Common fixed-points for Banach operator pairs in best approximation

Abstract

The notion of Banach operator pairs is introduced, as a new class of noncommuting maps. Some common fixed-point theorems for Banach operator pairs and the existence of the common fixed-points of best approximation are presented. These results are proved without the assumption of linearity or affinity for either f or g, which shows that the concept about Banach operator pairs is potentially useful in the study of common fixed-points.

Common fixed point and best approximation for Banach operator pairs in non-starshaped domain

“…Before proving the main results, a lemma is presented below, which extends and improves Lemma 3.1 of [3]: Lemma 3.1. Let M be a nonempty closed subset of a metric space (X , d), and (T , I) be Banach operator pair on M. Assume that clT (M) is complete, and T and I satisfy for all x, y ∈ M and 0 ≤ h < 1,…”

“…Theorem 3.1-Theorem 3.11 generalize Theorem 3.2-Theorem 4.2 in [3] by relaxing the starshaped condition of domain M or D and F (I), and using more generalized relatively nonexpansive mappings.…”

“…Further, definition providing the notion of Banach operator pair introduced by Chen and Li [3] may be written as: Definition 2.1. Banach Operator Pair.…”

“…The ordered pair (T , I) of two self-maps of a metric space (X , d) is called a Banach operator pair, if the set F (I) is T -invariant, namely T (F (I)) ⊆ F (I). Obviously commuting pair (T , I) is Banach operator pair but not conversely in general, see [3]. If (T , I) is Banach operator pair then (I, T ) need not be Banach operator pair (see [3,Example 1]).…”

“…Obviously commuting pair (T , I) is Banach operator pair but not conversely in general, see [3]. If (T , I) is Banach operator pair then (I, T ) need not be Banach operator pair (see [3,Example 1]). …”