journal of Difference Equations and Applications. This paper was previously published in online-only form as part of Taylor & Francis’ iFirst service. Date of online publication is 23 April 2012, http://www.tandfonline.com/doi/abs/10.1080/ 10236198.2012.662968. This article is now available as supplementary material at ,url of retraction statement. . The main result of the Note is incorrect, due to errors in the proof of the Theorem. First, in Case 1 of the proof, the assumption on the choice of epsilon just after (2.1) does not follow from the previous logic, and in fact is not true. Second, in the second paragraph of Case 2, the proof claims to reach a contradiction because the solution cannot converge to an unstable equilibrium. But solutions can converge to unstable equilibria if they are saddle equilibria, so there is no such contradiction. Either of these errors is sufficient to indicate that the Theorem remains unproved. Furthermore, the Theorem is false, and contradicts the published literature. Note that in both cases of the proof (a . 1 and a , 1), the equilibrium has one slightly positive eigenvalue, so there are solutions that do not oscillate, contradicting the main result of the Note. This was first proved in Theorem 2 of the article ‘A note on positive non-oscillatory solutions . . . ’ by K. Berenhaut and S. Stevic, JDEA 12, p. 495–9 (2006), http://www. tandfonline.com/doi/abs/10.1080/10236190500539543.

A note on the recursive sequence

In this paper, we give some remarks on the solution of the difference equation where and are arbitrary positive numbers. Moreover, this paper gives an answer to an Open Problem on the periodic behaviour of the difference equation where are arbitrary positive numbers.

Corrigendum: Corrigendum to Oscillation of a class of difference equations of second order [Math. Comput. Modelling 49 (2009) 912-917]

where n ∈ N0 := N∪{0},R, P,Q are bounded starting segments of positive integers, {ri(n)}n∈N0 and {f (n)}n∈N0 are sequences of real numbers, {pj(n)}n∈N0 and {qk(n)}n∈N0 are nonnegative sequences of real numbers, {a(n)}n∈N0 is a positive sequence, ρi, τj, σk ≥ 0 are integers for all i ∈ R, j ∈ P, k ∈ Q . Wehavediscovered that our paper includes some incorrect results. Below,wepartially salvage these results. Themistakes stem from the following invalid claim: