Some Results for Modular $b$-Metric Spaces and an Application to System of Linear Equations

Meltem Erden Ege, Cihagir Alaca

Abstract


In this paper, we define the modular $b$-metric space with some new notions and prove Banach fixed point theorem and its two generalizations for the new space. At the end of the paper, we give an application of Banach contraction principle to system of linear equations.

Keywords


Fixed point; modular $b$-metric; contraction principle

References


- M.A. Alghamdi, N. Hussain and P. Salimi, Fixed point and coupled fixed point theorems on $b$-metric-like spaces, J. Inequal. Appl., 2013:402.

- B. Azadifar, M. Maramaei and G. Sadeghi, Common fixed point theorems in modular $G$-metric spaces, J. Nonlinear Anal. Appl. 2013, 1-9 (2013).

- S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math., 3, 133-181 (1922).

- P. Chaipunya, Y.J. Cho and P. Kumam, Geraghty-type theorems in modular metric spaces with an application to partial differential equation, Adv. Difference. Equ., 83 (2012).

- V.V. Chistyakov, Modular metric spaces generated by $F$-modulars, Folia Math., 14, 3-25 (2008).

- V.V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal., 72, 1-14 (2010).

- V.V. Chistyakov, Fixed points of modular contractive maps, Dokl. Math., 86, 515-518 (2012).

- Y.J. Cho, R. Saadati and G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math., 907951 (2012).

- S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1, 5-11 (1993).

- S. Czerwik, K. Dlutek and S. L. Singh, Round-off stability of iteration procedures for operators in $b$-metric spaces, J. Nat. Phys. Sci., 11, 87-94 (1997).

- H. Dehghan, M.E. Gordji and A. Ebadian, Comment on fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl., 144 (2012).

- W.S. Du and E. Karapinar, A note on cone $b$-metric and its related results: generalizations or equivalence, Fixed Point Theory Appl., 210 (2012).

- O. Ege, Complex valued rectangular $b$-metric spaces and an application to linear equations, J. Nonlinear Sci. Appl., 8, 1014-1021 (2015).

- M.E. Ege and C. Alaca, Fixed point results and an application to homotopy in modular metric spaces, J. Nonlinear Sci. Appl., 8(6), 900-908 (2015).

- M.E. Ege and C. Alaca, Some properties of modular $S$-metric spaces and its fixed point results, Journal of Computational Analysis and Application, 20(1), 24-33 (2016).

- C. Alaca, M.E. Ege and C. Park, Fixed point results for modular ultrametric spaces, J. Comput. Anal. Appl., 20(7), 1259-1267 (2016).

- E. Kilinc and C. Alaca, A fixed point theorem in modular metric spaces, Adv. Fixed Point Theory, 4(2), 199-206 (2014).

- E. Kilinc and C. Alaca, Fixed point results for commuting mappings in modular metric spaces, J. Appl. Funct. Anal., 10(3-4), 204-210 (2015).

- M. Kir and H. Kiziltunc, On some well known fixed point theorems in $b$-metric spaces, Turkish J. Anal. Number Theory, 1(1), 13-16 (2013).

- P. Kumam, Fixed point theorems for nonexpansive mapping in modular spaces, Arch. Math., 40, 345-353 (2004).

- L. Maligranda, Orlicz spaces and interpolation: Semin. in Math., 5 (1989).

- C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl., 93 (2011).

- J. Musielak and W. Orlicz, On modular spaces, Studia Math., 18, 49-65 (1959).

- J. Musielak and W. Orlicz, Some remarks on modular spaces, Bull. Acad. Pol. Sci. Ser. Sci. Math., Astron. Phys., 7, 661–668 (1959).

- H. Nakano, Modulared semi-ordered linear spaces, In Tokyo Math. Book Ser, Maruzen Co, Tokyo, 1 (1950).

- M.O. Olatinwo and C.O. Imoru, A generalization of some results on multi-valued weakly Picard mappings in b-metric space, Fasc. Math., 40, 45-56 (2008).

- W. Orlicz, A note on modular spaces, Bull. Acad. Pol. Sci. Ser. Sci. Math., Astron. Phys., 9, 157–162 (1961).

- W. Orlicz, Collected Papers, 2, 851–1688 (1988).

- V. Ozturk and D. Turkoglu, Common fixed point theorems for mappings satisfying (E.A)-property in $b$-metric spaces, J. Nonlinear Sci. Appl., 8, 1127-1133 (2015).

- M. Pacurar, A fixed point result for $phi$-contractions on $b$-metric spaces without the boundedness assumption, Fasc. Math., 43, 127-137 (2010).

- J.R. Roshan, V. Parvaneh and I. Altun, Some coincidence point results in ordered $b$-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226, 725-737 (2014).

- P. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math. Tomus specialis in honorem Ladislai Orlicz I, 331-353 (1978).


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