Approximate Solutions of Common Fixed-Point Problems

Abstract

Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences.The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches.More information about this series at

The Combination Projection Method for Solving Convex Feasibility Problems

“…x, x k j − x k j +1 → 0, and {c l } m l=1 are w-lsc and bounded uniformly continuous, we can obtain c i (x) ≤ 0 by taking the limit in (15) as j → ∞. Hencex ∈ C and ω w (x k ) ⊂ C. This completes the proof.…”

“…which is very common problem in diverse areas of mathematics and physical sciences [15]. In the last twenty years, there has been growing interests in the CFP since it was found to have various applications in imaging science [16,17], medical treatments [18], and statistics [19].…”

Three extensions of Butnariu-Reich-Zaslavski theorem for inexact infinite products of nonexpansive mappings

“…It should be mentioned that in our proofs assumptions (A), (A3), (A4), (a), (3.3), (3.17) and (3.39) play a crucial role. Mappings for which assumption (A) holds are considered in [25,28]. Assumptions (A3), (A4) and (a) usually hold if the set K is bounded and the space X is Hilbert.…”

“…During more than fifty-five years now, there has been a lot of activity regarding the fixed point theory of nonexpansive (that is, 1-Lipschitz) mappings. See, for example, [3,5,11,13,14,17,18,19,21,22,23,24,25,28,29] and the references cited therein. This activity stems from Banach's classical theorem [1] concerning the existence of a unique fixed point for a strict contraction.…”

“…It also covers the convergence of (inexact) iterates of a nonexpansive mapping to one of its fixed points. Since that seminal result, many developments have taken place in this field including, in particular, studies of feasibility and common fixed point problems, which find important applications in engineering and medical sciences [8,12,15,26,27,28,29].…”