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

  • S. Takahashi
  • W. Takahashi
  • M. Toyoda


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)−10, where F (T) is the set of fixed points of T and (A+B)−10 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.


Nonexpansive mapping Maximal monotone operator Inverse strongly-monotone mapping Fixed point Iteration procedure Equilibrium problem 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Yokohama PublishersYokohamaJapan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  3. 3.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan
  4. 4.Faculty of EngineeringTamagawa UniversityTokyoJapan

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