Existence and stability behavior of random solutions of a system of nonlinear random equations
TL;DR: In this article, the authors studied the stability of a system of nonlinear random integral equations of the Volterra type of the form x(t;ω) = h(t, x (t, τ, x(τ, ω))dτ, is presented where t ϵ R+ = {t:t ≥ 0}, ωϵΩ, Ω being the underlying set of a complete probability measure space.
About: This article is published in Information Sciences.The article was published on 1975-01-01. It has received 2 citations till now.
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TL;DR: It is established that sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type are established by using admissibility theory and fixed point theorems.
Abstract: We establish sufficient conditions for the existence and
uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral
equations of mixed type by using admissibility theory and fixed point theorems. The
results obtained in this paper generalize the results of several papers.
5 citations
TL;DR: In this paper, the existence, uniqueness, and stability of random solutions of a general class of nonlinear stochastic integral equations were studied by using the Banach fixed point theorem, which generalizes the results of Szynal and Wedrychowicz (1993).
Abstract: We study the existence, uniqueness, and stability of random solutions of a general class of nonlinear stochastic integral equations by using the Banach fixed point theorem. The results obtained in this paper generalize the results of Szynal and Wedrychowicz (1993).
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Book•
26 Feb 1974
170 citations
TL;DR: In this article, the norm of solutions of systems of first order differential equations as well as theorems on global existence and boundedness and other useful results have been obtained by comparing solutions of the given system with those of a related (single) first-order differential equation, which is essentially due to Conti [5] and Wintner [9] and has been extended in scope by Brauer [2; 3] and Antosiewicz [ l ] to obtain many of the results for systems of differential equations referred to above.
Abstract: Upper and lower bounds for the norm of solutions of systems of first order differential equations as well as theorems on global existence and boundedness and other useful results have recently been obtained by comparing solutions of the given system with those of a related (single) first order differential equation. This technique, which is essentially due to Conti [5] and Wintner [9], has been extended in scope by Brauer [2; 3] and Antosiewicz [ l ] to obtain many of the results for systems of differential equations referred to above. In this paper, which will appear in complete form elsewhere, we present a similar technique and using it obtain results for systems of integral equations of the form
69 citations
Book•
01 Jan 1971
TL;DR: In this article, a random integral equation of the volterra type and a stochastic integral equation with fredholm type with application to systems theory are presented. But their solutions are approximate solutions of the random VOLTERRA integral equation.
Abstract: General introduction.- Preliminaries.- A random integral equation of the volterra type.- Approximate solutions of the random volterra integral equation.- A stochastic integral equation of the fredholm type with application to systems theory.- Random discrete fredholm and volterra equations.- The stochastic differential systems.- The stochastic differential systems.- The stochastic differential systems with lag time.
38 citations
TL;DR: In this paper, the authors studied the problem of existence and uniqueness of solutions of the nonlinear Volterra integral equation (1.1) and showed that convergence in B implies convergence in S. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg and Benes.
Abstract: In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation x =f+ a,g1(x) + * + angn(x), where the a, are continuous linear operators mapping a Frechet space J into itself and the g, are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of F having a stronger topology. The equations are studied first when the gi are of the form gi(x) = x + hi(x) where hi(x) is "small", and then when the g, are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Benes. I.Introduction. The purpose of this paper is to study the behavior of solutions of the nonlinear Volterra integral equation (t It (1.1) x(t) = f(t)+J a1(t-s)g1(x(s)) ds+ *** an(t-S)gn(x(s)) ds. In ?11 these are studied in the abstract form (1.2) x = f+ a, gl(X) + * * + angn(X) where x andf are elements of a Frechet space F the operators ai are linear continuous maps from Y -? Y and the gi are nonlinear maps from Y YJ We assume that gi(x) = x + hi(x) where the hi have certain "smallness" properties. Solutions are sought which lie in a Banach subspace B of F with a stronger topology. By this we mean that convergence in B implies convergence in S. In Theorem 2.1 we prove that the nonlinear problem (1.2) has a unique solution lying in B if f is in B with small norm and there exists a bounded linear operator co mapping B -? B, such that ai + coai is a bounded linear operator from B -? B and 11Z ai +Z cwai-w IIB < 1. Equations of this sort, with one kernel, have been studied by many authors; Presented to the Society, January 22, 1970 under the title Global existance and uniqueness of solutions to Volterra integral equations; received by the editors July 3, 1969 and, in revised form, December 10, 1969. AMS Subject Classifications. Primary 4513, 4530; Secondary 4780.
12 citations