Krzysztof Wesołowski
Czasopismo Techniczne, Nauki Podstawowe Zeszyt 1-NP (19) 2015, 2015, s. 5-14
https://doi.org/10.4467/2353737XCT.15.111.4148This paper is connected with the theory of a-nonexpansive mappings, which were introduced by K. Goebel and M. A. J. Pineda in 2007. These mappings are a natural generalisation of nonexpansive mappings from the point of view of the fixed point theory. In particular, they proved that in Banach spaces all α = ( α1,..., αn) -nonexpansive mappings with α1 big enough, namely α1≥2 (1/1-n), have minimal displacement equal to zero. This paper introduces some new results connected with this problem.
Krzysztof Wesołowski
Czasopismo Techniczne, Nauki Podstawowe Zeszyt 1-NP (19) 2015, 2015, s. 15-24
https://doi.org/10.4467/2353737XCT.15.112.4149This paper presents some results concerning the Generalized Banach Contraction Principle: In a complete metric space X if for some N ≥ 1 and 0 < M < 1 the mapping T : X → X satisfies min{d(Tj x,Tj y), 1 ≤ j ≤ N} ≤ Md(x, y) for any x, y ∈ X , then T has a unique fixed point. In some special cases, the above constant M can be replaced by a continuous, non- -increasing function 0 ≤ φ (d(x, y)) ≤ 1 such that φ (t) =1 if, and only if, t = 0.