Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space
- PDF / 272,309 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 39 Downloads / 167 Views
(0123456789().,-volV) (0123456789().,-volV)
ORIGINAL RESEARCH PAPER
Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space Shobha Jain1 • Shishir Jain1 Received: 8 May 2020 / Accepted: 31 August 2020 Ó Forum D’Analystes, Chennai 2020
Abstract The aim of this paper is to introduce fuzzy generalised weak contractive condition for a pair of self maps in a fuzzy metric space,which is in accordance with the theory of metric spaces given by Rhoades (Nonlinear Analysis 47:2683–2693, 2001). Our results generalize existing fuzzy contraction (by Gregori and Sapena (Fuzzy Sets System 125:245–252, 2002), which is for only one self map. Using this, we establish a unique common fixed point theorem for two self-maps through weak compatibility. The article includes an example, in support of our results. Also an application we established, for the existence and uniqueness of a solution of Fredholm non-linear integral equation in the setting of fuzzy metric space. Keywords Fuzzy metric space t-norm M-cauchy sequence common fixed points fuzzy generalized weak contraction weak compatibility.
Mathematics Subject Classification 47H10 54H25
1 Introduction In 1922, Stefan Banach [3], a Polish mathematician gave a fundamental result in fixed point theory, known as Banach contraction principle, in the setting of metric spaces which becomes a milestone in developing fixed point theory. Since then a lot of efforts to generalize this work has been made, in [1, 4, 7, 9–13, 18, 19] in various & Shobha Jain [email protected] Shishir Jain [email protected] 1
Shri Vaishnav Vidyapeeth Vishwavidyalaya, Indore, Madhya Pradesh, India
123
S. Jain, S. Jain
spaces. Weak contraction principle is one of these generalization of Banach’s contraction principle given by Alber and Guerre-Delabriere in Hilbert space in [2]. In [14], Rhoades subsequently extended the results to metric spaces with the help of a function /. He defines a weakly contractive map as follows: A self-map T in a metric space (X, d) is called weakly contractive, if it satisfies the condition dðTx; TyÞ dðx; yÞ /ðdðx; yÞÞ; 8x; y 2 X;
ð1Þ
for some continuous, non-decreasing function / : ½0; 1Þ ! ½0; 1Þ such that /ðtÞ [ 0 and /ð0Þ ¼ 0: From the definition, it is clear that weakly contractive maps lie between those which satisfy Banach’s contraction principle and contractive maps. Contractive mappings in fuzzy metric spaces have been studied by various authors (see e.g., [1, 5, 7–13, 17–20]) and they used it in establishing some fixed point theorems in fuzzy metric space in the sense of George and Veeramani. In fuzzy metric space Abbas et al. [1] introduced the notion of W-weak contraction, Gopal et al. [7] introduced (/-w) contraction and Vetro et al. [19] introduced cyclic weak / contraction and proved fixed point results using G-completeness of the space. In this paper we introduce a generalized weak contraction for two self map and we prove the existence and uniqueness of common fixed poin
Data Loading...
