Abstract
Very recently, Fang (Fuzzy Sets Syst. 267:86-99, 2015) gave some fixed point theorems for probabilistic φ-contractions in Menger spaces. Fang’s results improve the one of Jachymski (Nonlinear Anal. 73:2199-2203, 2010) by relaxing the restriction on the gauge function φ. In this paper, inspired by the results of Fang, we prove a new fixed point theorem for a probabilistic φ-contraction in Menger spaces in which a weaker condition on the function φ is required. Our result improves the corresponding one of Fang and some others. Finally, an example is given to illustrate our result.
Highlights
Let (X, F, Δ) be a probabilistic metric space and T : X → X be a mapping
Many authors investigated fixed point theorems for probabilistic φ-contractions in Menger spaces; see [ – ]
We give a positive answer to the question of Fang by proving a new fixed point theorem for a probabilistic φ-contraction in Menger spaces
Summary
Let (X, F, Δ) be a probabilistic metric space and T : X → X be a mapping. If there exists a gauge function φ : R+ → R+ such thatFTx,Ty φ(t) ≥ Fx,y(t) for all x, y ∈ X and t > , the mapping T is called a probabilistic φ-contraction. Many authors investigated fixed point theorems for probabilistic φ-contractions in Menger spaces; see [ – ]. On the fixed point theorems for other types of contractions in Menger or fuzzy metric spaces, please see [ – ].
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