A general inexact iterative method for monotone operators, equilibrium problems and fıxed point problems of semigroups in Hilbert spaces
Vittorio Colao, Giuseppe Marino, Daya Ram Sahu · Fixed Point Theory and Applications · 2012
Let H be a real Hilbert space. Consider on H a nonexpansive family with a common fixed point, a contraction f with the coefficient 0 < α < 1, and a strongly positive linear bounded self-adjoint operator A with the coefficient . Assume that and that is a family of nonexpansive self-mappings on H such that and has property with respect to the family . It is proved that the following schemes (one implicit and one inexact explicit): and converge strongly to a common fixed point , where denotes the set of common fixed point of the nonexpansive semigroup. The point x * solves the variational in-equality 〈(γf −A)x*, x−x*〉 ≤ 0 for all . Various applications to zeros of monotone operators, solutions of equilibrium problems, common fixed point problems of nonexpansive semigroup are also presented. The results presented in this article mainly improve the corresponding ones announced by many others. 2010 Mathematics Subject Classification: 47H09; 47J25.