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A continuation method for weakly Kannan maps. (English) Zbl 1196.54064

The well-known Banach fixed point theorem asserts that if \(D=X\), \(f\) is contractive and \((X,d)\) is complete, then \(f\) has a unique fixed point \(x\in X\), and for any \(x_0\in X\) the sequence \(\{ T^n(x_0)\}\) converges to \(x\). This result has been extended by several authors to some classes of maps which do not satisfy the contractive condition. For instance, one condition that can be used is the following: there exists \(\alpha\in [0,1)\) such that, for all \(x,y\in X\),
\[ d(f(x),f(y))\leq {\alpha\over 2}\,\left[d(x,f(x))+d(y,f(y))\right]. \]
This condition was introduced by R. Kannan [Bull. Calcutta Math. Soc. 60, 71–76 (1968; Zbl 0209.27104)].
A somewhat different way of generalizing Banach’s theorem was followed by Dugundji and Granas, who extended Banach’s theorem to the class of weakly contractive maps. The concept of weakly contractive map was introduced in [J. Dugundji and A. Granas, Bull. Greek Math. Soc. 19, 141–151 (1978; Zbl 0417.54010)] by replacing the constant \(\alpha\) in the contractive condition by a function \(\alpha=\alpha(x,y)\); we say that \(f:X\rightarrow X\) is weakly contractive if there exists \(\alpha :X\times X\rightarrow [0,1]\), satisfying that \(\theta(a,b):=\sup\{\alpha(x,y): a\leq d(x,y)\leq b\}<1\) for every \(0<a\leq b\), such that, for all \(x,y\in X\), \[ d(f(x),f(y))\leq \alpha(x,y)\,d(x,y). \]
In this interesting paper, the authors introduce the class of weakly Kannan maps and prove that Kannan’s fixed point theorem can be extended to this new class of maps.
The property of having a fixed point is an invariant under homotopy for some classes of nonlinear operators, such as contractive maps and compact maps. Usually, these maps are defined on a subset of a Banach space; the jump from the Banach space setting to the metric space setting was given by A. Granas [Topol. Methods Nonlinear Anal. 3, No. 2, 375–379 (1994; Zbl 0865.54041)], who gave a homotopy result known as a continuation method for contractive maps. M. Frigon [in: Recent advances on metric fixed point theory. Proceedings of the international workshop, Sevilla, Spain, September 25–29, 1995. Sevilla: Univ. de Sevilla, 19–30 (1996; Zbl 0883.47068)] gave a similar result for weakly contractive maps. In the last section of this paper, the authors prove the corresponding result for weakly Kannan mappings.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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References:

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