There have been many attempts to generalize the definition of a metric space in order to obtain possibilities for more general fixed point results. In this paper, we give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed point theorems on metric spaces and then investigate this fact in other generalized metric spaces. We show that many generalized metric spaces are topologically equivalent to certain metric spaces or to previously generalized metric spaces. Also, the fixed point theory in these generalized metric spaces may be a consequence of the fixed point theory in certain metric spaces or in previously generalized metric spaces.

Various generalizations of metric spaces and fixed point theorems

Pre-Requisites of Fixed Points.- The Picard Iteration.- The Krasnoselskij Iteration.- The Mann Iteration.- The Ishikawa Iteration.- Other Fixed Point Iteration Procedures.- Stability of Fixed Point Iteration Procedures.- Iterative Solution of Nonlinear Operator Equations.- Error Analysis of Fixed Point Iteration Procedures.

http://www.gbv.de/dms/goettingen/526955376.pdf

Iterative Approximation of Fixed Points

A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice xo E X. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself. 1. Definitions of contractive type mappings. (1) (Banach) There exists a number a, 0 6 a < 1, such that, for each x,y E X, d(f (x),f (y)) < ad(xY)(2) (Rakotch [21]) There exists a monotone decreasing function a: (0, oo) [0, 1) such that, for each x, y E= X, x # y, d(f (x)j(y)) <ad(x,Y)(3) (Edelstein [10]) For each x, y E X, x # y, d(f (x),f (y)) < d(x,y). (4) (Kannan [18]) There exists a number a, 0 < a < !, such that, for each x,y E X, d(f (x),f (y)) < a[d(x,f (x)) + d(y,f (y))]. (5) (Bianchini [3]) There exists a number h, 0 < h < 1, such that, for each x,,y E X, d(f (x),f (y)) < h max{d(x,f (x)), d(y,f (y))} (6) For each x, y E X, x # y, Received by the editors June 4, 1974 and, in revised form, March 5, 1975 and November 18, 1975. AMS (MOS) subject classifications (1970). Primary 47H10. c American Maithematical Society 1977 257 This content downloaded from 157.55.39.17 on Fri, 02 Sep 2016 04:42:04 UTC All use subject to http://about.jstor.org/terms

/pdf/a-comparison-of-various-definitions-of-contractive-mappings-502ily02qf.pdf

A comparison of various definitions of contractive mappings

We introduce the concept of a mixed g -monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We present a new generalization of the Banach contraction principle in the setting of Branciari metric spaces.

/pdf/a-new-generalization-of-the-banach-contraction-principle-5044561ehd.pdf

A new generalization of the Banach contraction principle

Let T: M -. M be a mapping of a metric space (M, d) into itself. A mapping T will be called a quasi-contraction iff d(Tx, Ty) ' q maxld(x, y); d(x, Tx); d(y, Ty); d(x, Ty); d(y, Tx)} for some q < 1 and all x, y E M. In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach's contraction principle. Multi-valued quasi-contractions are also discussed.

/pdf/a-generalization-of-banach-s-contraction-principle-2kizvkkua1.pdf

A generalization of Banach’s contraction principle

Popular Tasks used by Researchers

How can I help with your research?

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

We encountered an error while loading the panel. Please try refreshing the page.

Something went wrong

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

Anyone with the link can view this chat. You can switch it back to private anytime.

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Lj. B. Ćirić

Author Tools

Chat about Author

Papers

A generalization of Banach’s contraction principle