Common fixed points of Kannan, Chatterjea and Reich type pairs of self-maps in a complete metric space

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Common fixed points of Kannan, Chatterjea and Reich type pairs of self-maps in a complete metric space Pradip Debnath1

· Zoran D. Mitrovi´c2 · Sun Young Cho3

Accepted: 17 October 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020

Abstract In this paper, we establish existence and uniqueness of common fixed points for Kannan, Reich and Chatterjea type pairs of self-maps in a complete metric space. Keywords Common fixed point · Fixed point · Contraction map · Complete metric space Mathematics Subject Classification 47H10 · 54H25 · 54E50

1 Preliminaries In [12], Kannan proved the following theorem. Theorem 1.1 If  is a self-map on a complete metric space (MS, in short) (, ρ) satisfying   ρ(θ, ξ ) ≤ τ ρ(θ, θ ) + ρ(ξ, ξ ) ,

Communicated by H. M. Srivastava.

B

Pradip Debnath [email protected] Zoran D. Mitrovi´c [email protected] Sun Young Cho [email protected]

1

Department of Applied Science and Humanities, Assam University, Silchar, Cachar, Assam 788011, India

2

Faculty of Electrical Engineering, University of Banja Luka , Patre 5, 78 000 Banja Luka, Bosnia and Herzegovina

3

Department of Liberal Arts, Gyeongnam National University of Science and Technology, Jinju-Si 600-758, Gyeongsangnam-do, Korea

123

São Paulo Journal of Mathematical Sciences

where θ, ξ ∈  and 0 < τ < 21 , then  has a unique fixed point in . A mapping satisfying the above condition is known as a Kannan map and a Kannan map is not necessarily continuous. In [13], Kannan proved the above theorem by omitting the completeness of the space and by assuming continuity of the map at a point. Reich [17] generalized Banach and Kannan’s fixed point results as given next. Theorem 1.2 Let (, ρ) be a complete MS and  :  →  be a self-map. Suppose there exist nonnegative constants a, b, c satisfying a + b + c < 1 such that ρ(θ, ξ ) ≤ aρ(θ, ξ ) + bρ(θ, θ ) + cρ(ξ, ξ ) for all θ, ξ ∈ . Then  has a unique fixed point. In Reich’s theorem, b = c = 0 yields Banach’s result, whereas b = c, a = 0 produces Kannan’s theorem. Chatterjea [8] introduced a variation of Kannan’s theorem as follows. Theorem 1.3 If  is a self-map on a complete MS (, ρ) satisfying   ρ(θ, ξ ) ≤ τ ρ(θ, ξ ) + ρ(ξ, θ ) , where θ, ξ ∈  and 0 < τ < 21 , then  has a unique fixed point in . Common fixed point (CFP, in short) of several contractive mappings have been studied over the years [1–5,7,9,14–16,18,19]. Very recently, Debnath and Srivastava have presented new extensions of Kannan’s and Reich’s fixed point theorems using Wardowski’s technique [10] and some common best proximity point results for Kannan-type contractive pairs of mappings [11]. Further, Srivastava et al. [20] studied fixed points of F(ψ, ϕ)-contractions and their application to fractional differential equations. In the current paper, the authors present some CFP results for a pair of Kannan, Reich and Chattarjea type pairs of mappings. The following well known lemma will be used in the sequel. Lemma 1.4 [6] If {θn } is a sequence in a complete