The aim of this paper is to establish a random coincidence degree theory. This degree theory possesses all the usual properties of the deterministic degree theory such as existence of solutions, excision and Borsuk’s odd mapping theorem. Our degree theory provides a method for proving the existence of random solutions of the equation $Lx\in N(\omega ,x)$ where $L:\mathit{\text{dom}}L\subset X\to Z$ is a linear Fredholm mapping of index zero and $N:\Omega \times \overline{G}\to 2^Z$ is a noncompact Carathéodory mapping. Applications to random differential inclusions are also considered.

Let $(\Omega ,\Sigma )$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p>1$. We prove random fixed point theorems for a class of mappings $T\Omega \times C\to C$ satisfying: for each $x,y\in C$, $\omega \in \Omega$ and integer $n\ge 1$, $$\parallel T^n(\omega ,x)-T^n(\omega ,y)\parallel \le a\left(\omega \right)·\parallel x-y\parallel +b\left(\omega \right)\{\parallel x-T^n(\omega ,x)\parallel +\parallel y-T^n(\omega ,y)\parallel \}+c\left(\omega \right)\{\parallel x-T^n(\omega ,y)\parallel +\parallel y-T^n(\omega ,x)\parallel \},$$ where $a,b,c\Omega \to [0,\infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,·)$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p}$...

Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x) is the value...

Let $(\Omega ,\Sigma )$ be a measurable space, $(E,P)$ be an ordered separable Banach space and let $[a,b]$ be a nonempty order interval in $E$. It is shown that if $f:\Omega \times [a,b]\to E$ is an increasing compact random map such that $a\le f(\omega ,a)$ and $f(\omega ,b)\le b$ for each $\omega \in \Omega$ then $f$ possesses a minimal random fixed point $\alpha$ and a maximal random fixed point $\beta$.

In this paper we prove a general random fixed point theorem for multivalued maps in Frechet spaces. We apply our main result to obtain some common random fixed point theorems. Our main result unifies and extends the work due to Benavides, Acedo and Xu [4], Itoh [8], Lin [12], Liu [13], Tan and Yuan [20], Xu [23], etc.