*(English)*Zbl 1122.47055

The authors consider the following equilibrium problem: find $x\in C$ such that

where $C$ is a nonempty, closed and convex subset of a real Hilbert space $H$ and $f:C\times C\to \mathbb{R}$. The set of solutions is denoted by $EP\left(f\right)$. Within the present paper, the function $f$ is supposed to satisfy the following assumptions:

(A1) $f(x,x)=0$ for all $x\in C;$

(A2) $f$ is monotone;

(A3) $\forall x,y,z\in C$, ${lim\; sup}_{t\downarrow 0}f(tz+(1-t)x,y)\le f(x,y)$;

(A4) $f(x,\xb7)$ is convex and lower semicontinuous for all $x\in C$.

In their main result, the authors provide a strong convergence theorem which solves the problem of finding a common element of the set $EP\left(f\right)$ and the set of fixed points of a nonexpansive mapping $S:H\to H$. Indeed, given such a map $S$ whose fixed points are denoted by $F\left(S\right)$ and under the assumption $EP\left(f\right)\cap F\left(S\right)\ne \u2300$, they find a suitable sequence $\left\{{x}_{n}\right\}$, generated starting from a point $x\in H$, such that $\left\{{x}_{n}\right\}$ converges strongly to the projection of $x$ onto $F\left(S\right)\cap EP\left(f\right)$.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47J20 | Inequalities involving nonlinear operators |

49J40 | Variational methods including variational inequalities |

47H09 | Mappings defined by “shrinking” properties |

47N10 | Applications of operator theory in optimization, convex analysis, programming, economics |