## Abstract

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 *α*-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*)∩(*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.

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Communicated by J.-C. Yao.

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Takahashi, S., Takahashi, W. & Toyoda, M. Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces.
*J Optim Theory Appl* **147**, 27–41 (2010). https://doi.org/10.1007/s10957-010-9713-2

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DOI: https://doi.org/10.1007/s10957-010-9713-2