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

S. Takahashi (), W. Takahashi () and M. Toyoda ()
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S. Takahashi: Yokohama Publishers
W. Takahashi: Tokyo Institute of Technology
M. Toyoda: Tamagawa University

Journal of Optimization Theory and Applications, 2010, vol. 147, issue 1, No 2, 27-41

Abstract: 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)−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.

Keywords: Nonexpansive mapping; Maximal monotone operator; Inverse strongly-monotone mapping; Fixed point; Iteration procedure; Equilibrium problem (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (10)

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DOI: 10.1007/s10957-010-9713-2

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