On Some Fixed Point Theorems for $\mathcal{G} (\Sigma, \vartheta, \Xi )-$Contractions in Modular $b-$Metric Spaces

Year 2022, Volume: 5 Issue: 4, 210 - 227, 01.12.2022

Mahpeyker Öztürk , Abdurrahman Büyükkaya

https://doi.org/10.33401/fujma.1107963

Abstract

This article aims to specify a new $C-$class function endowed with altering distance and ultra altering distance function via generalized $\Xi -$contraction, which is called the $\mathcal{G}\left( {\Sigma ,\vartheta ,\Xi } \right) - $contraction in modular $b-$metric spaces. Regarding these new contraction type mappings, the study includes some existence and uniqueness theorems, and to indicate the usability and productivity of these results, some applications related to integral type contractions and an application to the graph structure.

Keywords

Common fixed point, $C-$class function, Modular $b-$metric space

References

• [1] S. Banach, Sur les operations dans les emsembles abstraits et leurs applications aux equations integrales, Fund. Math., 1 (1922), 133-181.
• [2] I. A. Bakhtin, The contraction mapping principle in quasi metric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst., 30 (1989), 26–37.
• [3] S. Czerwik, Contraction mappings in b􀀀metric spaces, Acta. Math. Inform. Univ. Ostrav, 1 (1) (1993), 5-11.
• [4] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 263-276.
• [5] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b􀀀metric spaces, Math. Slovaca, 64 (4) (2014), 941–960.
• [6] V. V. Chistyakov, Modular metric spaces generated by F-modulars, Folia Math., 15 (2008), 3-24.
• [7] V. V. Chistyakov, Modular metric spaces, I: Basic concepts. Nonlinear Anal., 72 (2010), 1-14.
• [8] V. V. Chistyakov, Modular metric spaces, II: Application to superposition operators, Nonlinear Anal., 72 (2010), 15-30.
• [9] V. V. Chistyakov, Fixed points of modular contractive maps, Dokl. Math., 86 (2012), 515-518.
• [10] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl., 93 (2011), 1-9.
• [11] M. E. Ege, C. Alaca, Some results for modular b􀀀metric spaces and an application to system of linear equations, Azerbaijan J. Math., 8 (1) (2018), 3-14.
• [12] V. Parvaneh, N. Hussain, M. Khorshidi, N. Mlaiki, H. Aydi, Fixed point results for generalized F-contractions in modular b-metric spaces with applications, Mathematics, 7 (10) (2019), 1-16.
• [13] A. H. Ansari, Note on ”j 􀀀y􀀀contractive type mappings and related fixed point, The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, September/2014, 11 (2014), 377–380.
• [14] M. S. Khan, M. Swalesh, S. Sessa, Fixed points theorems by altering distances between the points, Bull. Aust. Math. Soc., 30 (1984), 1–9.
• [15] A. H. Ansari, J. M. Kumar, N. Saleem, Inverse C􀀀class function on weak semi compatibility and fixed point theorems for expansive mappings in G-metric spaces, Math. Morav., 24 (1) (2020), 93-108.
• [16] A. H. Ansari, T. Dosenovic, S. Radenovic, N. Saleem, V. Sesum-Cavic, C􀀀class functions on some fixed point results in ordered partial metric space via admissible mappings, Novi Sad J. Math., 49 (1) (2019), 101-116. [17] N. Saleem, A. H. Ansari, M. K. Jain, Some fixed point theorems of inverse C􀀀class function under weak semi compatibility, J. Fixed Point Theory, 9 (2018), 2018.
• [18] A. H. Ansari, N. Saleem, B. Fisher, M. S. Khan, C􀀀class function on Khan type fixed point theorems in generalized metric space, Filomat, 31 (11) (2017), 3483-3494.
• [19] A. Fulga, A. M. Proca, Fixed point for Geraghty JE􀀀contractions, J. Nonlinear Sci. Appl., 10 (2017), 5125-5131. doi.org/10.22436/jnsa.010.09.48.
• [20] A. Fulga, A. M. Proca, A new Generalization of Wardowski fixed point theorem in complete metric spaces, Adv. Theory Nonlinear Anal. Appl., 1 (2017), 57-63.
• [21] A. Fulga, E. Karapınar, Some results on S􀀀contractions of Type E, Mathematics, 195 (6) (2018), 1-9.
• [22] B. Alqahtani, A. Fulga, E. Karapınar, A short note on the common fixed points of the Geraghty contraction of type ES;T , Demonstr. Math., 51 (2018), 233-240.
• [23] A. M. Proca, Fixed point theorem for jM􀀀Geraghty contraction, In: Proceedings of the Scientific Research and Education in the Air Force (AFASES2018), May 22-27/2018, Brasov, Romania, 1 (2018), 311-316. doi.org/10.19062/2247-3173.2018.20.41.
• [24] A. M. Proca, New fixed point theorem for generalized contractions, Bull. Transilv. Univ. Bras. III: Math. Inform. Phys, 12 (61) (2019), 435-442.
• [25] L. Ciric, On contraction type mappings, Math. Balcanica, 1 (1971), 52-57.
• [26] G. Jungck, Common fixed points for non-continuous nonself mappings on non-numeric spaces, Far East J. Math. Sci., 4 (2) (1996), 199-212.
• [27] H. Aydi, A. Felhi, E. Karapınar, S. Sahmim, A Nadler-type fixed point theorem in dislocated spaces and applications, Miskolc. Math. Notes, 19 (2018), 111-124. doi.org/10.18514/mmn.2018.1652.
• [28] E. Girgin, M. O¨ ztu¨rk, Modified Suzuki-Simulation type contractive mapping in non-Archimedean quasi modular metric spaces and application to graph theory, Mathematics, 7 (9) (2019), 1-14. doi.org/10.3390/math7090769.
• [29] D. Gopal, C. Vetro, M. Abbas, D. K. Patel, Some coincidence and periodic points results in a metric space endowed with a graph and applications, Banach J. Math. Anal., 9 (2015), 128–140. doi.org/10.15352/bjma/09-3-9.
• [30] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci., 29 (2002), 531-536.
• [31] B. Azadifar, G. Sadeghi, R. Saadati, C. Park, Integral type contractions in modular metric spaces, J. Inequalities Appl., 483 (2013), 1-14. doi.org/10.1186/1029-242x-2013-483.