A Generalization of b-Metric Space and Some Fixed Point Theorems

Kamran, Tayyab; Samreen, Maria; Ain, Qurat Ul

doi:10.3390/math5020019

Cited by 230 publications

(183 citation statements)

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“…By now there already exists considerable literature in b-metric spaces and for the work of this kind one can consult to [2][3][4][5][6][7][8] and similar others. In 2017, Kamran et al [9] introduced a new type of generalized b-metric space and termed it as extended b-metric space.…”

Section: Introductionmentioning

confidence: 99%

Extended Rectangular Mrξ-Metric Spaces and Fixed Point Results

et al. 2019

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In this paper, we enlarge the classes of rectangular M b -metric spaces and extended rectangular b-metric spaces by considering the class of extended rectangular M rξ -metric spaces and utilize the same to prove an analogue of Banach contraction principle in such spaces. We adopt an example to highlight the utility of our main result. Finally, we apply our result to examine the existence and uniqueness of solution for a system of Fredholm integral equation. Mathematics 2019, 7, 1136 2 of 15 class extended M b -metric spaces. Very recently, Asim et al. [18] introduced the class of rectangular M b -metric space and utilized the same to prove an analogue of Banach contraction principle. Soon, Asim et al. [19] generalized the class of rectangular M b -metric spaces by introducing the class of M ν -metric spaces.Inspired by foregoing observations, we introduce the class of extended rectangular M rξ -metric spaces and utilize the same to prove fixed point result in such spaces. We, also furnish an example to establish the genuineness of our newly proved result. Finally, we use our main result to examine the existence and uniqueness of solution for a Fredholm integral equation. Definition 6.[12] Let χ = ∅ and ξ : χ × χ → [1, ∞). A mapping r ξ : χ × χ → R + is said to be an extended rectangular b-metric on χ if, r ξ satisfies the following (for all ς, σ ∈ χ and all distinct ρ, ∈ χ \ {ς, σ}):

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Section: Introductionmentioning

confidence: 99%

Extended Rectangular Mrξ-Metric Spaces and Fixed Point Results

et al. 2019

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“…Mohanta [2] and Husain et al [3] utilized that space to prove existence and uniqueness of common fixed points. Recently, Kamran et al [4] in 2017 generalized b-metric to become extended b-metric for utilizing in fixed point results and Alqahtani et al [5] in 2018 utilized extended bmetric to prove the common fixed point. In 2015, George et al [6] introduced the notion of rectangular b-metric space as a generalization of rectangular metric space.…”

Section: Introductionmentioning

confidence: 99%

“…Let X be a nonempty set. A mappingfor all x, y, s ∈ X. e pair (X, d b ) is called a b-metric space.Definition 2 (see [4]). Let X be a nonempty set.…”

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confidence: 99%

Common Fixed Point Theorems on Generalized Ratio Contraction Mapping in Extended Rectangular b-Metric Spaces

Nurwahyu

2019

International Journal of Mathematics and Mathematical Sciences

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In this paper, we propose and prove the common fixed point theorems on generalized contraction mappings in extended rectangular b-metric spaces by utilizing the weakly compatible function property. PreliminariesIn the following section, we need some definitions to govern our theorems.Definition 1 (see [1]). Let X be a nonempty set. A mappingfor all x, y, s ∈ X. e pair (X, d b ) is called a b-metric space.Definition 2 (see [4]). Let X be a nonempty set. A mappingsatisfies the following conditions:for all x, y, s ∈ X. e pair (X, d b ) is called an extended b-metric space.Definition 3 (see [6]). Let X be a nonempty set. A mapping d b : X × X ⟶ [0, ∞) is called rectangular b-metric, if there exists b ≥ 1 such that d b satisfies the following conditions:(1) d b (x, y) � 0, if and only if x � y

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“…[2,[6][7][8], [11]- [14] and related references therein). Very recently, Kamran et al [18] extend the b-metric space and successfully prove the analog of Banach mapping principle in this new space.…”

Section: Introductionmentioning

confidence: 99%