ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.

GLOBAL EXISTENCE OF SOLUTION TO NONLINEAR INTEGRODIFFERENTIAL EQUATION IN A BANACH SPACE

Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.

NUMERICAL STUDY OF A NONLINEAR INTEGRODIFFERENTIAL EQUATION

In this Paper. by using the pseudo-spectral method of Fornberg and Whitham, a nonlinear integrodifferential equations is investigated numerically. It is found that for small ∈, the result is close to that of the KdV equation, whereas the effects of larger ∈ and the initial condition are significant.

Spectral Method for Semilinear Parabolic Integrodifferential Equations

Based on the discussion of the semidiscretization of a parabolic equation with asemilinear memory term,an error estimate is derived for the fully discrete scheme with spectral method in space and the backward Euler method in time The trapezoidal rule is adopted.for the quadrature of the memory term and the quadrature error isestimated.

PBVP of Integrodifferential Equations with Carathéodory Functions

We study the periodic boundary value problems for nonlinear integro-differential equa- tions of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β:αβ or β α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.

UNIQUE EXISTENCE OF PERIODIC SOLUTIONS OF NEUTRAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS

This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.

SOME NEW RESULTS FOR NONLINEAR BOUNDARY VALUE PROBLEMS OF MIXED TYPE INTEGRODIFFERENTIAL EQUATION

In this paper we give the existence and uniqueness of solutions for boundary value problems of the form u' = f(t, u,u', T1u,T2u), g(u(0),u(1)) = 0, h(w(0), u(1),u'(0), u'(1)) = 0 by means of the upper and lower solution method.

Approximate controllability results of semilinear integrodifferential equations with infinite delays

We study the approximate controllability of control systems governed by a class of semilinear integrod- ifferential equations with infinite delays. Sufficient conditions for approximate controllability of semilinear control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using fixed-point theorems and semigroup theory. As an illustration of the application of the approximate controllability results, a simple example is provided.

ON SOLUTIONS TO SEMILINEAR INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper, using the theory of resolvent operators, Banach,s contraction prin-ciple and Schauder,s fixed point theorem, we study the existence of integral solutions to semilinear integrodifferential equations under nonlocal conditions in Banach space. An example is provided to illustrate the results obtained.

ON THE DECOMPOSITION PROBLEM OF STABILITY FOR VOLTERRA INTEGRODIFFERENTIAL EQUATIONS

In this paper, we discuss the problem of stability of Volterra integrodifferential equationwith the decompositionwhere andin which is an matrix of functionB continuous forin which is an matrix of functionscontinuous for 0≤S≤ t<∞.According to the decomposition theory of large scale system and with the help of Liapunovfunctional, we give a criterion for concluding that the zero solution of (2) (i.e. large scale system(1)) is uniformly asymptotically stable.We also discuss the large scale system with the decompositionand give a criterion for determining that the solutions of (4) (i.e. large scale system (3)) areuniformly bounded and uniformly ultimately bounded.Those criteria are of simple forms, easily checked and applied.

HIGH ACCURACY ANALYSIS FORINTEGRODIFFERENTIAL EQUATIONS

In this paper we study the higher accuracy methods-the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations. Theglobal extrapolation and the correction approximations of third order, rather than the pointwiseextrapolation results, are derived.

Bell Polynomial Approach for the Solutions of Fredholm Integro- Differential Equations with Variable Coefficients

In this article,we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials.Using collocation points and treating the solution as a linear combination of Bell polynomials,the problem is reduced to linear system of equations whose unknown variables are Bell coefficients.The solution to this algebraic system determines the approximate solution.Error estimation of approximate solution is done.Some examples are provided to illustrate the performance of the method.The numerical results are compared with the collocation method based on Legendre polynomials and the other two methods based on Taylor polynomials.It is observed that the method is better than Legendre collocation method and as accurate as the methods involving Taylor polynomials.

EXTRAPOLATION AND DEFECT CORRECTION FOR DIFFUSION EQUATIONS WITH BOUNDARY INTEGRAL CONDITIONS

In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations with boundary integral conditions. The global extrapolation and the correction approximations, rather than the pointwise extrapolation results, are derived.

ON WINTNER TYPE GLOBAL EXISTENCE THEOREMS FOR CERTAIN INTEGRODIFFERENTIAL EQUATIONS

In this paper Wintner type global existence results for certain higher order integrodifferential equations are established. Our analysis is based on a simple and classical application of the Leray--Schauder alternative.

THE MAXIMUM AND MINIMUM SOLUTIONS OF NONLINEAR INTEGRODIFFERENTIAL EQUATIONS OF MIXED TYPE WITH IMPULSES IN BANACH SPACES

In this paper, we discuss the maximum and minimum solutions as will as generalized maximum and minimum solutions of nonlinear integrodifferential equations of mixed type with impulses at fixed moments in Banach soaces by the theorems of fixed points for inreasing operators.

Constrained controllability of nonlinear neutral fractional integrodifferential systems

In this paper,we study the complete controllability for a class of nonlinear neutral fractional integro-differential systems in a finite interval of time by means of controls whose initial and final values can be assigned in advance.The result is achieved using the Banach fixedpoint theorem and Schauder’s fixed-point theorem.Some numerical examples are provided to demonstrate the effectiveness of the main result.

The Viability Property of Controlled Jump Diffusion Processes

In this paper, we first give a comparison theorem of viscosity solution to some nonlinear second order integrodifferential equation. And then using the comparison theorem, we obtain a necessary and sufficient condition for the viability property of some controlled jump diffusion processes which can keep the solution within a constraint K.

Controllability of impulsive second-order nonlinear systems with nonlocal conditions in Banach spaces

In this paper,we are concerned with the controllability of damped second-order integrodifferential systems with impulses.Further the result is extended to study the controllability of nonlinear neutral systems with nonlocal conditions.The fixed point analysis approach is adopted in investigation.Sufficient conditions are formulated with a noncompact condition on the cosine family of operators.The results are obtained using the Banach fixed point theorem.An example is presented to illustrate the results.

A PERTURBATION THEOREM AND ITS APPLICATION

This paper is concerned with operators of abstract equations which are de-pendent on a multi-parameter. A general theorem, which may be viewed as aperturbation theorem, is developed first. The result is then applied to an abstractintegrodifferential equation and is used to study differentiability with respect toparameters of solutions of this integrodifferential equation. An application to aviscoelastic equation is also given.