Fixed point results of a new family of hybrid contractions in generalised metric space with applications

Abstract

<abstract><p>In this manuscript, a novel general family of contraction, called hybrid-interpolative Reich-Istr$ \breve{a}ţ $escu-type $ (G $-$ \alpha $-$ \mu) $-contraction is introduced and some fixed point results in generalised metric space that are not deducible from their akin in metric spaces are obtained. The preeminence of this class of contraction is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, a number of corollaries that reduce our result to other well-known results in the literature are highlighted and analysed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution of a family of integral equations.</p></abstract>

Advancements in Fixed Point Results of Generalized Metric Spaces: A Survey

“…Jiddah et al [71] obtained some fixed point results of a new family of hybrid contractions in G-metric space which complement some well-known contractions, including that of Reich and Istrȃt ¸escu. Their results cannot be reduced to any existence result in the manner of Jleli and samet [38] or Samet et al [39].…”

“…Definition 116. [71] Let (X, G) be a G-metric space and let α : X × X −→ R + be a function. A self-mapping T : X −→ X is called hybrid-interpolative Reich-Istrȃt ¸escutype (G-α-µ)-contraction if for some q ∈ [0, ∞), there exist constants µ ∈ (0, 1), δ ≥ 0 and λ i ≥ 0 with i = 1, 2, .…”

“…Theorem 125. [71] Let (X, G) be a complete G-metric space and let T : X −→ X be a hybrid-interpolative Reich-Istrȃt ¸escu-type (G-α-µ)-contraction satisfying the following conditions:…”

“…Theorem 127. [71] If in addition to the hypotheses of Theorem 126, we assume supplementary that α(x, y, Ty) ≥ 1 for any x, y ∈ Fix(T ), then the fixed point of T is unique.…”

Hybrid fixed point theorems of graphic contractions with applications

“…The chief role of hybrid contractions is that they allow for the presentation of contractive conditions involving a substantial amount of terms, including certain ones with self-composition of the mapping, while also admitting a number of parameters, allowing for extensions in various ways based on the parameters picked. We refer to [6] , [15] , [20] , [27] , [29] , [33] and the references therein for various improvements on this matter. In accordance with the existing findings, we notice that hybrid Fp concepts in MS equipped with graph have not been exhaustively studied.…”