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Some new existence results and stability concepts for fractional partial random differential equations

Abbas S., Benchohra M., Darwish M. A.

Journal of Mathematics and Applications

2016

Vol. 39

5--22

In the present paper we provide some existence results and Ulam’s type stability concepts for the Darboux problem of partial fractional random differential equations in Banach spaces, by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.

On potential kernels associated with random dynamical systems

Hmissi M., Mokchaha F., Hmissi A.

Opuscula Mathematica

2015

Vol. 35, no. 4

499--515

Let (Θ, φ) be a continuous random dynamical system defined on a probability space (Ω, F, P) and taking values on a locally compact Hausdorff space E. The associated potential kernel V is given by [formula]. In this paper, we prove the equivalence of the following statements: 1. The potential kernel of (Θ, φ) is proper, i.e. V ƒ is x-continuous for each bounded, x-continuous function with uniformly random compact support. 2. (Θ, φ) has a global Lyapunov function, i.e. a function L : Ω x E → (0, ∞) which is x-continuous and L,(Θ tω, φ(t, ω)x) ↓0 as t ↑ ∞. In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.

Existence and attractivity results for nonlinear first order random differential equations

Dhage B. C., Ntouyas S. K

Opuscula Mathematica

2010

Vol. 30, no. 4

411-429

In this paper, the existence and attractivity results are proved for nonlinear first order ordinary random differential equations. Two examples are provided to demonstrate the realization of the abstract developed theory.