Common fixed points via asymptotic contraction and application to matrix equations
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Common fixed points via asymptotic contraction and application to matrix equations Ashis Bera1 · Lakshmi Kanta Dey1 · Hiranmoy Garai1 · Sayandeepa Raha2 Received: 21 July 2020 / Revised: 23 September 2020 / Accepted: 9 October 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we employ the notion asymptotic contraction of Kirk (J Math Anal Appl 277:645–650, 2003) to obtain some common fixed point results of a pair of mappings and a finite family of mappings. To execute this, we introduce the notion of asymptotic contractions in pair for a finite family of mappings. Then, we obtain our prescribed common fixed point results using these asymptotic contractions in pair concepts. After this, we apply these results to analyze the unique common solution to a certain kind of pair of matrix equations. Finally, we provide some numerical examples to authenticate our results using figurative presentations. Keywords Asymptotic contration · Common fixed point · Matrix equation · Positive definite matrix Mathematics Subject Classification 47H10 · 54A20
1 Introduction In the last hundred years, many mathematicians have achieved a numerous number of contractions, and they have utilized these contractions to obtain several important fixed points, coincidence points, and common fixed points results. Among all these existing contractions,
Communicated by Jinyun Yuan.
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Lakshmi Kanta Dey [email protected] Ashis Bera [email protected] Hiranmoy Garai [email protected] Sayandeepa Raha [email protected]
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Department of Mathematics, National Institute of Technology, Durgapur, India
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Department of Biotechnology, National Institute of Technology, Durgapur, India 0123456789().: V,-vol
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some are very interesting and easy to apply in different applicable areas. The asymptotic contraction due to Kirk (Kirk 2003) is one of such contractions. Definition 1.1 A self-map T defined on a metric space (X , d) is called an asymptotic contraction if for all n ∈ N: d(T n (x), T n (y)) ≤ ϕn (d(x, y)) for all x, y ∈ X , where ϕn : [0, ∞) → [0, ∞) are functions, such that ϕn → ϕ ∈ uniformly on the range of d, is the collection of all functions ϕ : [0, ∞) → [0, ∞) satisfying the following: (i) ϕ is continuous; (ii) ϕ(t) < t for all t > 0. Due to the simplicity of the asymptotic contraction, many renowned mathematicians have studied this contraction in different ways and different structures to obtain a number of impor¯ tant and fascinating fixed point results (see Arandelovi´ c 2007; Razani et al. 2007; Kirk and Xu 2008; Singh et al. 2011). It is well known that in the literature of fixed point theory, besides the fixed point results, there are several interesting common fixed point results involving different contraction conditions. Most of these contractions involving common fixed points are basically obtained by modifying some specific contractions which are previously used to obtain fixed point results. Therefore, it will be
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