Abstract
The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature.
Highlights
Introduction and PreliminariesLet ðZ, dÞ be a metric space
The symbol fix Ω = fπ ∈ Z : x ∈ Ωπg denotes the fixed point set of Ω
By getting inspiration from this, in this paper, we prove fixed point results for contractive conditions involving functions F, not necessarily continuous and belongs to ΔðϜ Þ by taking support of a continuous function from P
Summary
Let ðZ, dÞ be a metric space (in short MS). The set of all nonempty subsets of Z is denoted by PðZÞ, the set of all nonempty closed subsets of Z is denoted by CLðZÞ, the set of all nonempty closed and bounded subsets of Z is denoted by CBðZÞ, and the set of all nonempty compact subsets of X is denoted by KðZÞ. Let ðZ, dÞ be a complete MS and Ω : Z ⟶ KðZÞ be a multivalued F -contraction, and Ω has a fixed point in Z. Assume that there exists χ ∈ Φ, a nondecreasing real valued function F1 on ð0, ∞Þ and a real valued function F2 on ð0, ∞Þ satisfying condition ðϜ 2′Þ and ðϜ 3Þ such that (N1) and the following condition holds: HðΩπ, ΩωÞÞ > 0 implies χðdðπ, ωÞÞ + F2ðHðΩπ, ΩωÞÞ. Assume that there exist χ ∈ Φ, a non decreasing real valued function F1 on ð0, ∞Þ and a real valued function F2 on ð0, ∞Þ satisfying condition ðϜ 2′Þ and ðϜ 3Þ such that (N1) and the following condition holds: HðΩπ, ΩωÞÞ > 0 implies χðdðπ, ωÞÞ + F2ðHðΩπ, ΩωÞÞ. It implies nk + 1 < mk < mk + 1 for all k: ð60Þ
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