Cyclic quasi-contractions of Ciric type in b-metric spaces

Abstract

In this paper, we give a negative

Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property

“…Recently, Amini-Harandi [4] proved a generalization ofĆirić fixed point theorem in the setting of b-metric spaces with Fatou property. Later, He et al [8] and Zhao et al [12] respectively improved the results of Amini-Harandi without Fatou property assumption.…”

“…Theorem 2 (see [8,12]). Let (W, ρ) be a complete b-metric space with s ≥ 1, T : W → W be aĆirić type contraction with contraction constant λ ∈ 0, 1 s .…”

“…Later, in 1998, Czerwik improved and generalized this notion in [3] from the constant s = 2 to a constant s ≥ 1. In the last years, a lot of fixed point results in the framework of b-metric space were studied by many authors, see e.g., [4][5][6][7][8][9][10][11][12] and references therein.…”

“…Clearly,Ćirić fixed point theorem is an actually generalization of the Banach contraction principle [15], Kannan's fixed point theorem [16] and Chatterjea's fixed point theorem [17]. Due to the importance and application potential to quantitative sciences, the generalizations ofĆirić fixed point theorem have been investigated heavily by many authors in various distinct directions over the past 20 years; see, e.g., [4,5,8,12,[18][19][20][21] and the related references therein. Recently, Amini-Harandi [4] proved a generalization ofĆirić fixed point theorem in the setting of b-metric spaces with Fatou property.…”

“…In fact, the ranges of the contraction constants are almost limited to [0, 1 s ) used in all known fixed point results forĆirić type contractions in the setting of b-metric spaces with s ≥ 1; see, e.g., [4,8,9,12,13]. In [6], Dung and Hang successfully generalized the Banach contraction principle from metric spaces to b-metric spaces with contraction constant λ ∈ [0, 1).…”