Cyclic $(\alpha ,\beta )$-Admissible Mappings in Modular Spaces and Applications to Integral Equations

Year 2019, , 85 - 93, 28.06.2019

Müzeyyen Sangurlu Sezen

https://doi.org/10.32323/ujma.543824

Cited By: 3

Abstract

The main concern of this study is to present a generalization of Banach's fixed point theorem in some classes of modular spaces, where the modular is convex and satisfying the $\Delta _{2}$-condition. In this work, the existence and uniqueness of fixed point for $(\alpha ,\beta )-(\psi ,\varphi )-$ contractive mapping and $\alpha -\beta -\psi -$weak rational contraction in modular spaces are proved. Some examples are supplied to support the usability of our results. As an application, the existence of a solution for an integral equation of Lipschitz type in a Musielak-Orlicz space is presented.

Keywords

Modular space, Cyclic $(\alpha ;\beta )$-admissible mapping, $(\alpha ;\beta )-(\psi ;\phi )$-contractive mapping

References

• [1] A.A.N. Abdou, M.A. Khamsi, On common fixed points in modular vector spaces, Fixed Point Theory and Appl., 2015(229) (2015).
• [2] H. Abobaker, R. A. Ryan, Modular Metric Spaces, Irish Math. Soc. Bulletin, 80 (2017), 35-44.
• [3] U. Aksoy, E. Karapinar, I. M. Erhan, Fixed point theorems in complete modular metric spaces and an application to anti-periodic boundary value problems, Filomat 31(17) (2017), 5475-5488.
• [4] S. Alizadeh, F. Moradlou, P. Salimi, Some fixed point results for (a;b)􀀀(y;f)-contractive mappings, Filomat, 28(3) (2014), 635-647.
• [5] A. H. Ansari, Marta Demma, Liliana Guran, Jung Rye Lee, Choonkil Park, Fixed point results for C-class functions in modular metric spaces, J. Fixed Point Theory Appl. https://doi.org/10.1007/s11784-018-0580-z
• [6] S. Banach, Sur les op´erations dans les ensembles abstraits et leurs applications aux ´equations int´egrales, Fund. Math., 24(3) (1922), 133-182.
• [7] M. Beygmohammadi, A. Razani, Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space, Int. J. Mathematics and Mathematical Sci., Article ID 317107, 2010.
• [8] Vyacheslav V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Analysis 72 (2010), 1-14.
• [9] Vyacheslav V. Chistyakov, A fixed point theorem for contractions in modular metric spaces, arXiv:1112.5561v1Sˇmath.FAC´ 23Dec2011.
• [10] H. Dehghan, M. E. Gordji, A. Ebadian, Comment on ”Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2012(144) (2012).
• [11] P. N. Dutta, B.S. Coudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., Article ID 406368, 2018.
• [12] M. E. Ege, C. Alaca, Some results for modular b-metric spaces and an application to system of linear equations, Azerbaijan Journal of Mathematics, 8(1) (2018).
• [13] E. Girgin, M. Öztürk $\alpha_k$- implicit contraction in non-AMMS with some applications, Fundam. J. Math. Appl., 1(2) (2018), 212-219.
• [14] A. Hajji, E. Hanebaly, Perturbed integral equations in modular function spaces, Fund. Math., 2003(20) (2003), 1-7.
• [15] M. A. Khamsi, Nonlinear semigroups in modular function space these d’etat, Departement de Mathematiques, Rabat, 1994.
• [16] M. A. Khamsi, A convexity property in modular function spaces, Math. Jpn., 44(2) (1996), 269-279.
• [17] S. M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 1990(14) (1990), 935-953.
• [18] F. Khojasteha, S. Shuklab, S. Radenovic, A new approach to the study of fixed point theory in modular spaces with simulation functions, Filomat 29(6) (2015), 1189–1194.
• [19] M. Kiftiah, Fixed Point Theorems for Modular Contraction Mappings on Modulared Spaces, Int. Journal of Math. Analysis, 7(22) (2013), 965 - 972.
• [20] W. M. Kozlowski, Modular Function Spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, 122 (1922), Dekker, New York 1988.
• [21] H. Lakzian, Best proximity points for weak MT-cyclic Kannan contractions, Fundam. J. Math. Appl., 1(1) (2018), 43-48.
• [22] Z. Mitrovic, S. Radenovic, H. Aydi, On a two new approach in modular spaces, Italian J. Pure Appl. Math. 2019.
• [23] C. Mongkolkeha, P. Kumam, Some fixed point results for generalized weak contraction mappings in modular spaces, Int. J. Anal. , 5, Article ID 247378, 2013.
• [24] C. Mongklkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011(93) (2011).
• [25] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.
• [26] H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Ltd., Tokyo, 1950.
• [27] H. Nakano, Modulared sequence spaces, Proc. Japan Acad., 27(9) (1951), 508-512.
• [28] M.O. Olatinwo, Some non-unique fixed point theorems of $\acute{C}iri\acute{c}$ type using rational-type contractive conditions, Georgian Math. J., 24 (2016), 455-461.
• [29] M. Paknazar, M. Eshaghi, Y.J. Cho, S. M. Vaezpour, A Pata-type fixed point theorem in modular spaces with application, Fixed Point Theory Appl. 2013(239) (2013).
• [30] M. Paknazar, M. A. Kutbi, M. Demma, P. Salimi, On non-Archimedean modular metric spaces and some nonlinear contraction mappings, J. Nonlinear Sci and Appl. (in press).
• [31] P. Salimi, A. Latif, N. Hussain, Modified $\alpha-\varphi $-contractive mappings with applications, Fixed Point Theory and Appl., 2013(151) (2013).
• [32] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha -\varphi $-contractive type mappings, Nonlinear Anal, 2012(75) (2012), 2154-2165.
• [33] A.A. Taleb, E. Hanebaly, A fixed point theorem and its application to integral equations in modular spaces, Proceedings of the American Math. Soci., 128(2) (1999), 419-426.
• [34] B. Zlatanov, Best proximity points in modular function spaces, Arab. J. Math. 128 (2015), 215-227.