Proximal point algorithms involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces

Abstract

In this paper, a new modified proximal point algorithm involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces is proposed. Under suitable conditions, some weak convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the nonspreading-type multivalued mappings in Hilbert space are proved. The presented results in the paper are new.

Approximating Common Solution of Minimization Problems Involving Asymptotically Quasi-Nonexpansive Multivalued Mappings

“…Since asymptotically quasi-nonexpansive multivalued mapping is much more general than asymptotically nonexpansive multivalued mapping , asymptotically nonspreading-type multivalued mapping, quasi-nonexpansive multivalued mapping, nonexpansive multivalued mapping, nonspreading-type multivalued mapping, the problem studied in our paper is quite general and includes problems with optimization, variational inequality and fixed point as its special cases. Thus, it follows that Theorem 2 in our paper improves, extends and generalizes the results obtained in [10,12,[14][15][16][17][18]33], and many more others currently existing in the literature.…”

A novel approach for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators

“…Remark 3.2. Many already studied problems in the literature can be considered as special cases of this paper; see, for example, [6,12] and the references therein. Our results are applicable for finding a common solution of inclusion problems, convex optimization problems and fixed point problems involving set-valued operators in real Hilbert spaces (see, for example, [4] for more details).…”

A Modified Proximal Point Algorithm for Finite Families of Minimization Problems and Fixed Point Problems of Asymptotically Quasi-nonexpansive Multivalued Mappings

“…Also, the class of asymptotically quasi-nonexpansive multivalued mapping is a superclass of the classes of asymptotically nonexpansive multivalued mappings and quasi-noexpansive multivalued mappings (Recall that a multivalued mapping V : [34], and subsequently Phuengrattan and Lerkchaiyaphum [33], obtained weak and strong convergence to the common solution of minimzation problem and fixed point problem using the modified version of (1. 19 ) in the setting of real Hilbert spaces More recently, Chang,Wu and Wang [9], using the scheme…”