Approximation of Common Solutions for Monotone Inclusion Problems and Equilibrium Problems in Hilbert Spaces

Abstract

Let $H$ be a real Hilbert space and let $C$ be a nonempty closed convex subset of $H$. Let $\alpha >0$ and let $A$ be an $\alpha$-inverse-strongly monotone mapping of $C$ into $H$. Let $B$ be a maximal monotone operator on $H$ and let $F$ be a maximal monotone operator on $H$ such that the domain of $F$ is included in $C$. Let $(A+B)^{-1}0$ and $F^{-1}0$ be the sets of zero points of $A+B$ and $F$, respectively. Let $0< k<1$ and let $g$ be a $k$-contraction of $H$ into itself. Let $G$ be a strongly positive bounded linear self-adjoint operator on $H$ with coefficient $\overline{\gamma}>0$ and let $0< \gamma <\frac{\overline{\gamma}}{k}$. In this paper, under the assumption $(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset$, In this paper, we prove a strong convergence theorem for finding a point $z_0\in (A+B)^{-1}0\cap F^{-1}0$ which is a unique fixed point of a nonlinear operator and also a unique solution of a variational inequality. $z_0\in (A+B)^{-1}0\cap F^{-1}0$ is a unique fixed point of $P_{(A+B)^{-1}0\cap F^{-1}0}(I-G+\gamma g)$. This point $z_0\in (A+B)^{-1}0\cap F^{-1}0$ is also a unique solution of a variational inequality. Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in Nonlinear Analysis and Optimization.