The generalized contraction proximal point algorithm with square-summable errors

Abstract

Let $(x_n)$ be a sequence generated by $x_{n+1}=\alpha _nu+\gamma _nx_n+\delta _nJ_{\beta _n}x_n+e_n$ for $n\ge 0$, where $J_{\beta _n}$ is the resolvent of a maximal monotone operator A with $\beta _n\in (0,\infty )$, $u,x_0\in H$, $(e_n)$ is a sequence of errors and $\alpha _n\in (0,1)$, $\gamma _n\in (-1,1)$, $\delta _n\in (0,2)$ are real numbers such that $\alpha _n+\gamma _n+\delta _n=1$ for all $n\ge 0$. We present strong convergence results for the sequence generated by the generalized contraction proximal point algorithm defined above under weaker accuracy conditions and mild conditions on the parameters $\alpha _n, \beta _n$ and $\delta _n$. Our results generalize and unify many known results in the literature.