An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings

作者：Colao Vittorio; Lopez Acedo Genaro; Marino Giuseppe^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2009, 71(7-8): 2708-2715.

DOI：10.1016/j.na.2009.01.115

• Equilibrium problem
• Fixed point
• Nonexpansive mapping
• Variational inequality
• Implicit method
• Minimization problem

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings {T(n)} with common fixed points, a finite family of equilibrium functions {G(i)}(i=1,...,K), a contraction f with coefficient 0 < alpha < 1 and a strongly positive linear bounded operator A with coefficient (gamma) over bar > 0.
Let 0 < gamma < (gamma) over bar/alpha. Assuming there are common equilibrium points of the family {G(i)}i=(1,...,K) which are also fixed points for {T(n)}, we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality <(A - gamma f)x*, x - x*> >= 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence {T(n)}.

• 出版日期2009-10-1