I.K. Purnaras

Positive solutions for systems of nonlinear discrete boundary value problems

Existence of eigenvalues yielding positive solutions for systems of second order discrete boundary value problems (two-point, three-point and generalized three-point) are established. The results are obtained by the use of a Guo–Krasnoselskii fixed point theorem in cones.

An asymptotic result for some delay difference equations with continuous variable

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.

Some results on the behavior of the solutions of a linear delay difference equation with periodic coefficients

Some new results on the asymptotic behavior, the nonoscillation and the stability of a class of linear delay difference equations with periodic coefficients and constant delays are given. The coefficients have a common period and the delays are multiples ,of this period. The results are obtained via a positive root of an associated equation, which is in a sense the corresponding characteristic equation. The application in the special case of constant coefficients of the results obtained is also presented.

On linear Volterra difference equations with infinite delay

Linear neutral, and especially non-neutral, Volterra difference equations with infinite delay are considered and some new results on the behavior of solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.

Oscillations in higher-order neutral differential equations

Consider the n-th order (n ≥ 1) neutral differential equation ... (formule)... where δ ∈ {0,+1,−1}, ζ ∈ {+1,−1}, −∞ < τ 1 < τ 2 < ∞ WITH τ 1 t 2 ¬= 0, −∞ < σ 1 < σ 2 < ∞ AND μ AND η ARE INCREASING REAL-VALUED FUNCTIONS ON [τ 1 , τ 2 ] and [σ 1 , σ 2 ] respectively. The function μ is assumed to he not constant on [τ 1 , τ] and [τ, τ 2 ] for every τ ∈ (τ 1 , τ 2 ); similarly, for each σ ∈ (σ 1 , σ 2 ), it is supposed that η is not constant on [σ 1 , σ] and [σ, σ 2 ]

Oscillations in linear difference equations with variablecoefficients

A class of linear difference equations with variable coefficients is considered. Sufficient conditions and necessary conditions for the oscillation of the solutions are established. In the special cases where the coefficients are constant or periodic the conditions become both necessary and sufficient.

On non-autonomous linear difference equations with continuous variable

This article is concerned with non-autonomous linear neutral (and, especially, non-neutral) delay difference equations with continuous variable, in which the coefficients vary and the delays are constant. An asymptotic criterion as well as a useful inequality for the solutions are established, by the use of an appropriate solution of the so called generalized characteristic equation.

On the Behavior of the Solutions for Certain Linear Autonomous Difference Equations

Certain linear autonomous delay as well as neutral delay difference equations are considered. A class of linear autonomous delay difference equations with continuous variable is also considered. Some results on the behavior of the solutions are established via two distinct positive roots of the corresponding characteristic equation.

Sufficient Conditions for the Oscillation of Delay Difference Equations

The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [J. Appl. Math. Simulation 2 (1989), 101–111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations.

On the oscillation of some linear difference equations with periodic coefficients

Abstract Some linear difference equations with periodic coefficients (not necessarily nonnegative) are considered. Necessary conditions and sufficient conditions for the oscillation of the solutions are established. Conditions under which all nonoscillatory solutions tend to zero at ∞ are also presented. The results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients, which were given earlier by the second author [Oscillations of some delay differential equations with periodic coefficients, J. Math. Anal. Appl. 162 (1991) 452–475].