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Jleli, Mohamed; Samet, Bessem

A new generalization of the Banach contraction principle. (English) Zbl 1322.47052

J. Inequal. Appl. 2014, Paper No. 38, 8 p. (2014).
Denote by \(\Theta\) the class of all functions \(\theta:(0,\infty)\to (1,\infty)\) satisfying
(i)
\(\theta\) is non-decreasing,
(ii)
\(\theta(t_n)\to 1\) if and only if \(t_n\to 0\),
(iii)
\(\exists r\in (0,1)\), \(\exists s\in (0,\infty]\), such that \(\lim_{t\to 0+}(\theta(t)-1)/(t^r)=s\).
The main result is the following.
{ Theorem.} Let \((X,d)\) be a complete Branciari metric space and \(T:X\to X\) be a map. Suppose that there exist a function \(\theta\in \Theta\) and a number \(k\in (0,1)\) such that \[ x,y\in X,\;d(Tx,Ty)> 0\;\Longrightarrow\;\theta(d(Tx,Ty))\leq [\theta(d(x,y))]^k. \] Then \(T\) has a unique fixed point in \(X\).
Some comments involving related statements in the area are also given.
Reviewer: Mihai Turinici (Iaşi)

Cited in 19 Reviews
Cited in 179 Documents

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)

Keywords:

Branciari metric space; Banach contraction; fixed point

Citations:

Zbl 0963.54031
Cite Review PDF
Full Text: DOI

References:

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