Cyclic Contractions and Related Fixed Point Theorems on G-Metric Spaces

Abstract

Very recently, Jleli and Samet [53] and Samet et. al. [52] reported that some fixed point result in G-metric spaces can be derived from the fixed point theorems in the setting of usual metric space. In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings in the context of G-metric spaces that can not be obtained by usual fixed point results via techniques used in [53, 52]. We also give an example to illustrate our statements.

Related fixed point results for cyclic contractions on G-metric spaces and application

“…Otherwise, assume that T n x 0 T n+1 x 0 for all n ≥ 0. By (11) in the proof of Theorem 2.1, we know that…”

“…In 2012, Karapınar [19] obtained a unique fixed point of cyclic weak φcontraction mappings and studied well-posedness problem for such mappings (for other results on cyclic contractions, see also [2], [3], [13], [15], [34] and [44]). Very recently, Bilgili et al [10,11] presented some new fixed point results involving cyclic contractions in the setting of G-metric spaces where two variables x and y in the space X are considered.…”

On Fixed Point Theorem of C Class Functions - B Weak Cyclic Mappings

“…Definition 2 (Bilgili et al, 2014) The element Definition 3 (Jleli et al, 2014) B is U-cyclic Φ-weak contraction on X if and only if there are A∈U and a continuous function ϕ∈Φ satisfying the two conditions:…”

“…Bilgili et al (2014;Karapinar and Sadarangani, 2012;Du and Karapinar, 2013) discussed the concept of Φ-weakly cyclic contraction mappings and proved fixed point theorems for mappings on Banach spaces. While Harjani et al (2013) considered cyclic weak Φ-contraction on compact metric spaces, the considered mapping φ need not be continuous, another fixed point treating the concept is given in (Karapinar and Sadarangani, 2012).…”

Fixed point of single-valued cyclic weak φF-contraction mappings

“…Then, T has a unique fixed point z ∈ m i=1 A i . There are another results on the existence of fixed point for cyclic mappings, see for example [3], [7], [8], [9] and [10].…”