"Equation missing"-commuting maps and invariant approximations

Abstract

We obtain common fixed point results for generalized I-nonexpansive C q -commuting maps. As applications, various best approximation results for this class of maps are derived in the setup of certain metrizable topological vector spaces.

On common fixed point and approximation results of Gregus type

“…, then the maps are called 1-subcommuting [7]; (5) R-subweakly commuting on M (see [8,9]) if for all x ∈ M and for all p α ∈ A * (τ ), there exists a real number R > 0 such that…”

Comments on the Papers Arch. Math. Brno, 422006, 51-58 - Thai J. Math., 32005, 63-70 and Math. Communications 132008, 85-96

“…The measurability of T n is still an open problem(see [2,13] and references therein). Thus all the results, Theorems 3.1-3.3 in [17], are deterministic in nature and hence are simple corollaries to more general results in [5,6,10,11].…”

“…(f ) For more general and comprehensive results for noncommuting maps namely, R-subweakly commuting, R-subcommuting and C q -commuting maps defined on the set M satisfying the Dotsonś convexity condition (or the so called (S)-convex structure), we refer the reader to [5,6,10,11].…”

“…Clearly, commuting maps are R-subweakly commuting, R-subweakly commuting maps are R-subcommuting and R-subcommuting maps are C q -commuting but the converse, in each case, does not hold in general (see [8,11] and references therein).…”

A Result in Best Approximation Theory for Family of Maps

“…By introducing the concept of noncommuting maps to this area Shahzad [15], Al-Thagafi and Shahzad [2,3], Hussain and Jungck [8], Hussain [7], Hussain and Rhoades [9], Jungck and Hussain [10], O'Regan and Hussain [12] and Pathak and Hussain [13] extended and improved the above mentioned results to noncommuting maps such as pointwise Rsubweakly commuting maps, compatible maps, -commuting maps and Banach operator pairs.…”