Muhammad N. Islam

Uniform asymptotic stability in nonlinear volterra discrete systems

Abstract We employ the notion of total stability to obtain new criteria for uniform asymptotic stability of the zero solution of a nonlinear Volterra discrete system. Resolvent equation methods are employed, and a summability criterion on the resolvent kernel is obtained. Also, we obtain a new difference equation that the resolvent R(n, s) satisfies.

Exponential Stability in Non-linear Difference Equations

Non-negative definite Lyapunov functionals are employed to obtain sufficient conditions that guarantee exponential stability of the zero solution of a non-linear discrete system. The theory is illustrated with several examples.

Periodic solutions of nonlinear integral equations

SummaryThe existence of a continuous periodic solution of the system

Periodic solutions of neutral nonlinear system of differential equations with functional delay

x(t) = f(t) + \int\limits_{ - \infty }^t {q(t,s,x(s))ds} ,

Separate Contraction and Existence of Periodic Solutions in Totally Nonlinear Delay Differential Equations FULL TEXT

, is studied using Horns fixed point theorem as the basic tool. First it is assumed that the solutions are bounded in some sense and that they depend continuously on initial functions. Then the required boundedness of solutions are obtained for special cases of q. Also, a few sufficient conditions are provided to ensure the continuous dependence of solutions on initial functions.

Nonlinear integrodifferential equations anda priori bounds on periodic solutions

In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel’skii’s fixed point theorem, and a combination of Krasnosel’skii’s and Schaefer’s fixed point theorems are employed in the analysis. The combination theorem of Krasnosel’skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov’s direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes. Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA e-mail: muhammad.islam@notes.udayton.edu Received by the editors March 5, 2010; revised August 18, 2010. Published electronically June 24, 2011. AMS subject classification: 45D05, 45J05.