Abstract
Abstract An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent. MSC:47H05, 47H10, 47H17.
Highlights
1 Introduction Construction of fixed points of nonlinear mappings is a classical and active area of nonlinear functional analysis due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonlinear mappings
As a matter of fact, the well-known Banach contraction principle states that the Picard iterates {Tnx} converge to the unique fixed point of T whenever T is a contraction of a complete metric space
Where {αn} is a sequence in the unit interval [, ], T is a self-mapping of a closed convex subset C of a Hilbert space H, and the initial guess x is an arbitrary point of C
Summary
Construction of fixed points of nonlinear mappings is a classical and active area of nonlinear functional analysis due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonlinear mappings. Mann’s algorithm fails to converge for Lipschitzian pseudocontractions (see the counterexample of Chidume and Mutangadura [ ]) It is an interesting question of inventing iterative algorithms which generate a sequence converging in the norm topology to a fixed point of a Lipschitzian pseudocontraction (if any). It is an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. For the existing literature on iterative methods for pseudocontractions, the reader can consult [ , – ]; for finding minimum-norm solutions of nonlinear fixed point and variational inequality problems, see [ – ]; and for related iterative methods for nonexpansive mappings, see [ , , , ] and the references therein. Recall that the nearest point (or metric) projection from H onto C is defined as follows: For each point x ∈ H, PCx is the unique point in C with the property x – PCx ≤ x – y , y ∈ C
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