Ioannis K. Purnaras

Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations

In this paper, we study a nonlinear fractional q-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach’s contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed.MSC:34A08, 39A05, 39A12, 39A13.

Global attractivity in a nonlinear difference equation

Abstract We consider the (nonlinear) difference equation x n = a + ∑ k=1 m b k x n −k , n = 0,1,2… where a and bk (k = 1, 2, …, m) are nonnegative numbers with B ≡ ∑mk = 1bk > 0, and we are interested in whether all positive solutions are attracted by the positive equilibrium L = ( a 2 ) + ( a 2 ) 2 + B .

Periodic first order linear neutral delay differential equations

Some new asymptotic and stability results are given for a first order linear neutral delay differential equation with periodic coefficients and constant delays. The asymptotic behavior of the solutions and the stability of the trivial solution are described by the use of an appropriate real root of an equation, which is in a sense the corresponding characteristic equation.

Oscillations in superlinear differential equations of second order

Abstract A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t) + a(t)f[x(t)] = 0, where a ϵ C([t 0 , ∞)), f ϵ C(R) with yf(y) > 0 for y ≠ 0 and ∝ ±1 ±∞ [ 1 f(y) ] dy and f is continuously differentiable on R − {0} with f ′( y ) ⩾ 0 for all y ≠ 0. The coefficient a is not assumed to be eventually nonnegative and the oscillation cirterion obtained involves the average behavior of the integral of a . In the special case of the differential equation x″(t) + a(t) ¦x(t)¦ λ sgn x(t) = 0 (λ > 1) this criterion improves a recent oscillation result due to Wong [Oscillation theorems for second-order nonlinear differential equations, Proc. Amer. Math. Soc. 106 (1989), 1069–1077].

A boundary value problem on the whole line to second order nonlinear differential equations

Abstract Second order nonlinear ordinary differential equations are considered, and a certain boundary value problem on the whole line is studied. Two theorems are obtained as main results. The first theorem is established by the use of the Schauder theorem and concerns the existence of solutions, while the second theorem is concerned with the existence and uniqueness of solutions and is derived by the Banach contraction principle. These two theorems are applied, in particular, to the specific class of second order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of second order linear ordinary differential equations, respectively.

Oscillations in a class of linear difference equations with periodic coefficients

A class of linear difference equations with periodic coefficients is considered, and it is established that all solutions are oscillatory if and only if an associated equation (which is in a sense thecharacteristic equation) has no roots in(I,∞).

An asymptotic property of the solutions to second order linear nonautonomous delay differential equations

Second order linear nonautonomous delay differential equations are considered, and a fundamental asymptotic criterion for the solutions is established, by the use of the concept of generalized characteristic equation.

Asymptotic behavior of the oscillatory solutions to first order non-autonomous linear neutral delay differential equations of unstable type

This article is concerned with the asymptotic behavior of the oscillatory solutions to a class of first order unstable type linear neutral delay differential equations with variable coefficients and constant delays. A sufficient condition for all oscillatory solutions to tend to zero at ~ is established. This condition considerably improves a previous one given by Ladas and the third author [G. Ladas, Y.G. Sficas, Asymptotic behavior of oscillatory solutions, Hiroshima Math. J. 18 (1988) 351-359].