A proof of the Generalized Banach Contraction Conjecture

Arvanitakis, Alexander D.

doi:10.1090/s0002-9939-03-06937-5

Cited by 61 publications

(30 citation statements)

References 6 publications

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“…This principle has been generalized in many ways over the years [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Coupled best proximity point theorem in metric Spaces

Sintunavarat

2012

Fixed Point Theory Appl

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In this article the concept of coupled best proximity point and cyclic contraction pair are introduced and then we study the existence and convergence of these points in metric spaces. We also establish new results on the existence and convergence in a uniformly convex Banach spaces. Furthermore, we give new results of coupled fixed points in metric spaces and give some illustrative examples. An open problems are also given at the end for further investigation.

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“…This principle has been generalized in many ways over the years [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Coupled best proximity point theorem in metric Spaces

Sintunavarat

2012

Fixed Point Theory Appl

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“…At about the same time as the last of these papers appeared, a different approach was discovered by Arvanitakis, and has now been published in [1]; this is also highly combinatorial.…”

Section: Introductionmentioning

confidence: 97%

On Contractive Families and a Fixed‐Point Question of Stein

Austin

2005

Mathematika

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The following conjecture generalizing the Contraction Mapping Theorem was made by Stein in [9]:Let (X, ρ) be a complete metric space and let F = {T 1 , . . . , T n } be a finite family of self-maps of X. Suppose there is a constant γ ∈ (0, 1) such that for any x, y ∈ X there exists T ∈ F with ρ(T (x), T (y)) ≤ γρ(x, y). Then some composition of members of F has a fixed point.In this paper we disprove this conjecture. We also show that it does hold for a (continuous) commuting F in the case n = 2. We conjecture that it holds for commuting F for any n.

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“…, N 0 }} ≤ φ(d(x, y)) for all x, y ∈ X a Jachymski-Schröder-Stein contraction (with respect to φ). Such mappings with φ(t) = γt for some γ ∈ (0, 1) have recently been of considerable interest [1,[7][8][9][10][11]. In the present paper we study general Jachymski-Schröder-Stein contractions and prove two fixed point theorems for them (Theorems 2.1 and 3.1 below).…”

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confidence: 93%