Some Applications of the WeakContraction Principle
Keywords:
fixed point, quasi-metric, Rus-Hicks-Rhoades (RHR) map, T-orbitally completeAbstract
Recently we obtained several extensions of the Banach contraction principle. One of them is the weak contraction principle or the Rus-Hicks-Rhoades contraction principle or Theorem P. There are a large number of examples or applications of Theorem P in the literature. Recently, Romaguera stated two corollaries of Theorem P are false. Our main aim of this paper is to clarify his claimReferences
[1] S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022) 1–51.
[2] S. Park, History of the metatheorem in ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 62 (2023) 373–410.
[3] S. Park, All metric fixed point theorems hold for quasi-metric spaces, Results in Nonlinear Analysis 6 (2023) 116–127.
[4] I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.
[5] T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327–330.
[6] S.B. Nadler, Jr. Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488.
[7] H. Covitz, S.B. Nadler, Jr. Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5–11.
[8] S. Park, The realm of the Rus-Hicks-Rhoades maps in the metric fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 63(1) (2024) 1–50.
[9] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3 (1922) 133–181.
[10] V. Berinde, A. Petrusel, J.A. Rus, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory 24(2) (2023) 525–540. DOI: 10.24193/fpt-ro.2023.2.05
[11] S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Letters Nonlinear Anal. Appl. 1 (2023) 57–63.
[12] S. Park, Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Adv. Th. Nonlinear Anal. Appl. 7(2) (2023) 455–471. DOI 0.31197/atnaa.1185449
[13] S. Park, The use of quasi-metric in the metric fixed point theory, J. Nonlinear Convex Anal. 25(7) (2024) 1553–1564.
[14] S. Park, Improving many metric fixed point theorems, Letters Nonlinear Anal. Appl. 2(2) (2024) 35–61.
[15] H. Aydi, M. Jellali, E. Karapinar, On fixed point results for α-implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal. Model. Control, 21(1) (2016) 40–56.
[16] M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. (2012) 2012:210.
[17] S. Park, Remarks on the metatheorem in ordered fixed point theory, Advanced Mathematical Analysis and Its Applications (Edited by P. Debnath, D.F.M. Torres, Y.J. Cho) CRC Press (2023), 11–27. DOI : 10.1201/9781003388678-2
[18] W. Oettli, M. Th´era, Equivalents of Ekeland’s principle, Bull. Austral. Math. Soc. 48 (1993) 385–392.
[19] V.I. Istr˘at¸escu, Fixed Point Theory: An Introduction, D. Reidel Publ. Co., Dortrecht, Holland, 1981.
[20] F.F. Bonsall, Lectures on some fixed point theorems of functional analysis, Tata Institute of Fundamental Research, Bombay, 1962.
[21] 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.
[22] S. Park, On the Rus-Hicks-Rhoades contraction principle, Manuscript.
[23] J.T. Markin, A fixed point theorem for set-valued mappings. Bull. Amer. Math. Soc. 74 (1968) 639–640.
[24] S. Park, Several recent episodes on the metric completeness, Mathematical Analysis: Theory and Applications (Editors: P. Debnath, H. M. Srivastava, D. F. M. Torres, and Y. J. Cho), CRC Press. Jan. 11, 2024
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