Remarks on the fixed point theory for quasi-metric spaces: Remarks on the fixed point theory for quasi-metric spaces

Salvador  Romaguera Bonilla

Authors

Salvador Romaguera Bonilla

Abstract

Motivated by a recent and interesting article by S. Park [Results in Nonlinear Analysis 6 (2023) No. 4, 116–127], we recall several different notions of quasi-metric completeness that appear in the literature and revise how they influence on the fixed point theory in quasi-metric spaces. In particular, we point out that there are several classical fixed point theorems that cannot be directly transferred to the quasi-metric setting without extra conditions, when Park's approach is considered. We also recall some emblematic examples that can help to clarify some aspects of the fixed point theory for these spaces.

References

S. Al-Homidan, Q. H. Ansari, J.-C. Yao, Some generalizations of Ekeland-type variational principle with applications

to equilibrium problems and fixed point theory, Nonlinear Anal. 69 (2008) 126–139.

C. Alegre, H. Dağ, S. Romaguera, P. Tirado, On the fixed point theory in bicomplete quasi-metric spaces, J. Nonlinear

Sci. Appl. 9 (2016) 5245–5251

C. Alegre, H. Dağ, S. Romaguera, P. Tirado, Characterizations of quasi-metric completeness in terms of Kannan-type

fixed point theorems, Hacettepe J. Math. Stat. 46 (2017) 67–76.

H. Aydi, M. Jleli, E. Karapinar, On fixed point results for -implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal. Model. Control 21 (2016) 40–56.

D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464.

F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad.

Wetensch. Proc. Ser. A Math. 30 (1968) 27–35.

J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976)

–251.

L. Ćirić, A generalization of Banach’s contraction principle, Proc. Am. Math. Soc. 45 (1974) 267–273.

L. Ćirić, Semi-continuous mappings and fixed point theorems in quasi metric spaces, Publ. Math. Debrecen 54 (1999)

–261

S. Cobzaş, Completeness in quasi-metric spaces and Ekeland Variational Principle, Topol. Appl. 158 (2011) 1073–1084.

S. Cobzaş, Functional Analysis in Asymmetric Normed Spaces; Frontiers in Mathematics, Birkhäuser/Springer Basel

AG: Basel, Switzerland, 2013.

S. Cobzaş, Completeness in quasi-pseudometric spaces—a survey, Mathematics 2020, 8, 1279.

D. Downing and W.A. Kirk, A generalization of Caristi’s theorem with applications to nonlinear mapping theory,

Pacific J. Math. 69 (1977) 339–345.

P.N. Dutta, B.S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. 2008

(2008) 8 pages.

I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324–353.

Fletcher, P.; Lindgren, W.F. Quasi-Uniform Spaces; Marcel Dekker, Inc. New York, NY, USA; Basel, Switzerland,

J. Jachymski, A contribution to fixed point theory in quasi-metric spaces, Publ. Math. Debrecen 43 (1993) 283–288.

M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. 2012:210, 2012.

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968) 71–76.

E. Karapinar, S. Romaguera, On the weak form of Ekeland’s variational principle in quasi-metric spaces, Topology

Appl. 184 (2015) 54–60.

E. Karapinar, S. Romaguera, P. Tirado, Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points, Demonstratio Mathematica 55 (2022) 939–951.

M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math.

Soc. 30 (1984) 1–9.

A. W. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. 36 (1976) 81–86.

M.A. Krasnoselskii, V.Y. Stetsenko, About the theory of equations with concave operators, (Russian) Sib. Mat. Zh. 10

(1969) 565–572.

J. Matkowski, Integrable solutions of functional equations, Diss. Math. (Rozprawy Mat.) 127 (1975) 68 pages.

A. Meir, E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl. 28 (1969) 326–329.

S.B. Nadler, Jr. Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488.

S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000) 881–889.

S. Park, All metric fixed point theorems hold for quasi-metric spaces, Results Nonlinear Anal. 6 (2023) 116–127.

Reilly, I.L.; Subrahmanyam, I.L.; Vamanamurthy, M.K. Cauchy sequences in quasi-pseudo-metric spaces. Monatsh.

Math. 93 (1982) 127–140.

S. Romaguera, Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics 2022,

, 3931.

S. Romaguera, Generalized Ćirić contraction in quasi-metric spaces, Lett. Nonlinear Anal. Appl. 1 (2023) 30–38.

S. Romaguera, P. Tirado, The Meir–Keeler fixed point theorem for quasi-metric spaces and some consequences,

Symmetry 2019, 11, 741.

T. Suzuki, A generalized Banach contraction principle that generalizes metric completeness, Proc. Amer. Math. Soc.

(2008) 1861–1869.

Remarks on the fixed point theory for quasi-metric spaces