Some Results for Modular b-Metric Spaces and an Ap- plication to System of Linear Equations

Meltem E. Ege

Abstract


In this paper, we dene the modular b-metric space with some new notions and prove Banach xed 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 a system of linear equations.


Keywords


Fixed point, modular b-metric space, contraction principle

References


M.A. Alghamdi, N. Hussain and P. Salimi, Fixed point and coupled xed point the-orems on b-metric-like spaces, J. Inequal. Appl., 2013:402 (2013).

B. Azadifar, M. Maramaei and G. Sadeghi, Common xed 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,

{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., 2012, Article ID 907951 (2012).

S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Os-

traviensis, 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 xed point theorems for

contraction mappings in modular metric spaces, Fixed Point Theory Appl., 2012:144

(2012).

W.S. Du and E. Karapinar, A note on cone b-metric and its related results: general-

izations or equivalence, Fixed Point Theory Appl., 2012:210 (2012).

O. Ege, Complex valued rectangular b-metric spaces and an application to linear

equations, J. Nonlinear Sci. Appl., 8(6), 1014{1021 (2015).

M.E. Ege and C. Alaca, Fixed point results and an application to homotopy in mod-

ular 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 xed

point results, J. Comput. Anal. Appl., 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 xed 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 xed 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, 1{14 (1989).

C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contrac-

tion mappings in modular metric spaces, Fixed Point Theory Appl., 2011:93 (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 xed point theorems for mappings satisfying

(E.A)-property in b-metric spaces, J. Nonlinear Sci. Appl., 8, 1127{1133 (2015).

M. Pacurar, A xed point result for ϕ-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. Com-

put., 226, 725{737 (2014).

P. Turpin, Fubini inequalities and bounded multiplier property in generalized mod-

ular spaces, Comment. Math. Tomus specialis in honorem Ladislai Orlicz I, 331{353

(1978).


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