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 $T\phantom{\rule{0.222222em}{0ex}}\Omega \times C\to C$ satisfying: for each $x,y\in C$, $\omega \in \Omega $ and integer $n\ge 1$, $$\parallel {T}^{n}(\omega ,x)-{T}^{n}(\omega ,y)\parallel \le a\left(\omega \right)\xb7\parallel x-y\parallel +b\left(\omega \right)\{\parallel x-{T}^{n}(\omega ,x)\parallel +\parallel y-{T}^{n}(\omega ,y)\parallel \}+c\left(\omega \right)\{\parallel x-{T}^{n}(\omega ,y)\parallel +\parallel y-{T}^{n}(\omega ,x)\parallel \},$$
where $a,b,c\phantom{\rule{0.222222em}{0ex}}\Omega \to [0,\infty )$ are functions satisfying certain conditions and ${T}^{n}(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,\xb7)$. 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 ${H}^{k,p}$...