Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an alpha-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F(T) boolean AND (A + B)(-1)0, where F(T) is the set of fixed points of T and (A + B)(-1)0 is the set of zero points of A + B. Then, we prove a strong convergence theorem, which is different from the results of Halpern's type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.