Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.

Iterative Algorithms for Nonlinear Operators

well known that many problems in a Hilbert space H are solved by applying the following identity: IlAx + (1 - A)YI12 = a412 + (1 - ~)llYl12 - A(1 - Mx - Yl12 (1.1) for all x, y in H and 0 5 I 5 1. Therefore, one natural method to solve problems in a Banach space Xis to establish equalities and (usually) inequalities in Xanalogous to (1.1). This is in fact shown by some authors’ recent work, see for example, Reich [l], Poffald and Reich [2], Lim [3], Lim et af. [4], Prus and Smarzewski [5], Smarzewski [6], and Reich [21]. The purpose of this paper is to continue carrying out investigations in this line. More precisely, in Section 3, we establish some inequalities analogous to (1 .l) in either uniformly convex or uniformly smooth Banach spaces. As corollaries, we obtain new characteristics of p-uniform convexity and q-uniform smoothness of a Banach space X via the functionals I] . lip and II * llq respectively. We give in Section 4 applications of the inequalities obtained in Section 3 to fixed points of uniformly Lipschitzian mappings and to best approximation theory.

Inequalities in Banach spaces with applications

Several methods for solving systems of equilibrium problems in Hilbert spaces – and for finding best approximations thereof – are presented and their convergence properties are established. The proposed methods include proximal-like block-iterative algorithms for general systems, as well as regularization and splitting algorithms for single equilibrium problems. The problem of constructing approximate equilibria in the case of inconsistent systems is also considered.

Equilibrium programming in Hilbert spaces

well known that many problems in a Hilbert space H are solved by applying the following identity: IlAx + (1 - A)YI12 = a412 + (1 - ~)llYl12 - A(1 - Mx - Yl12 (11) for all x, y in H and 0 5 I 5 1 Therefore, one natural method to solve problems in a Banach space Xis to establish equalities and (usually) inequalities in Xanalogous to (11) This is in fact shown by some authors’ recent work, see for example, Reich [l], Poffald and Reich [2], Lim [3], Lim et af [4], Prus and Smarzewski [5], Smarzewski [6], and Reich [21] The purpose of this paper is to continue carrying out investigations in this line More precisely, in Section 3, we establish some inequalities analogous to (1 l) in either uniformly convex or uniformly smooth Banach spaces As corollaries, we obtain new characteristics of p-uniform convexity and q-uniform smoothness of a Banach space X via the functionals I]  lip and II * llq respectively We give in Section 4 applications of the inequalities obtained in Section 3 to fixed points of uniformly Lipschitzian mappings and to best approximation theory

Inequalities in banach spaces with applications

This paper presents a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space. The algorithmic solution is named the hybrid steepest descent method, because it is constructed by blending important ideas in the steepest descent method and in the fixed point theory, and generates a sequence converging strongly to the solution of the problem. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems.

Kazimierz Goebel

Papers

Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings

Iterating holomorphic self-mappings of the Hilbert ball