Abstract
We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type yields the fixed point and endpoint existence theorem for these contractive maps.MSC:47H10, 54C60, 47H09, 54E15, 46A03, 54E50, 46B40.
Highlights
We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances, and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type [ , Th
Nadler [ ] extended Banach’s fixed point theorem [ ] for set-valued maps in complete metric spaces.Theorem . ([, Th. ]) Let (X, d) be a complete metric space, let Cl(X) denote the class of all nonempty closed subsets of X, and let H : (Cl(X)) → [, ∞] be defined by∀A,B∈Cl(X){H(A, B) = max sup d(u, B), sup d(v, A), u∈A v∈B where, for each u ∈ X and V ∈ Cl(X), d(u, V ) = infv∈V d(u, v)
A number of authors introduce the new concepts of set-valued contractions of Nadler type and study the problem concerning the existence of fixed points for such contractions; see, e.g., Aubin and Siegel [ ], de Blasi et al [ ], Ćirić [ ], Eldred et al [ ], Feng and Liu [ ], Frigon [ ], Al-Homidan et al [ ], Jachymski [ ], Kaneko [ ], Klim and Wardowski [ ], Latif and Al-Mezel [ ], Mizoguchi and Takahashi [ ], Pathak and Shahzad [ ], Quantina and Kamran [ ], Reich [, ], Reich and Zaslavski [, ], Sintunavarat and Kumam [ – ], Suzuki [ ], Suzuki and Takahashi [ ], Takahashi [ ] and Zhong et al [ ]
Summary
We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances, and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type [ , Th.
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