Quasi-contractions restricted with linear bounded mappings in cone metric spaces

Zhilong Li,Shujun Jiang

doi:10.1186/1687-1812-2014-87

Abstract

In this paper, we introduce the notion of a quasi-contraction restricted with a linear bounded mapping in cone metric spaces, and prove a unique fixed point theorem for this quasi-contraction without the normality of the cone. It is worth mentioning that the main result in this paper could not be derived from ´ Ciri´ c’s result by the scalarization method, and hence indeed improves many recent results in cone metric spaces. MSC: 06A07; 47H10

Highlights

Let (X, d) be a complete metric space
We introduce the notion of a quasi-contractions restricted with a linear bounded mapping in cone metric spaces
Without using the normality of P, we show the unique existence of fixed point of quasi-contractions restricted with linear bounded mappings at the expense that (H) is satisfied

Summary

Introduction

Let (X, d) be a complete metric space. Recall that a mapping T : X → X is called a quasicontraction, if there exists λ ∈ ( , ) such that d(Tx, Ty) ≤ λ max d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) , ∀x, y ∈ X.Ćirić [ ] introduced and studied Ćirić’s quasi-contractions as one of the most general classes of contractive-type mappings. Without using the normality of the cone, we prove the unique existence of fixed point for this quasi-contraction at the expense of un →w θ ⇒ Aun →w θ , ∀{un} ⊂ P. It is not hard to conclude that P is a non-normal cone of a normed vector space (E, · ) if and only if there exist sequences {un}, {vn} ⊂ P such that un + vn →· θ un →· θ , which implies that the Sandwich theorem does not hold.

Results

Conclusion