In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T-periodic solution of



(0.1) x(t)= a(t) + tt-h D(t,s)g(s,x(s))ds



if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T-periodic solution of



(0.2) x(t) = f(t,x(t)) + tt-h D(t,s)g(s,x(s))ds



if f defines a contraction and if D and g are small enough.



We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T-periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.

A Fixed Point Theorem of Krasnoselskii—Schaefer Type

The first article by Turunen and Friberg deals with a class of fields with propagation-invariant properties such as the optical intensity distribution. Coherent and partially coherent stationary and pulsed solutions are treated in view of scalar and electromagnetic approaches. Approximations of ideal propagation-invariant fields and methods for their generation are discussed. Finally, some application areas are covered.

Propagation-Invariant Optical Fields

This note deals with impulsive control of discrete systems with time delay. It's the first time that an impulsive controller was introduced for discrete systems with time delay. Firstly, some stability criteria for delay discrete systems with impulse effects are obtained by using Lyapunov functionals and Razumikhin technique. These stability criteria show that if the impulses satisfy some conditions, then the original stable systems can keep their stability property even with impulse effects. Then by using Lyapunov functionals or Lyapunov functions together with Razumikhin technique, two stability criteria for the discrete systems with time delay under impulsive control are given. These two results show that unstable systems can be stable under impulsive control. Some examples are also presented to illustrate the efficiency of the obtained results.

Impulsive Control of Discrete Systems With Time Delay

In this paper we provide sufficient conditions for the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.

On fractional integro-differential equations with state-dependent delay

Stability in neutral nonlinear differential equations with functional delays using fixed-point theory

Carry-free addition and borrow-free subtraction of modified signed-digit trinary numbers with optical symbolic substitution are presented. The proposed two-step and three-step algorithms can be easily implemented by using phase-only holograms, optical content-addressable memories, a multichannel correlator, or a polarization-encoded optical shadow-casting system.

Modified signed-digit trinary arithmetic by using optical symbolic substitution

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

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.

Uniform asymptotic stability in nonlinear volterra discrete systems

In this paper we study stability and boundedness of the nonlinear dierence equation x(t + 1) = a(t)x(t) + c(t) x(t g(t)) + q(x(t); x(t g(t)) : In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis. Liapunov's method is normally used to study the stability properties of the zero solution of dieren tial and dierence equations. Certain diculties arise when Liapunov's method is applied to equations with unbounded de- lay or equations containing unbounded terms (11), (20). It has been found that some of these diculties can be eliminated if xed point theory is used instead (3). In the present paper we study certain type of boundedness and stability properties of solutions of linear and nonlinear dierence equations with delay using xed point theory as the main mathematical tool. In particular we study equi-boundedness of solutions and stability of the zero solution of

/pdf/boundedness-and-stability-in-nonlinear-delay-difference-2vuqbdwvyk.pdf

Muhammad N. Islam

Papers

Modified signed-digit trinary arithmetic by using optical symbolic substitution

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

Uniform asymptotic stability in nonlinear volterra discrete systems

Boundedness and stability in nonlinear delay difference equations employing fixed point theory.

Uniform Asymptotic Stability in Linear Volterra Integrodifferential Equations with Application to Delay Systems