Common fixed points of Geraghty-Suzuki type contraction maps in b-metric spaces

Year 2020, Volume: 2 Issue: 1, 26 - 47, 30.06.2020

Dasari Ratna Babu

Abstract

In this paper, we prove the existence and uniqueness of common fixed points for two pairs of
selfmaps satisfying a Geraghty-Suzuki type contraction condition in which one pair is compatible,
b-continous and the another one is weakly compatible in complete b-metric spaces. Further, we
prove the same with different hypotheses on two pairs of selfmaps which satisfy b-(E.A)-property.
We draw some corollaries from our results and provide examples in support of our results

Keywords

Common fixed points, b-metric space, weakly compatible maps, b-(E.A)-property

Supporting Institution

PSCMR CET, Vijayawada, Andhra Pradesh, India

Thanks

To The Executive Editor, Proceedings of International Mathematical Sciences I am herewith submitting my research paper(PDF file) entitled ``Common fixed points of Geraghty-Suzuki type contraction maps in b-metric spaces" Authors : G. V. R. Babu and D. Ratna Babu for favor of publication in the journal `Proceedings of International Mathematical Sciences'. Thanking you Your sincerely D. Ratna Babu

References

• [1] M. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270(2002), 181-188.
• [2] A. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64(4)(2014), 941-960.
• [3] H. Aydi, M-F. Bota, E. Karapınar and S. Mitrovic, A fixed point theorem for set-valued quasi contractions in b-metric spaces, Fixed Point Theory Appl., 88(2012), 8 pages.
• [4] G. V. R. Babu and G. N. Alemayehu, A common fixed point theorem for weakly contractive mappings satisfying property (E.A), Applied Mathematics E-Notes, 24(6)(2012), 975-981.
• [5] G. V. R. Babu and T. M. Dula, Common fixed points of two pairs of selfmaps satisfying (E.A)- property in b-metric spaces using a new control function, Inter. J. Math. Appl., 5(1-B)(2017), 145-153.
• [6] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Func. Anal. Gos. Ped. Inst. Unianowsk, 30(1989), 26-37.
• [7] V. Berinde, Iterative approximation of fixed points, Springer, 2006.
• [8] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math., 4(3)(2009), 285-301.
• [9] M. Boriceanu, M. Bota and A. Petrusel, Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8(2)(2010), 367-377.
• [10] N. Bourbaki, Topologie Generale, Herman: Paris, France, 1974.
• [11] L. Ciric, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45(1974), 267-273.
• [12] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1(1993), 5-11.
• [13] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti del Seminario Matematico e Fisico (DellUniv. di Modena), 46(1998), 263-276.
• [14] B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions, Indian J. Pure and Appl. Math., 6(1975), 1455-1458.
• [15] D. Dukic, Z. Kadelburg and S. Radenovic, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstr. Appl. Anal.,(2011), Article ID 561245, 13 pages.
• [16] H. Faraji, D. Savic and S. Radenovic, Fixed point theorems for Geraghty contraction type mappings in b-metric spaces and applications, Axioms, 8(34)(2019), 12 pages.
• [17] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40(1973), 604-608.
• [18] H. Huang, G. Deng and S. Radenovic, Fixed point theorems for C-class functions in b-metric spaces and applications, J. Nonlinear Sci. Appl., 10(2017), 5853-5868.
• [19] N. Hussain, V. Paraneh, J. R. Roshan and Z. Kadelburg, Fixed points of cycle weakly ( ; '; L; A;B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013(2013), 256, 18 pages.
• [20] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. and Math. Sci., 9(1986), 771-779.
• [21] G. Jungck and B. E. Rhoades, Fixed points of set-valued functions without continuity, Indian J. Pure and Appl. Math., 29(3)(1998), 227-238.
• [22] P. Kumam and W. Sintunavarat, The existence of fixed point theorems for partial q-set valued quasi-contractions in b-metric spaces and related results, Fixed point theory appl., 2014(2014): 226, 20 pages.
• [23] A. Latif, V. Parvaneh, P. Salimi and A. E. Al-Mazrooei, Various Suzuki type theorems in b-metric spaces, J. Nonlinear Sci. Appl., 8(2015), 363-377.
• [24] B. T. Leyew and M. Abbas, Fixed point results of generalized Suzuki-Geraghty contractions on f-orbitally complete b-metric spaces, U. P. B. Sci. Bull., Series A, 79(2)2017, 113-124.
• [25] V. Ozturk and D. Turkoglu, Common fixed point theorems for mappings satisfying (E.A)-property in b-metric spaces, J. Nonlinear Sci. Appl., 8(2015), 1127-1133.
• [26] V. Ozturk and S. Radenovic, Some remarks on b-(E.A)-property in b-metric spaces, Springer Plus, 5(2016), 544, 10 pages.
• [27] V. Ozturk and A. H. Ansari, Common fixed point theorems for mapping satisfying (E.A)-property via C-class functions in b-metric spaces, Appl. Gen. Topol., 18(1)(2017), 45-52.
• [28] J. R. Roshan, V. Paraneh and Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 7(4)(2014), 229-245.
• [29] W. Shatanawi, Fixed and common fixed point for mappings satisfying some nonlinearcontractions in b-metric spaces, J. Math. Anal., 7(4)(2016), 1-12.
• [30] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136(2008), 1861-1869.