Haydar Akça

ON A MILD SOLUTION OF A SEMILINEAR FUNCTIONAL-DIFFERENTIAL EVOLUTION NONLOCAL PROBLEM

The existence, uniqueness, and continuous dependence of a mild solution of a nonlocal Cauchy problem for a semilinear functional-differential evolution equation in a general Banach space are studied. Methods of a C0 semigroup of operators and the Banach contraction theorem are applied. The result obtained herein is a generalization and continuation of those reported in references [2-8].

IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

1. Introduction. In this paper, we study the existence, uniqueness, and continuous dependence of a mild solution of a nonlocal Cauchy problem for impulsive functionaldifferential evolution equation. Such problems arise in some physical applications as a natural generalization of the classical initial value problems. The results for semilinear functional-differential evolution nonlocal problem [2] are extended for the case of impulse effect. We consider the nonlocal Cauchy problem in the form

Difference approximations for impulsive differential equations

A convergent difference approximation is obtained for a nonlinear impulsive system in a Banach space.

Global Exponential Stability of Impulsive Cohen-Grossberg-Type BAM Neural Networks with Time-Varying and Distributed Delays

—The purpose of this paper is to investigate the global exponential stability of a class of impulsive bidirectional associative memories (BAM) neural networks that possesses Cohen-Grossberg dynamics. By constructing and using some inequality techniques and a fixed point theorem sufficient conditions are obtained to ensure the existence and global exponential stability of the solutions for impulsive Cohen-Grossberg neural networks with time delays and distributed delays.

Discrete-time Cohen-Grossberg neural networks with transmission delays and impulses

Abstract A discrete-time analogue is formulated for an impulsive Cohen- -Grossberg neural network with transmission delay in a manner in which the global exponential stability characterisitics of a unique equilibrium point of the network are preserved. The formulation is based on extending the existing semidiscretization method that has been implemented for computer simulations of neural networks with linear stabilizing feedback terms. The exponential convergence in the p-norm of the analogue towards the unique equilibrium point is analysed by exploiting an appropriate Lyapunov sequence and properties of an M-matrix. The main result yields a Lyapunov exponent that involves the magnitude and frequency of the impulses. One can use the result for deriving the exponential stability of non-impulsive discrete-time neural networks, and also for simulating the exponential stability of impulsive and non-impulsive continuous-time networks.

On Existence of Solutions of Semilinear Impulsive Functional Differential Equations with Nonlocal Conditions

The existence, uniqueness and continuous dependence of a mild solution of a semilinear impulsive functional-differential evolution nonlocal Cauchy problem in general Banach spaces are studied. Methods of fixed point theorems, of a C0 semigroup of operators and the Banach contraction theorem are applied.

Improved Stability Estimates for Impulsive Delay Reaction-Diffusion Cohen-Grossberg Neural Networks via Hardy-Poincaré Inequality

Abstract An impulsive Cohen-Grossberg neural network with time-varying and S-type distributed delays and reaction-diffusion terms is considered. By using Hardy-Poincaré inequality instead of Hardy-Sobolev inequality or just the nonpositivity of the reaction-diffusion operators, under suitable conditions in terms of M-matrices which involve the reaction-diffusion coefficients and the dimension and size of the spatial domain, improved stability estimates for the system with zero Dirichlet boundary conditions are obtained. Examples are given.

Exponential Stability of Neural Networks with Time-Varying Delays and Impulses

We present sufficient conditions for the uniqueness and exponential stability of equilibrium points of impulsive neural networks which are a generalization of Cohen-Grossberg neural networks.

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.

ASYMPTOTIC PROPERTIES OF THE SOLUTIONS OF A CLASS OF OPERATOR-DIFFERENTIAL EQUATIONS

Some asymptotic properties of the nonoscillating solutions of opreator-differential equations of arbitrary order are investigated.