Fixed point theorems for weakly contractive multivalued maps

By Jong-Sook Bae. 2003

76 citations recorded by the source corpus.

Recently Lim [9] proved that every weakly inward multivalued contraction with clo values defined on a nonempty closed subset of a Banach space has a fixed point. H includes results of Deimling [7], Xu [15], Martinez-Yanez [10], and Yi and Zhao [14] prove the result, Lim used transfinite induction argument instead of Caristi’s theore In [9], the author says that he had attempted unsuccessfully to use Caristi’s theorem One of generalizations of contractions on metric spaces is the concept of weakl tractive maps appeared in [1]. Actually in [1], the authors defined such maps for s valued maps on Hilbert spaces and proved the existence of fixed points. Rhoad showed that most results of [1] are still true for any Banach spaces. In fact, weakly tractive maps are closely related to maps of Boyd and Wong type ones [4] and Reich ones [12]. Naturally we can extend this concept to multivalued maps. The purpose of this paper is to define weakly contractive multivalued maps and various fixed point theorems for weakly contractive multivalued maps with inward (weakly inwardness) conditions. Our proofs are based on Caristi’s theorem and consequently we give an anothe of Lim’s theorem [9] by using Caristi’s theorem directly. In Section 2, we give some variations of Caristi’s theorem in order to use provin existence of weakly contractive multivalued maps with inwardness or weakly inwar conditions. In Section 3, we prove fixed point theorems for such maps.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0109a9ee4fd7de31fa2054f8cf429598a58d8efb

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

FIXED POINT THEOREMS FOR GENERALIZED WEAKLY CONTRACTIVE MAPPINGS (2009, background): “Rhoades proved that most of the results in [1] hold in a Banach space setting, and Bae considered these type of multi-valued mappings (cf. [6]).”
Common fixed points for Ćirić type $f$-weak contraction with applications (2010, background): “Therefore f -weakly contractive maps are closely related to maps of Reich [30] type, which are also generally researched by Bae [4] and Mizoguchi and Takahashi [23].”
Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory (2008, unknown): “2 in [3], Theorem (2.”