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Fixed points of the contractive or expansive type for multivalued mappings: toward a unified approach. (English) Zbl 1104.54019

Several common fixed point theorems for multivalued maps are proved under weak contraction or expansion conditions. Many authors dealing with the common fixed point problem replaced the contraction constant \(\alpha\in (0,1)\) in the Banach contraction principle by a function \(\phi:\mathbb R^+\to\mathbb R^+\) satisfying suitable monotonicity and continuity assumptions and such that \(\phi(t)<t\) for every \(t>0\). Following two previous papers of the author: J. Rodríguez-Montes and J. A. Charris [Int. J. Appl. Math. 7, No. 2, 121–138 (2001; Zbl 1030.54034)] and [Southwest J. Pure Appl. Math. 2001, No. 1, 93–101 (2001; Zbl 0985.54036)], a weaker assumption on \(\phi\) is proposed and applied to the case of multivalued maps. Namely, the author assumes that: For any strictly decreasing sequence \((t_n)\) in \(\mathbb R^+\) such that \(\lim t_n=\lim \phi(t_n)=t\), it follows that \(t=0\). This type of assumption allows the author to study, using a similar approach, multivalued expanding maps.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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