Random fixed point theorems in Banach algebras with applications to random integral equations

Abstract

The present paper studies the random versions of some deterministic fixed point theorems of Dhage [5] and Dhage and Regon [7]. Applications are given to a certain nonlinear functional random integral equation for proving the existence of random solution under the generalized Lipschitzicity and Caratheodory conditions.

Existence and Attractivity Theorems for Nonlinear Hybrid Fractional Integrodifferential Equations with Anticipation and Retardation

“…Our essential tool used in the chapter is the following hybrid fixed point theorem of Dhage [9,16] for a quadratic operator equation involving three operators in a Banach algebra X which uses arguments from analysis and topology. See also Dhage [6,7,9,16] and Dhage and O'Regan [22] for some related results and applications.…”

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

“…When C ≡ 0 in Theorem 2.3 and Corollary 2.2 we obtain the following random versions of the fixed point theorems of Dhage [9] and Dhage and Regan [12]. See Dhage [13].…”

“…To finish, it is enough to prove that F is measurable on Ω. Let C be a closed subset of X. Denote 13) where…”

Applications of Krasnoselskii-Dhage Type Fixed-Point Theorems to Fractional Hybrid Differential Equations

“…- [11]). An operator A : X → X is called a nonlinear D-contraction on X if there exists a D-function ψ A ∈ D such that ∥Ax − Ay∥ ≤ ψ A ∥x − y∥ for all elements x, y ∈ X, where 0 < ψ A (r) < r for all r > 0.…”

“…Throughout this paper X := (X, ∥ • ∥) stands for a real Banach space. Next, we will borrow some definitions and facts from Dhage [7]- [11] which is well-known in the metric fixed point theory and has been widely used in the literature on applications to the theory of nonlinear differential and integral equations.…”