Abstract
The symmetry concept is a congenital characteristic of the metric function. In this paper, our primary aim is to study the fixed points of a broad category of set-valued maps which may include discontinuous maps as well. To achieve this objective, we newly extend the notions of orbitally continuous and asymptotically regular mappings in the set-valued context. We introduce two new contractive inequalities one of which is of Geraghty-type and the other is of Boyd and Wong-type. We proved two new existence of fixed point results corresponding to those inequalities.
Highlights
A great deal of information about recent developments in fixed point theory of single and set-valued maps may be found in the monographs by Kirk and Shahzad [8] and Pathak [9]
Most of the contractive conditions existing in literature produce fixed points but they force the map under consideration to be continuous as well
Let Ω denote the family of functions ψ : R+ → R+ satisfying the following conditions: 1
Summary
In 1968, Markin [1] extended Browder’s fixed point theorem to its set-valued counterpart, whereas, in 1969, Nadler [2] proved the set-valued version of Banach’s contraction principle with the help of the Hausdorff metric. Jleli et al [4] have studied existence of fixed points for multi-valued maps under some. 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 ) =. Orbital sequence is one of the important components in the investigation of fixed points for set-valued maps (see [13,14]). Most of the contractive conditions existing in literature produce fixed points but they force the map under consideration to be continuous as well.
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