Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces

Abstract

Let $(X, d)$ be a quasi-metric space. A Rus-Hicks-Rhoades (RHR) map $f : X \to X$ is the one satisfying $d(fx, f^2x) \le \alpha d(x, fx)$ for every $x\in X$, where $\alpha \in [0,1)$. In our previous work [37], we collected various fixed-point theorems closely related to RHR maps. In the present article, we collect almost all the things we know about RHR maps and their examples. Moreover, we derive new classes of generalized RHR maps and fixed point theorems on them. Consequently, many of the known results in metric fixed point theory are improved and reproved in an easy way.

Keywords

• fixed point theorem
• metric space
• fixed point
• stationary point
• maximal element

Kaynakça

1. [1] S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Fractional Differential and Integral Equations (Existence and Stability). Gruyter, Berlin (2018).
2. [2] S. Abbas, M. Benchohra, J. Henderso, Existence and oscillation for coupled fractional q-difference systems. Fract. Calc. Appl. 12 (1) (2021), pp. 143-155.
3. [3] S. Abbas, M. Benchohra, N. Laledj and Y. Zhou, Existence and Ulam Stability for implicit fractional q-difference equation. Adv. Differ. Equ. 2019 (480) (2019), pp. 1-12.
4. [4] B. Ahmad, Boundary value problem for nonlinear third order q-difference equations. Electron. J. Dier. Equ. 2011 (94) (2011), pp. 1-7.
5. [5] B. Ahmad, S. K. Ntouyas and I. K. Purnaras, Existence results for nonlocal boundary value problems of nonlinear fractional q-dierence equations. Adv. Differ. Equ. 2011 (140) (2012), pp. 1-15.
6. 6 Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semicontinuity. Bull. Korean Math. Soc. 39 (2002) 535--541.
7. 7 L.B. Ciric, On some maps with a non-unique fixed point, Publ. Inst. Math. 17 (1974) 52--58.
8. 8 S. Cobzaz, Fixed points and completeness in metric and generalized metric spaces, J. Math. Sci. (N.Y.) 250 (2020) 475--535.

9. 9 H. Covitz, S.B. Nadler, Jr., Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5--11.
10. 10 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
11. 11 R. Fierro, B. S. Pizarro, Fixed points of set-valued mappings satisfying a Banach orbital condition, CUBO, A Math. Jour. 25(1) (2023) 151--159, DOI: 10.56754/0719-0646.2501.151
12. 12 G.E. Hardy, T.D. Rogers, A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16 (1973) 201--206.
13. 13 T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327--330.
14. 14 R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71--76.
15. 15 E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl. 2(2) (2018) 85--87. 16 E. Karapinar, C‚iric type non unique fixed point results: A review, Appl. Comput. Math. 18(1) (2019) 3-21.
16. 17 E. Karapinar, R. Agarwal, H. Aydi, Interpolative Reich-Rus-‚Ciric type contractions on partial metric spaces, Mathematics 6 (2018) 256; doi:10.3390/math6110256
17. 18 E. Karapinar, O. Alqahtani, H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry 11(1) (2018) 8.
18. 19 F. Khojasteh, M. Abbas, S. Costache, Two new types of fixed point theorems in complete metric spaces, Abstr. Appl. Anal., 2014, Art. ID 325840, 5pp.
19. 20 M. Kikkawa, T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory Appl. 8 (2008) Article ID 649749. 21 M. Kikkawa, T. Suzuki, Some similarity between contractions and Kannan mappings, II, Bull. Kyushu Inst. Tech. Pure Appl. Math. 55 (2008), 1--13. 22 M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (2008) 2942--2949. 23 M. Kikkawa, T. Suzuki, Some notes on fixed point theorems with constants, Bull. Kyushu Inst. Tech. Pure Appl. Math. 56 (2009) 11--18.
20. 24 W. A. Kirk, Caristi's fixed point theorem and metric convexity, Colloq. Math. 36(1) (1976) 81--86.
21. 25 W.A. Kirk, L.M. Saliga, The Brézis-Browder order principle and extensions of Caristi's theorem, Nonlinear Anal. TMA 47 (2001) 2765--2778.
22. 26 Z. Liu, On Park's open questions and some fixed-point theorems for general contractive type mappings, J. Math. Anal. Appl. 234 (1999) 165--182.
23. 27 S.B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475--488.
24. 28 M. Nakanishi, T. Suzuki, An observation on Kannan mappings, Cent. Eur. J. Math. 8(1) (2010) 170--178.
25. 29 R. Pant, D. Khantwa, Fixed points of single and multivalued mappings in metric spaces with applications, (2023) Preprint. DOI: https://doi.org/10.21203/rs.3.rs-2844971/v2
26. 30 S. Park, A unified approach to fixed points of contractive maps, J. Korean Math. Soc. 16 (1980) 95--105.
27. 31 S. Park, On general contractive-type conditions, J. Korean Math. Soc. 17 (1980) 131--140.
28. 32 S. Park, Fixed points and periodic points of contractive pairs of maps, Proc. Coll. Natur. Sci., SNU 5 (1980) 9--22.
29. 33 S. Park, On extensions of the Caristi-Kirk fixed point theorems, J. Korean Math. Soc. 19 (1983) 143--151.
30. 34 S. Park, Characterizations of metric completeness, Colloq. Math. 49(1) (1984) 21--26.
31. 35 S. Park, Recollecting joint works with B. E. Rhoades, Indian J. Math. 56(3) (2014) 263--277.
32. 36 S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022) 247--287.
33. 37 S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Letters Nonlinear Anal. Appl. 1(2) (2023) 57--63.
34. 38 S. Park, The use of quasi-metric in the metric fixed point theory, Manuscript.
35. 39 S. Park, Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Manuscript.
36. 40 S. Park, B.E. Rhoades, Some general fixed point theorems, Acta Sci. Math. 42 (1980), 299--304. Notices Amer. Math. Soc. Abstract 79T-B95. MR 82a:54089. Zbl 449.54045. 41 S. Park, B.E. Rhoades, Comments on characterizations for metric completeness, Math. Japon. 31 (1986) 95--97.
37. 42 S. Park, B.E. Rhoades, Some fixed point theorems for expansion mappings, Math. Japonica 33(1) (1988) 129--132.
38. B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257-290.
39. B.E. Rhoades, A biased discussion of fixed point theory, Carpathian J. Math. 23(1-2) (2007) 11--26.
40. S. Romaguera, Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics 2022, 10, 3931. https://doi.org/ 10.3390/math10213931
41. 43 I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.
42. 44 I.A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.
43. 45 N. Shobkolaei, S. Sedghi, J.R. Roshan, N. Hussain, Suzuki-type fixed point results in metric-like spaces, J. Function Spaces Appl. 2013, Article ID 143686, 9pp. http://dx.doi.org/10.1155/ 2013/143686
44. 46 T. Suzuki, Generalized distance and existence theorems in complete metric spaces, Jour. Math. Anal. Appl. 253 (2001), 440--458. doi:10.1006.jmaa.2000.7151.
45. 47 T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness. Proc. Amer. Math. Soc. 136 (2008) 1861--1869.
46. 48 T. Suzuki, A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71 (2009) 5313--5317.
47. 49 S.Z. Wang, B.Y. Li, Z.M. Gao, K. Iséki, Some fixed point theorems on expansion mappings, Math. Japonica 29 (1984) 631--636.
48. 50 S.S. Yesilkaya, On interpolative Hardy-Rogers contractive of Suzuki type mappings, Topol. Algebra Appl. 9 (2021) 13--19.