All metric fixed point theorems hold for quasi-metric spaces


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Authors

  • Sehie Park The National Academy of Sciences, Republic of Korea, Seoul 06579; Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

Keywords:

Banach contraction, fixed point, fixed point theorem, metric space, quasi- metric, maximal element. Mathematics Subject Classification: 06A75, 47H10, 54E35, 54H25, 58E30, 65K10

Abstract

Our aim in this article is to show that all metric fixed point theorems hold for quasi-metric spaces (X, δ) (without symmetry). In fact, we show some well-known theorems on metric spaces hold for quasi-metric spaces from the beginning. We check this fact for the Banach contraction principle, the Covitz-Nadler fixed point theorem, the Rus-Hicks-Rhoades fixed point theorem, and others. In
these theorems the concepts of continuity and completeness can be replaced by orbital continuity and T-orbital completeness for a selfmap T, respectively.Consequently, we improve and generalize the basic known theorems in the metric fixed point theory.

References

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

H. Covitz, S.B. Nadler, Jr. Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5–11.

B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290.

S. Park, On general contractive-type conditions, J. Korean Math. Soc. 17 (1980) 131–140.

S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022) 1–51.

I. A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.

T. L. Hicks, B. E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327–330.

S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Letters Nonlinear Anal. Appl. 1 (2023) 57–63. ISSN 2958-874X

S. Park, Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Adv. Th. Nonlinear Anal. Appl. 7(2) (2023) 455–471. doi: 10.31197/atnaa.1185449

H. Aydi, M. Jellali, E. Karapinar, On fixed point resultsfor α-implicit contractions in quasi-metric spaces and consequences}, Nonlinear Anal. Model. Control, 21(1) (2016) 40–56.

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

S. Park, History of the metatheorem in ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 62(2)

W. A. Kirk, Contraction mappings and extensions, Chapter 1, Handbook of Metric Fixed Point Theory (W.A. Kirk and B. Sims, eds.), Kluwer Academic Publ. (2001) 1–34.

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

S. K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727–730.

S. Willard, General Topology, Addison-Wesley, 1970.

L. B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12–26 (1971) 19–26.

V. Berinde, Approximating fixed points of weak contractions using the picard iteration, Nonlinear Anal. Forum 9(1) (2004) 43–53.

R. P. Pant, V. Rakoc˘ević, D. Gopal, A. Pant, M. Ram, A general fixed point theorem, Filomat 35:12 (2021) 4061–4072. doi: 10.2298/FIL2112061P

T. Suzuki, A generalized Banach contraction principle that generalizes metric completeness, Proc. Amer. Math. Soc. 136(5) (2008) 1861–1869.

N. Chandra, B. Joshi, M. C. Joshi, Generalized fixed point theorems on metric spaces, Mathematica Moravica 26(2) (2022) 85–101. doi: 10.5937/MatMor2202085C

S. Park, Remarks on the metatheorem in ordered fixed point theory, Advanced Mathematical Analysis and Its Applications, Chapter 2 (Edited by P. Debnath, D.F.M. Torres, Y.J. Cho), CRC Press (2023) 11–27. doi: 10.1201/9781003388678-2

O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica 44 (1996) 381–391.

S. Park, A unified approach to fixed points of contractive maps, J. Korean Math. Soc. 16 (1980) 95–105.

E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Students 63 (1994) 123–145.

W. Oettli, M. Théra, Equivalents of Ekeland’s principle, Bull. Austral. Math.Soc. 48 (1993) 385–392.

S. Park, B.G. Kang, Generalizations of the Ekeland type variational principles, Chinese J. Math. 21 (1993) 313–325.

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

T. Suzuki, The strong Ekeland variational principle, J. Math. Anal. Appl. 320 (2006) 787–794.

J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

D. R. Smart, Fixed point theorems, Cambridge Univ. Press, 1974.

V. I. Istratescu, Fixed Point Theory, D. Reidel Pub. Co. 1981

J. Dugundji, A. Granas, Fixed Point Theory I, Polish Scientific Publishers, Warszawa, 1982.

K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Univ. Press 1990.

S. Park, Equivalents of ordered fixed point theorems of Kirk, Caristi, Nadler, Banach, and others, Adv. Th. Nonlinear Anal. Appl. 6(4) (2022) 420–439.

S. Park, Equivalents of some ordered fixed point theorems, J. Advances Math. Comp, Sci. 38(1) (2023) 52–67.

T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (basel) 23 (1972) 292–298.

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Published

2023-11-03

How to Cite

Sehie Park. (2023). All metric fixed point theorems hold for quasi-metric spaces. Results in Nonlinear Analysis, 6(4), 116–127. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/356