Abstract
The symmetry concept is an intrinsic property of metric spaces as the metric function generalizes the notion of distance between two points. There are several remarkable results in science in connection with symmetry principles that can be proved using fixed point arguments. Therefore, fixed point theory and symmetry principles bear significant correlation between them. In this paper, we introduce the new definition of the eventually Δ -restrictive set-valued map together with the concept of p-orbital continuity. Further, we introduce another new concept called the Δ ( ϵ ) -restrictive set-valued map. We establish several fixed point results related to these maps and proofs of these results also provide us with schemes to find a fixed point. In a couple of results, the stronger condition of compactness of the underlying metric space is assumed. Some results are illustrated with examples.
Highlights
Inspired by the work of Edelstein on fixed and periodic points, in the current paper we introduce the new notions of eventually ∆-restrictive and ∆(e)-restrictive set-valued maps in metric spaces
The main motivation of the present study is to provide an alternative approach for the investigation of fixed points of certain class of set-valued mappings by developing some new hypothesis which we call restrictive conditions
We study fixed points of certain set-valued maps by introducing new conditions with respect to the function ∆ in an metric space (MS)
Summary
A set-valued map R : X → Γ( X ) is said to be P H-continuous at a point μ0 , if for each sequence {μn } ⊂ X, such that limn→∞ δ(μn , μ0 ) = 0, we have limn→∞ P H( Rμn , Rμ0 ) = 0 (i.e., if μn → μ0 , Rμn → Rμ0 as n → ∞). We first define an eventually ∆-restrictive set-valued map and introduce the notion of p-orbital continuity.
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