Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. (English) Zbl 1198.65100
Let \(E\) be a real Banach space and \(C\subset E\) closed, convex and nonempty. A mapping \(A:D(A)\subset E \rightarrow E'\) is said to be \(\gamma\)-inverse strongly monotone if there exists \(\gamma >0\) such that
\[ (Ax-Ay,x-y)\geq\gamma \parallel Ax-Ay\parallel^{2},\quad \forall x,y\in D(A). \]
The authors introduce an iterative process of the type:
\[ x_{n-1}=\Pi_{C_{n-1}}(x_{0}); \quad x_{0}\in C_{0}(\equiv C) \quad {\text{arbitrary}} \quad n\geq 0, \]
converging strongly to a common element of the set of common fixed points of the countably infinite family of closed relatively quasi-nonexpensive mappings, the solution set of a generalized equilibrium problem and the solution set of a variational inequality problem for a \(\gamma\)-inverse strongly monotone mapping in Banach spaces. The theorems of the paper improve, generalize, unify and extend several known results.
\[ (Ax-Ay,x-y)\geq\gamma \parallel Ax-Ay\parallel^{2},\quad \forall x,y\in D(A). \]
The authors introduce an iterative process of the type:
\[ x_{n-1}=\Pi_{C_{n-1}}(x_{0}); \quad x_{0}\in C_{0}(\equiv C) \quad {\text{arbitrary}} \quad n\geq 0, \]
converging strongly to a common element of the set of common fixed points of the countably infinite family of closed relatively quasi-nonexpensive mappings, the solution set of a generalized equilibrium problem and the solution set of a variational inequality problem for a \(\gamma\)-inverse strongly monotone mapping in Banach spaces. The theorems of the paper improve, generalize, unify and extend several known results.
Reviewer: Erwin Schechter (Moers)
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 47H10 | Fixed-point theorems |
| 47J25 | Iterative procedures involving nonlinear operators |
| 47J40 | Equations with nonlinear hysteresis operators |
| 65K15 | Numerical methods for variational inequalities and related problems |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
equilibrium problems; strong convergence; variational inequalities; common fixed points; iterative methods; Banach space; quasi-nonexpensive mappingsReferences:
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