Abstract
The metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan’s fixed point theorem to the case of multivalued maps using Wardowski’s F-contraction. We show that our result is applicable to a class of mappings where neither the multivalued version of Kannan’s theorem nor that of Wardowski’s can be applied to determine the existence of fixed points. Application of our result to the solution of integral equations has been provided. A multivalued Reich type generalized version of the result is also established.
Highlights
Introduction and PreliminariesKannan [1] generalized the Banach contraction principle in the following manner which assured that even certain discontinuous functions might possess fixed points.Theorem 1. [1] Let (=, ζ ) be a complete metric space
Nadler [5] started the research on fixed points for multivalued maps with the help of Hausdorff concept, i.e., by considering the distance between two arbitrary sets in the following manner
First we introduce a proper generalization of Kannan’s theorem for multivalued maps via F-contraction and further introduce a Reich-type generalization of the same
Summary
Kannan [1] generalized the Banach contraction principle in the following manner which assured that even certain discontinuous functions might possess fixed points. Nadler [5] started the research on fixed points for multivalued maps with the help of Hausdorff concept, i.e., by considering the distance between two arbitrary sets in the following manner. [8] A multivalued map Γ : = → Cl (=) (where Cl (=) is the family of nonempty closed subsets of =) is called a Reich-type multivalued (l, m, n)-contraction if there are constants l, m, n ∈ R+ satisfying l + m + n < 1 such that. It was proved in [8] that a Reich-type multivalued (l, m, n)-contraction in a complete MS possesses a fixed point. First we introduce a proper generalization of Kannan’s theorem for multivalued maps via F-contraction and further introduce a Reich-type generalization of the same. We present an application of our multivalued Kannan-type F-contraction to the solution of integral equations
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