WEAK AND STRONG CONVERGENCE THEOREMS FOR STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACES

By Giuseppe Marino, Hong-Kun Xu. 2007

553 citations recorded by the source corpus.

Abstract Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ -strict pseudo-contraction on C with a fixed point, for some 0 ⩽ κ 1 . Given an initial guess x 0 ∈ C and given also a real sequence { α n } in ( 0 , 1 ) . The Mann's algorithm generates a sequence { x n } by the formula: x n + 1 = α n x n + ( 1 − α n ) T x n , n ⩾ 0 . It is proved that if the control sequence { α n } is chosen so that κ α n 1 and ∑ n = 0 ∞ ( α n − κ ) ( 1 − α n ) = ∞ , then { x n } converges weakly to a fixed point of T . However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] from nonexpansive mappings to strict pseudo-contractions.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/0c6d93d61b5ee7378f7bb683d527525aca6b67c1

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Citations in the corpus

An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings (2009, methodology): “Very recently, Marino and Xu [11] extended Reich’s result [13] to strict pseudo-contraction mappings in the setting of Hilbert spaces.”
An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces (2018, background): “We know that the set of fixed point of a demi-contractive mapping is closed and convex [46].”
An iterative method for solving split monotone variational inclusion and fixed point problems (2015, unknown): “It is a common knowledge that if F(T ) = ∅, then it is closed and convex (see, for example, [1,5,21,34] and references therein, for more details on strictly pseudocontrative mappings).”
Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces (2008, unknown): “In this case, duality mapping J is also called weak sequentially continuous, that is, for each { x n } ⊂ X with x n ⇀ x , we have J ( x n )⇀* J ( x ); see, for example, [ 13 , 26 ].”
Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense (2009, background): “8 extends the demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [5], Gornicki [6], Marino and Xu [7] and Xu [8].”
The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space (2018, background): “We know the following lemmas obtained by Marino and Xu [9] and Kocourek, Takahashi and Yao [6]; see also [18, 19].”