Article

Journal of Optimization Theory and Applications

, Volume 149, Issue 2, pp 239-253

Strong Convergence of an Iterative Scheme by a New Type of Projection Method for a Family of Quasinonexpansive Mappings

  • Y. KimuraAffiliated withDepartment of Mathematical and Computing Sciences, Tokyo Institute of Technology
  • , W. TakahashiAffiliated withDepartment of Mathematical and Computing Sciences, Tokyo Institute of TechnologyDepartment of Applied Mathematics, National Sun Yat-sen University
  • , J. C. YaoAffiliated withDepartment of Applied Mathematics, National Sun Yat-sen University Email author 

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Abstract

We deal with a common fixed point problem for a family of quasinonexpansive mappings defined on a Hilbert space with a certain closedness assumption and obtain strongly convergent iterative sequences to a solution to this problem. We propose a new type of iterative scheme for this problem. A feature of this scheme is that we do not use any projections, which in general creates some difficulties in practical calculation of the iterative sequence. We also prove a strong convergence theorem by the shrinking projection method for a family of such mappings. These results can be applied to common zero point problems for families of monotone operators.

Keywords

Quasinonexpansive mapping Nonexpansive mapping Monotone operator Inverse-strongly monotone operator Fixed point Metric projection Shrinking projection method