On a nonlinear Hammerstein integral equation with a parameter

This paper, following theories for asymptotically linear operators, the Schaefer fixed point theorem, decomposition of operators, and critical point theory, is mainly concerned with the existence and multiplicity of solutions to a nonlinear Hammerstein integral equation with a parameter. The results show that when the nonlinearity satisfies certain conditions, different parametric intervals lead to different existence results; however, in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem. Our results can be applied to some well known boundary value problems, and some examples are given.     

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.Research supported by National Science Foundation under Grant No. ECS-8005960.     

On a class of nonlinear stochastic integral equations

We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution.     

On a stochastic nonlinear equation in one-dimensional viscoelasticity

In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.

     

Moment equation methods for nonlinear stochastic systems

One method of approaching models represented by systems of stochastic ordinary differential equations is to consider the moment equations. This approach can be far more efficient than a Monte Carlo simulation or a finite-difference solution of the associated Fokker-Plank equation. However, a nonlinear system generates an infinite hierarchy of moment equations, which requires the adoption of some hierarchy truncation technique to facilitate solution. This paper considers a method of hierarchy truncation, based on the quasi-moments of the state-variables.     

On a nonlinear volterra-hammerstein integral equation in two variables

Using a fixed point theorem of Krasnosel'skii type, this article proves the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation in two variables.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

On a nonlinear Volterra integral equation in a Banach space

The equation u(t) + ∝₀^tk(t ? s)g(s) ds?f(t), t ? 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g(t) and the function f. The case when k is realvalued, g(t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy.     

Optimal control over nonlinear stochastic systems

The synthesis of optimal control over nonlinear stochastic systems that are described by the Itô equations is reduced to the solution of recurrence relations derived from the Bellman stochastic equation.     

Abstract stochastic integral equation involving a vector generalized Stochastic integral

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 153–155, March, 1991.     

Newton-Kantorovich approximations to nonlinear singular integral equation with shift

The paper is concerned with the applicability of some new conditions for the convergence of Newton-kantorovich approximations to solution of nonlinear singular integral equation with shift of Uryson type. The results are illustrated in generalized Holder space.