A. Zafer

On the oscillation of fractional differential equations

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form

Oscillation criteria for even order neutral differential equations

D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1

Oscillation criteria for third-order nonlinear functional differential equations

, where Daq denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

Oscillation of a neutral difference equation

Oscillation criteria are given for even order neutral type differential equations of the following form [x(t)+a(t)x(τ(t))](n)+ƒ(t,x(t),x(σ(t)))=0 , where f(t, x, y) ∈ C([0, ∞) × R2, R) and a, τ, σ ∈ C([0, ∞), R) such that 0 ≤ a(t) < 1, τ(t) < t, σ(t) ≤ t, and limt→∞ τ(t) = limt→∞σ(t) = ∞.

Oscillatory and asymptotic behavior of higher order difference equations

Abstract In this work, we are concerned with oscillation of third-order nonlinear functional differential equations of the form ( r 2 ( t ) ( r 1 ( t ) y ′ ) ′ ) ′ + p ( t ) y ′ + q ( t ) f ( y ( g ( t ) ) ) = 0 . By using a Riccati type transformation and integral averaging technique, we establish some new sufficient conditions under which every solution y ( t ) either oscillates or converges to zero as t → ∞ . Unlike ones from the known works in the literature, our results are applicable to nonlinear functional differential equations of the above form when f ( u ) = | u | α − 1 u , α > 0 .

On oscillation and nonoscillation of second-order dynamic equations

Abstract This paper is concerned with the oscillation of the bounded solutions of neutral difference equation where Δ is the forward difference operator defined by Δ n = n+1 - n .

Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities

This paper is concerned with the oscillation of solutions of neutral difference equation @D[a(t)@D^n^-^1(x(t)+p(t)x(r(t)))] + F(t,x(@s(t))) = 0, t@eI, and the asymptotic behavior of solutions of delay difference equation @D^nx(t) + F(t, x(g(t))) = h(t), t@eI, where I is the discrete set {0, 1, 2,...} and @D is the forward difference operator @Dx(t) = x(t+1)-x(t).

Oscillation of integro-dynamic equations on time scales

Abstract New oscillation and nonoscillation criteria are established for second-order linear equations with damping and forcing terms. Examples are given to illustrate the results.

Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects

We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form , with , , where is a time scale interval with , the functions are right-dense continuous with , is the forward jump operator, , and . All results obtained are new even for and . In the special case when and our theorems reduce to (Y. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 337 (2007), 549–560). Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale.

Forced oscillation of super-half-linear impulsive differential equations

Abstract In this paper, the authors initiate the study of oscillation theory for integro-dynamic equations on time-scales. They present some new sufficient conditions guaranteeing that the oscillatory character of the forcing term is inherited by the solutions.