## Nihonkai Mathematical Journal

- Nihonkai Math. J.
- Volume 23, Number 2 (2012), 115-134.

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

Mayumi Hojo and Wataru Takahashi

#### 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.

#### Article information

**Source**

Nihonkai Math. J., Volume 23, Number 2 (2012), 115-134.

**Dates**

First available in Project Euclid: 12 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.nihmj/1363096201

**Mathematical Reviews number (MathSciNet)**

MR3060230

**Zentralblatt MATH identifier**

1372.47072

**Subjects**

Primary: 47H05: Monotone operators and generalizations

Secondary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 58E35: Variational inequalities (global problems)

**Keywords**

Equilibrium problem fixed point inverse-strongly monotone mapping iteration procedure maximal monotoone operator resolvent strict pseudo-contraction

#### Citation

Hojo, Mayumi; Takahashi, Wataru. Approximation of Common Solutions for Monotone Inclusion Problems and Equilibrium Problems in Hilbert Spaces. Nihonkai Math. J. 23 (2012), no. 2, 115--134. https://projecteuclid.org/euclid.nihmj/1363096201