Improving Many Metric Fixed Point Theorems
Keywords:
Quasi-metric space, fixed point, RHR contraction principle, orbitally complete, orbitally continuousAbstract
We found that many metric fixed point theorems hold for orbitally complete quasi-metric spaces. In order to show this, we obtain several basic principles extending the Banach contraction principle and new fixed point theorems for the Rus-Hicks-Rhoades maps with a large number of their consequences. Moreover, we improve theorems on mappings with contractive conditions with some auxiliary functions and on mappings with asymptotically regularity. Consequently, a large number of known metric fixed point theorems are extended with almost trivial short proofs.
References
[1] S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Lett. Nonlinear Anal. Appl. 1(2) (2023) 57–63.
[2] S. Park, All metric fixed point theorems hold for quasi-metric spaces, Results in Nonlinear Analysis 6(4) (2023) 116–127. https://doi.org/10.31838/rna/2023.06.04.012 (Dec. 1, 2023). A revised and corrected version appears in Research Gate.
[3] S. Park, The use of quasi-metric in the metric fixed point theory, to appear.
[4] 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
[5] S. Park, Comments on the Suzuki type fixed point theorems, Adv. Theory Nonlinear Anal. Appl. 7(3) (2023) 67–78. https://doi.org/10.17762/atnaa.v7.i3.275
[6] 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) 1–45.
[7] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290.
[8] S.-S. Chang, On Rhoades’ open questions and some fixed point theorems for a class of mappings, Proc. Amer. Math. Soc. 97(2) (1986) 343–346.
[9] M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. 2012:210, 2012.
[10] H.H. Alsulami, E. Karapinar, F. Khojasteh, A.-F. Roldan-Lopez-de-Hierro, A proposal to the study of contractions in quasi-metric spaces, Discrete Dynamics in Nature and Society (2014), Article ID 269286, 10pp.
[11] 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.
[12] S. Park, On general contractive type conditions, J. Korean Math. Soc. 17 (1980) 131–140.
[13] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861–1869.
[14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71–76.
[15] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14(1) (1971) 121–124.
[16] G.E. Hardy, T.D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16(2) (1973) 201–206.
[17] Lj. B. ´Ciri´c, On some maps with a non-unique fixed point, Publ. Inst. Math. 17 (1974) 52–58.
[18] Lj. B. ´Ciri´c, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45(2) (1974) 267- 273.
[19] P.V. Subramanyam, Remarks on some fixed point theorems related to Banach’s contraction principle, J. Malh. Phys. Sri. 8 (1974) 445-457: Erratum, 9 (1975) 195.
[20] B.K. Dass, S. Gupta, An extension of Banach contraction principle through rational expressions, Indian J. Pure Appl. Math., 6 (1975) 1455–1458.
[21] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8 (1977) 223–230.
[22] B.E. Rhoades, A fixed point theorem for some non-self-mappings, Math. Japon. 23(4) (1978/79) 457–459.
[23] T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327–330.
[24] J. Jachymski, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Analysis 52 (2003) 1455–1463.
[25] B.G. Pachpatte, On ´ Ciri´c type maps with a non-unique fixed point, Indian J. Pure Appl. Math. 10 (1979) 1039–1043.
[26] S. Park, A unified approach to fixed points of contractive maps, J. Korean Math. Soc. 16 (1980) 95–105.
[27] S. Park, B.E. Rhoades, Some general fixed point theorems, Acta Sci. Math. 42 (1980) 299-304.
[28] R. Srivastava, Fixed point theorems in a T-orbitally complete metric space, Math. Student 64 (1995) 15–20.
[29] B.E. Rhoades, A biased discussion of fixed point theory, Carpathian J. Math. 23(1-2) (2007) 11–26.
[30] W.A. Kirk, L.M. Saliga, Some results on existence and approximation in metric fixed point theory, J. Comp. Appl. Math. 113 (2000) 141–152.
[31] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carphathian J. Math. 19(1) (2003) 7–22.
[32] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Analysis Forum 9(1) (2004) 43–53.
[33] T. Suzuki, Contractive mappings are Kannan mappings, and Kannan mappings are contractive mappings in some sense, Comment. Math. Prace Mat. 45 (2005) 45–58.
[34] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. TMA 69(9) (2008) 2942–2949.
[35] Y. Enjouji, M. Nakanishi, T. Suzuki, A generalization of Kannan’s fixed point theorem, Fixed Point Theory Appl. (2009), Article ID 192872, 10pp. doi:10.1155/2009/192872
[36] M. Nakanishi, T. Suzuki, An observation on Kannan mappings, Cent. Eur. J. Math. 8(1) (2010) 170-178. DOI: 10.2478/s11533-009-0065-9
[37] I. Altun, A. Erduran, A Suzuki type fixed point theorem, Inter. J. Math. Math. Sci. 2011, Article ID 736063, 9pp. doi:10.1155/2011/736063
[38] E. Karapinar, A new non-unique fixed point theorem, J. Appl. Funct. Anal. 7(1-2) (2012) 92–97.
[39] S. Chandok, T.D. Narang, M.-A. Taoudi, Fixed point theorem for generalized contractions satisfying rational type expressions in partially ordered metric spaces, Gulf J. Math. 2(4) (2014) 87–93.
[40] P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A generalization of ´ Ciri´c fixed point theorems, Filomat 29(7) (2015) 1549– 1556. DOI 10.2298/FIL1507549K
[41] V.V. Nalawade, U.P. Dolhare, Another Kannan version of Suzuki fixed point theorem, Inter. J. Math. Trends and Technology (IJMTT) 36(2) (2016) 110–115. ISSN: 2231-5373
[42] I.A. Rus, Some problems in the fixed point theory, Adv. Theory Nonlinear Anal. Appl. 2(1) (2018) 1–10. doi.org/10.31197/atnaa.379280
[43] I.A. Rus, Relevant classes of weakly Picard operators, An. Univ. Vest Timioara, Mat.-Inform., 54(2) (2016) 3–9.
[44] J.-J. Mi˜nana, O. Valero, Are fixed point theorems in G-metric spaces an authentic generalization of their classical counterparts? J. Fixed Point Theory Appl. (2019) 21:70. doi.org/10.1007 /s11784-019-0705-z
[45] V. Berinde, M. P˘acurar, Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences, Arab. J. Math., Published online 29 September 2022. https://doi.org/10.1007/s40065-022-00398-6
[46] N. Chandra, B. Joshi, M.C. Joshi, Generalized fixed point theorems on metric spaces, Mathematica Moravica 26(2) (2022) 85–101.
[47] E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962) 459–465.
[48] F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad.Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968) 27–35.
[49] D. Boyd, J. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. https://doi.org/10.1090/S0002- 9939-1969-0239559-9
[50] M.A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973) 604–608,
[51] S. Kasahara, Generalizations of Hegedus’ fixed point theorem, Math. Sem. Notes, 7 (1979) 107–111.
[52] R.K. Bisht, An overview of the emergence of weaker continuity notions, various classes of contractive mappings and related fixed point theorems, J. Fixed Point Theory Appl. (2023) 25:11. https://doi.org/10.1007/s11784-022-01022-y
[53] M.R. Taskovi´c, Some results in the fixed point theory-II, Publ. Inst, Math. 41 (1980) 249–258.
[54] W. Walter, Remarks on a paper by F. Browder about contraction, Nonlinear Anal. TMA 5 (1981) 21–25.
[55] B.E. Rhoades, Using general principles to prove fixed point theorems, Ann. Univ. Mariae Curie-Sk lodowska LI.2, 22 A (1997) 241–260.
[56] M. Akkouch, On a result of W.A. Kirk and L.M. Saliga, J. Comp. Appl. Math. 142 (2002) 445–448.
[57] P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis 67 (2007) 2361–2369.
[58] Lj. B. ´Ciri´c, On contraction type mappings. Math. Balkanica 1 (1971) 52–57.
[59] P.D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64(3) (2006) 546–557.
[60] J. G´ornicki, Fixed point theorems for Kannan type mappings, J. Fixed Point Theory Appl. 19 (2017) 2145–2152. DOI 10.1007/s11784-017-0402-8
[61] J. G´ornicki, Remarks on asymptotic regularity and fixed points, J. Fixed Point Theory Appl. 21(29) (2019)
[62] R.K. Bisht, A note on the fixed point theorem of G´ornicki, J. Fixed Point Theory Appl. (2019) 21:54. https://doi.org/10.1007/s11784-019-0695-x
[63] 10 S. Panja, K. Roy, M. Saha, R.K. Bisht, Some fixed point theorems via asymptotic regularity, Filomat 34(5) (2020) 1621–1627.
[64] B.E. Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis, Contemporary Math., Vol. 21, Amer. Math. Soc, Providence, R. I. (1983) 189–205.
[65] A. Pant, R.P. Pant, Fixed points and continuity of contractive maps. Filomat Filomat 31(11) (2017) 3501–3506.
[66] P.D. Proinov, Fixed point theorems for generalized contractive mappings in metric spaces, JFPTA 22(1) (2020) p.21.
[67] R. Pant, D. Khantwal, Fixed points of single and multivalued mappings in metric spaces with applications, Preprint. 2023
[68] K. Roy, Fixed point approach through simulation function and asymtotic regularity, U.P.B. Sci. Bull., Series A 86(1) (2024) 87-94. ISSN 1223-7027
[69] I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.
[70] V. Berinde, A. Petrus¸el, I.A, Rus, Remarks on the terminology of the mappings in fixed point iterayive in metric spaces, Fixed Point Theory 24(2) (2023) 525–540. DOI: 10.24193/fpt-ro.2023.2.05
[71] S. Cobza¸s, Fixed points and completeness in metric and generalized metric spaces, Jour. Math. Sci. 250(3) (2020) 475–535. DOI 10.1007/s10958-020-05027-1
[72] S. Park, Several recent episodes on the metric completeness, Research Gate on Jan. 11, 2024
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