System Of Nonlinear Random Equations Research Articles

Normal limiting distribution of the normalized number of extraneous solutions of a compatible system of nonlinear random equations over the field GF(2)

Normal limiting distribution of the normalized number of extraneous solutions of a compatible system of nonlinear random equations over the field GF(2)

The normal limit distribution of the normalized number of false solutions of a one system of nonlinear random equations over the field GF(2)

The theorem on a normal limit distribution of the normalized number of false solutions of a beforehand consistent system of nonlinear random equations over the field GF(2) is proved. The results with the additional condition on the number of nonzero components both false solutions and true solution of the solutions are obtained.

Uniqueness conditions for the solution of an inhomogeneous system of nonlinear random equations over the field GF(3)

An inhomogeneous consistent system of second-order nonlinear random equations over a field consisting of three elements is considered. The necessary and sufficient condition is found under which the system has a unique solution belonging to a given set of n-dimensional vectors as n ? ?.

An estimate for the rate of convergence of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$

An estimate for the rate of convergence of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$

SOLUTIONS FOR A SYSTEM OF RANDOM OPERATOR EQUATIONS AND SOME APPLICATIONS

The present paper is a continuation of [1]. By using more general type of random contractor than that defined by Lee and Padgett, we show several existence and uniqueness theorems for solutions to a system of nonlinear random operator equations with stochastic domain. Next, we obtain a random coincidence theorem and a random implicit function theorem, which improves and generalizes the corresponding results of Ding, Reddy and Subrahmanyam, Altman, Eng, Krasnoselskii and others. Finally, some applications of our results are given for the solutions of a system of nonlinear random integral equations.

Existence and stability behavior of random solutions of a system of nonlinear random equations

A study of a system of nonlinear random integral equations of the Volterra type of the form x(t;ω) = h(t, x(t;ω)) + ∝ 0 t k,(t, τ, x(τ;ω); ω)ƒ(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 (Ω, A, P). The random process x( t; ω) is the unknown random vector { x i ( t; ω)} i = 1 n , each component defined on R + × Ω; h(t, x(t, ω)) is the stochastic free vector { h i ( t, x i ( t; ω))} i = 1 n ; the nonlinear stochastic kernel is a matrix k( t, τ, x( τ, ω); ω) = { k ij ( t, τ, x( τ, ω); ω)}; defined, for each component, on R + × R + × R × Ω for 0 ≤ τ ≤ t < ∞ and i, j = 1, 2,…, n; and the random scalar function ƒ(t, x (t;ω)) = {ƒ i(t, x i(t; ω))} i = 1 n is defined on R + × R. The object of this paper is to study the following types of stability: 1. (i) stability in the mean; 2. (ii) asymptotic stability in the mean and; 3. (iii) stochastic asymptotic exponential stability of the above equation. In addition, sufficient hypotheses, utilizing specific Banach spaces, will be developed for the existence of a unique random solution, a second-order stochastic process, for the Volterra system described above. The present results generalize the recent results of Ahmed and Teo.