Table Of ContentFixed Point Results in
W-Distance Spaces
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Fixed Point Results in W-Distance Spaces
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Fixed Point Results in
W-Distance Spaces
Vladimir Rakočević
Faculty of Sciences and Mathematics
University of Niš, Serbia
First edition published 2022
by CRC Press
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and by CRC Press
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© 2022 Vladimir Rakočević
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Library of Congress Cataloging-in-Publication Data
Names: Rakočević, Vladimir, author.
Title: Fixed point results in W-distance spaces / Vladimir Rakočević,
Faculty of Sciences and Mathematics, University of Niš, Serbia.
Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2022.
| Series: Chapman & Hall/CRC monographs and research notes in
mathematics | Includes bibliographical references and index.
Identifiers: LCCN 2021030274 (print) | LCCN 2021030275 (ebook) | ISBN
9781032081465 (hardback) | ISBN 9781032100586 (paperback) | ISBN
9781003213444 (ebook)
Subjects: LCSH: Fixed point theory. | Metric spaces.
Classification: LCC QA329.9 .R35 2022 (print) | LCC QA329.9 (ebook) | DDC
515/.7248--dc23
LC record available at https://lccn.loc.gov/2021030274
LC ebook record available at https://lccn.loc.gov/2021030275
ISBN: 9781032081465 (hbk)
ISBN: 9781032100586 (pbk)
ISBN: 9781003213444 (ebk)
DOI: 10.1201/9781003213444
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface vii
Author xi
Symbol Description xiii
1 Introduction 1
1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Banach contraction principle . . . . . . . . . . . . . . . . . . 5
1.3 Kannan contraction . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 C´iri´c quasicontraction . . . . . . . . . . . . . . . . . . . . . . 9
2 Some basic properties of W-distances 13
2.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . 13
2.2 Basic properties of W-distances . . . . . . . . . . . . . . . . 17
2.3 More results on W-distances . . . . . . . . . . . . . . . . . . 20
3 Fixed point results in the framework of W-distances 23
3.1 Basic fixed point results . . . . . . . . . . . . . . . . . . . . . 23
3.2 Banach contraction principle . . . . . . . . . . . . . . . . . . 28
3.3 Rakotch theorem . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Meir and Keeler theorem . . . . . . . . . . . . . . . . . . . . 36
3.5 Kannan mappings . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 C´iri´c quasicontraction . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Fisher quasicontraction . . . . . . . . . . . . . . . . . . . . . 45
4 Some common fixed point results using W-distances 51
4.1 Some results of Ume and Kim . . . . . . . . . . . . . . . . . 51
4.2 Das and Naik contraction . . . . . . . . . . . . . . . . . . . . 56
4.3 Common coupled fixed point results . . . . . . . . . . . . . . 65
4.4 Some of Mohanta’s results . . . . . . . . . . . . . . . . . . . 69
4.5 Second Fisher theorem . . . . . . . . . . . . . . . . . . . . . 71
v
vi Contents
5 Best proximity points and (ϕ,ψ,p)-contractive mappings 79
5.1 Best proximity points involving simulation functions . . . . 79
5.2 Best proximity points with R-functions . . . . . . . . . . . . 85
5.3 (ϕ,ψ,p)-contractive mappings . . . . . . . . . . . . . . . . 89
5.4 (ϕ,ψ,p)-weakly contractive mappings . . . . . . . . . . . . . 95
5.5 Generalized weak contraction mappings . . . . . . . . . . . 99
5.6 w-φ-Kannan contractions . . . . . . . . . . . . . . . . . . . 107
6 Miscellaneous complements 119
6.1 Multivalued mappings . . . . . . . . . . . . . . . . . . . . . 119
6.2 C´iri´c type contractions at a point . . . . . . . . . . . . . . 128
6.3 Extension of a result by Ri . . . . . . . . . . . . . . . . . . . 133
6.4 Weaker Meir-Keeler function . . . . . . . . . . . . . . . . . 137
6.5 Contractive mappings of integral type . . . . . . . . . . . . 141
6.6 Ekeland’s variational principle . . . . . . . . . . . . . . . . . 151
6.7 Some generalizations and comments . . . . . . . . . . . . . 155
Bibliography 159
Index 175
Preface
Let (X,d) be a metric space. A mapping T : X → X is called a contrac-
tion if there exists a constant k ∈ [0,1) such that for any x,y ∈ X, we have
d(Tx,Ty) ≤ kd(x,y). The famous Banach contraction theorem of 1922 [15]
states that a contraction mapping T on a complete metric space has a unique
fixed point, that is, there exists a point x in X such that x = Tx. Banach’s
contraction principle and its variants have various applications in many areas
of mathematics, such as differential equations and optimization theory, and
in related disciplines, such as computer science, economics and telecommuni-
cations (see, e.g., [6, 69, 97, 102, 122, 155, 156, 197]). This intrigued several
mathematicians to extend the Banach contraction theorem to different direc-
tions. One such way is to extend the domain of a mapping.
In 1996, the Japanese mathematicians Kada, Suzuki and Takahashi [91] in-
troduced the new concept of w-distance on a metric space.
LetX beametricspacewithmetricd.Thenafunctionp:X×X →[0,∞)
is called a w-distance on X if the following are satisfied:
(1) p(x,z)≤p(x,y)+p(y,z), for any x,y,z ∈X,
(2) for any x∈X, p(x,·):X →[0,∞) is lower semicontinuous,
(3) for any (cid:15)>0, there exists δ >0 such that p(z,x)≤δ and p(z,y)≤δ
imply d(x,y)≤(cid:15).
They proved a new nonconvex minimization theorem, and by using this
theorem, the authors obtained some generalizations of Caristi’s fixed point
theorem and Ekeland’s (cid:15)-variational principle. As applications, some fixed
point theorems due to C´iri´c, Kannan and Subrahmanyam were also deduced.
Fundamental results in these directions are the papers of Suzuki and Taka-
hashi ([170, 171, 175]). There exists a large literature devoted to this subject.
Other distances were defined and a lot of results, mainly connected to fixed
points,wereproved.Therearemanyexcellenttextbooksonfixedpointtheory.
However,tothebestofmyknowledgethereisnotanybookonfixedpointre-
sultsinw-distancespaces.Inwritingthisbook,mygoalwastoproduceatext
on this subject. The technique of using this theory, which would be of great
usetospecialistsinthefieldoffixedpointtheory,hasnotyetbeensufficiently
developed. The purpose of this monograph is to fill this gap. Specifically, we
present new topics, applications and recent results that are not mentioned in
[6, 69, 97, 102, 122, 155, 156, 197]. Also these notes are addressed to non-
experts who want to learn something about w-distance. Our emphasis is on
vii
viii Preface
the applicability to many problems. As an application, the obtained results
are used to guarantee the existence and uniqueness of the solution for nonlin-
earFredholmintegralequationsandVolterraintegralequationstogetherwith
nonlinear fractional differential equations of Caputo type.
These are the lecture notes on fixed point methods and their applications.
They are based on courses taught by the author in the master’s and PhD
programsinmathematicsatFacultyofSciencesandMathematics,University
ofNiˇs,Serbia.ThreePhDthesesweredefendedattheFacultyofSciencesand
Mathematics related to w-distance spaces (Cvetkovi´c [30], Ili´c [82], Kosti´c
[107]). The material of the lecture notes could be covered in one semester in
a course of four hours per week. The presented topics would also serve as a
reference for further work, and could be used as a basis for seminar work,
master’s and PhD theses. The author took care for the lecture notes to be
self-contained and comprehensive. Only a solid background in real analysis is
needed.
The book contains six chapters with a great number of illustrating ex-
amples and remarks concerning the related results. The book is organized as
follows:
Chapter 1 provides some basic definitions and notations used throughout
the book necessary for understanding the main topics and ideas discussed
in the remaining chapters. For this reason, fundamental concepts and results
concerning metric spaces are given. Furthermore, three famous fixed point
theorems of Banach, Kannan and C´iri´c are given with their proofs.
Chapter 2 introduces the w-distance and some of the basic results of w-
distance. Several examples are included. Those results were mainly obtained
by Kada, Suzuki and Takahashi [91], Suzuki and Takahashi ([170, 171, 175])
and Liu, Hou, Kang and Ume [119], and Wu [195].
Chapter 3 deals with results and problems in fixed point theory to which
the theory of w-distances can be applied. So, relevant theorems of Banach,
Rakotch, Kannan, C´iri´c, Fisher, Meir and Keeler, and Subrahmanyam are
studied in the framework of w-distances. Mainly these results were obtained
by Kada, Suzuki and Takahashi [91], Suzuki [171], Morales [128, 129], Ume
[181], and Kocev, Karapinar and Rakoˇcevi´c [105].
Chapter 4 studies some common fixed point results using w-distances.
Jungck [89] described the interdependence of common fixed points and com-
muting mappings, and his results allow us to investigate common fixed point
theoremsforcommutingandnoncommutingpairsofmappingssatisfyingcon-
tractive conditions. Generalizations of Jungck’s contraction condition have
been extensively used to study common fixed points of contractive mappings.
The main results in this chapter were obtained by Ume and Kim [184], Ili´c
and Rakoˇcevi´c [83], Abbas, Ili´c and Khan [1], Mohanta [126], and Kocev and
Rakoˇcevi´c [104].
ThefirsttwosectionsofChapter5dealwithsomeresultsonbestproximity
points using w-distances involving simulation and R-functions. These results
were mainly obtained by Kosti´c, Rakoˇcevi´c and Radenovi´c [109] and Kosti´c,
Preface ix
KarapinarandRakoˇcevˇi´c[108].TheothersectionsofChapter5studyvarious
(ϕ,ψ,p)-contractivemaps.Asanapplication,theobtainedresultsareusedto
guaranteetheexistenceanduniquenessofthesolutionfornonlinearFredholm
integral equations and Volterra integral equations together with nonlinear
fractional differential equations of Caputo type. These results were mainly
obtained by Razani, Mazlumi Nezhad and Boujary [146], Lakzian and Lin
[111], Lakzian, Aydi and Rhoades [112], Wongyat and Sintunavarat [194],
Dobri¸toiu [47], and Lakzian and Rakoˇcevi´c [114].
Chapter 6 contains various complements that have not been incorporated
inthemainbodyofthebookinordertokeepthepresentationmorecompact.
TheyareconnectedtoChapters2-5.ThemainresultswereobtainedbySuzuki
and Takahashi [175], Garcia-Falset, Guran and Llorens-Fuster [64], Guran
andPetru¸sel[74],Kocev,KarapinarandRakoˇcevi´c[105],Aydi,Wongyatand
Sintunavarat,[13],LakzianandRhoades[115],Liu,Hou,KangandUme[119],
Liu,Zhang,KangandUme[120],Kada,SuzukiandTakahashi[91],andPark
[136]. Several proofs have been omitted, and the interested reader is invited
to consult the references.
Iowethankstomanypeopleforreadingthefirstdraftsofthesenotesand
givingmefeedback.AmongthemareMujahidAbbas,LilianaGuran,Dragana
Cvetkovi´c Ili´c, Zoran Kadelburg, Erdal Karapinar, and Eberhard Malkowsky.
Also, I am very grateful to my colleagues and friends for their cooperation on
manyjointlypublishedworks,andspecialthanksgotoRajendraBisht,Marija
Cvetkovi´c, Moosa Gabeleh, Ljiljana Gaji´c, Dejan Ili´c, Darko Kocev, Aleksan-
dar Kosti´c, Hossein Lakzian, Gradimir V. Milovanovi´c, Zoran D. Mitrovi´c,
MohammadMursaleen,R.P.Pant,VladimirPavlovi´c,AbdolrahmanRazani,
Stojan Radenovi´c, and Billy E. Rhoades.
