A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

Abstract

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Hölder continuity of the bifunctions involved.

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Acknowledgements

The authors would like to thank the associate editor and anonymous referee for their constructive comments and helpful remarks. This work is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.01-2017.315.

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Correspondence to Le Dung Muu.

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Muu, L.D., Le, X.T. A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems. J. Fixed Point Theory Appl. 20, 130 (2018). https://doi.org/10.1007/s11784-018-0612-8

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Keywords

  • Monotone equilibria
  • Fixed point
  • Common solution
  • Splitting algorithm

Mathematics Subject Classification

  • 47H05
  • 47H10
  • 90C33