A fixed point theorem for cyclic generalized contractions in metric spaces

Maryam A Alghamdi,Adrian Petruşel,Naseer Shahzad

doi:10.1186/1687-1812-2012-122

Maryam A Alghamdi, Adrian Petruşel + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2012-122

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Abstract

Abstract In this paper, we extend a recent result of V. Pata (J. Fixed Point Theory Appl. 10:299-305, 2011) in the frame of a cyclic representation of a complete metric space.

Highlights

One of the fundamental result in fixed point theory is the Banach contraction principle
In terms of Picard operator theory, Banach contraction principle asserts that if f is a contraction and (X, d) is complete, f is a Picard operator. This result has been extended to other important classes of maps
(where ψ : [, ] → [, ∞) is an increasing function vanishing with continuity at zero and x := d(x, x ), with arbitrary x ∈ X), f has a unique fixed point in X

Summary

Introduction

One of the fundamental result in fixed point theory is the Banach contraction principle. In terms of Picard operator theory (see [ ]), Banach contraction principle asserts that if f is a contraction and (X, d) is complete, f is a Picard operator This result has been extended to other important classes of maps. Pata [ ] proved that if (X, d) is a complete metric space and f : X → X is an operator such that there exists fixed constants γ ≥ , α ≥ and β ∈ [ , α] such that, for every ε ∈ [ , ] and every x, y ∈ X, d f (x), f (y) ≤ ( – ε)d(x, y) + γ εαψ(ε) + x + y β (where ψ : [ , ] → [ , ∞) is an increasing function vanishing with continuity at zero and x := d(x, x ), with arbitrary x ∈ X), f has a unique fixed point in X. Srinivasan and Veeramani [ ] obtained an extension of Banach’s fixed point theorem for mappings satisfying cyclical contractive conditions.

Ai are nonempty sets for each i

The case j