Nonlinear Random Integral Equations Research Articles

A common random fixed point theorem and Application to Random Integral Equations

The aim of this paper is to prove a flxed point theorem via contraction mappings of a pair of weakly increasing mappings using an altering function in a partially ordered complete separable metric spaces. Our theorem is useful to determine a large of nonlinear problems. We discuss the existence of a common solution to a system of nonlinear random integral equations.

Some Algebraic and topological random fixed point theorems with applications to nonlinear random integral equations

In this paper some algebraic and topological random fixed point theorems are proved involving the three random operators on a Banach algebra and they are further applied to a certain nonlinear functional random integral equation of mixed type for proving the existence as well as existence of extremal random solutions under the generalized Lipschizicity, Carath´eodory and monotonicity conditions.

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 theorems for non-linear random integral equations with time lags

Two basic forms of non-linear random integral equations are studied and where being the underlying set of a complete probability measure space (fi, A, P). The random process x(t; ω) is the unknown random function is the stochastic free term; the stochastic kernels are of convolution type, of linear and non-linear form, respectively, defined for and the processes are scalar functions defined for tsR+ and x£R.The object of this paper is to develop sufficient conditions for the existence of a, random solution, a second-order stochastic process, to the above equations. We utilize the methods of successive stochastic approximation and Banach's fixed point theorem to fulfil the objective.

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.