Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $TK\to H$ a $k$-strict pseudo-contraction for some $0\le k<1$ such that $F\left(T\right)=\{x\in Kx=Tx\}\ne \varnothing$. Consider the following iterative algorithm given by $$\forall x_1\in K,x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+((1-{\beta }_n)I-{\alpha }_{n}A)P_{K}S{x}_n,n\ge 1,$$ where $SK\to H$ is defined by $Sx=kx+(1-k)Tx$, $P_K$ is the metric projection of $H$ onto $K$, $A$ is a strongly positive linear bounded self-adjoint operator, $f$ is a contraction. It is proved that the sequence $\left\{x_n\right\}$ generated by the above iterative algorithm converges strongly to a fixed point of $T$, which solves a variational inequality related...

We show that the set of those Markov semigroups on the Schatten class ₁ such that in the strong operator topology $li{m}_{t\to \infty }T\left(t\right)=Q$, where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\&lt;p\&lt;\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\&lt;q\&lt;\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups ${\left(T_t\right)}_{t\ge 0}$ of positive...