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.
Citation
Mayumi Hojo. Wataru Takahashi. "Approximation of Common Solutions for Monotone Inclusion Problems and Equilibrium Problems in Hilbert Spaces." Nihonkai Math. J. 23 (2) 115 - 134, 2012.
Information