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Some remarks on the generalized Banach contraction principle

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Wesołowski K.

Czasopismo Techniczne

This 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.

Fixed points of α-nonexpansive mappings

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Wesołowski K.

Czasopismo Techniczne

This 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.

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2 Czasopismo Techniczne

2 2016

2 Wesołowski K.

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Some remarks on the generalized Banach contraction principle

Fixed points of α-nonexpansive mappings