\(C\)-class and \(F(\psi,\varphi)\)-contractions on generalized metric spaces
Keywords:
Generalized metric space, C-class functionsAbstract
In this work, we introduce the class of \(F(\psi,\varphi)\)-contractions and investigate the existence and uniqueness of fixed points for the new class \(\mathcal{C}\) in the setting of generalized metric space. Our theorems improve very recent results in the literature.
References
[1] A. H. Ansari, “Note on” φ –ψ -contractive type mappings and related fixed point”, The 2nd Regional Conference on Mathematics And Applications, PNU, September 2014, 377-380.
[2] A. Hojat Ansari, Sumit Chandok and Cristiana Ionescu, Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, Journal of Inequalities and Applications 2014, 2014:429,17 pages.
[3] M. Asadi, Erdal Karapınar and Anil Kumar, α−ψ-Geraghty contractions on generalized metric spaces, Journal of Inequalities and Applications 2014, 2014:423. Doi:10.1186/1029-242X-2014-423.
[4] H. E. Karapınar and H. Lakzian, Fixed point results on the class of generalized metric spaces, Math. Sci. 2012, 6:46
[5] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math. 3, 133–181 (1922).
[6] N. Bilgili, E. Karapınar, K. Sadarangani, A generalization for the best proximity point of Geraghty-contractions, J. Inequal. Appl. 2013, 2013:286.
[7] R. M. Bianchini , M. Grandolfi, Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti Acad. Naz. Lincei, VII. Ser., Rend., Cl. Sci. Fis. Mat. Natur. 45, 212–216 (1968).
[8] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57, 31–37 (2000).
[9] J. Caballero, J. Harjani and K. Sadarangani, A best proximity point theorem for Geraghty-contractions, Fixed Point Theory and Appl. 2012, 2012:231
[10] C. M. Chen, and W. Y. Sun, Periodic points for the weak contraction mappings in complete generalized metric spaces, Fixed Point Theory and Appl. 2012, 2012:79.
[11] S.H. Cho, J.S. Bae, Common fixed point theorems for mappings satisfying property (E.A) on cone metric spaces, Math. and Comput. Model. 53, 945–951 (2011).
[12] P. Das, BK. Lahiri, Fixed point of contractive mappings in generalized metric space, Math. Slovaca 59, 499–504 (2009)
[13] I. Erhan, E. Karapinar, T. Sekuli´c, Fixed points of (ψ, ϕ) contractions on rectangular metric spaces, Fixed Point Theory and Appl. 2012, 2012:138
[14] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40, 604–608 (1973).
[15] M. E. Gordji, M. Ramezani. Y. J. Cho, S. Pirbavafa, A generalization of Geraghty theorem in partially ordered metric space and application to ordinary differential equations, Fixed Point Theory and Appl. 2012, 2012:74.
[16] R. H. Haghi, Sh. Rezapour, N.Shazad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74, 1799–1803 (2011).
[17] E. Hille, R.S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, American Mathematical Society, Providence, RI, 1957.
[18] E. Hoxha, A. Hojat Ansari, K. Zoto, Some common fixed point results through generalized altering distances on dislocated metric spaces, Proceedings of EIIC, September 1-5, 2014, pages 403-409
[19] Z. Kadelburg, S. Radenovi´c, Fixed point results in generalized metric spaces without Hausdorff property, Math. Sci. 2014, 8:125
[20] E. Karapınar, On best proximity point of psi-Geraghty contractions, Fixed Point Theory and Appl. 2013, 2013:200
[21] M. Jleli and B. Samet, The Kannan’s fixed point theorem in a cone rectangular metric space, J. Nonlinear Sci. Appl. 2(3), 161–167 (2009)
[22] L. Kikina and K. Kikina, A fixed point theorem in generalized metric space, Demonstrateio Mathematica 46(1), 181–190 (2013)
[23] W. A. Kirk and N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory and Appl. 2013, 2013:129
[24] M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bulletin of the Australian Mathematical Society, 30 (1) (1984) 1–9.
[25] H. Lakzian and B. Samet, Fixed points for(ψ, φ)-weakly contractive mapping in generalized metric spaces, Appl. Math. Lett. 25, 902–906 (2012)
[26] D. Mihet, On Kannan fixed point principle in generalized metric spaces, J. Nonlinear Sci. Appl. 2(2), 92–96 (2009)
[27] Z. M. Fadail, A. Gh. Bin Ahmad, A. Hojat Ansari, Stojan Radenovic , Miloje Rajovic, Some Common Fixed Point Results of Mappings in 0-σ-Complete Metric-like Spaces via New Function, Applied Mathematical Sciences, Vol. 9, 2015, no. 83, 4109 - 4127
[28] P. D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Anal. TMA 67, 2361–2369 (2007)
[29] P. D. Proinov, New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems, J. Complexity 26, 3–42 (2010)
[30] S. Rathee and A. Kumar, Some common fixed point and invariant approximation results with generalized almost contractions, Fixed Point Theory and Appl. 2014, 2014:23
[31] B.E. Rhoades, A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc. 226, 257–290 (1977)
[32] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.
[33] T. Suzuki, Generalized metric spaces do not have the compatible toplogy, Abstr, Appl. Anal. 2014, Article ID 458098
[34] B. Samet, Discussion on: a fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces by A. Branciari, Publ. Math. (Debr.) 76(4), 493–494 (2010)
[35] WA. Wilson, On semimetric spaces, Am. J. Math. 53(2), 361–373 (1931) [36] S.K. Yang, J.S. Bae and S.H. Cho, Coincidence and common fixed and periodic point theorems in cone metric spaces. Comput. Math. Appl. 61, 170–177 (2011)
[36] S.K. Yang, J.S. Bae and S.H. Cho, Coincidence and common fixed and periodic point theorems in cone metric spaces. Comput. Math. Appl. 61, 170–177 (2011)
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