Skip to main content

Existence Theory on Modular Metric Spaces

  • Chapter
  • First Online:
Advances in Mathematical Inequalities and Applications

Part of the book series: Trends in Mathematics ((TM))

Abstract

Since the year 1922, Banach’s contraction principle, due to its simplicity and usability, has become a popular tool in modern analytics, particularly in nonlinear analysis, including the use of equations, differential equations, variance, equilibrium problems, and much more (see, e.g., [1,2,3,4,5,6,7,8,9,10]).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Z. Golubović, Z. Kadelburg, S. Radenović, Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012, Article ID 192581, 18 p (2012)

    Google Scholar 

  2. M.A. Khamsi, A. Latif, H. Al-Sulami, KKM and Ky Fan theorems in modular function spaces. Fixed Point Theory Appl. 2011, 57 (2011)

    Article  MathSciNet  Google Scholar 

  3. M.A. Khamsi, Quasicontraction mappings in modular spaces without \(\Delta _{2}\)-condition. Fixed Point Theory Appl. 2008, Article ID 916187, 6 p (2008)

    Google Scholar 

  4. K. Kuaket, P. Kumam, Fixed points of asymptotic pointwise contractions in modular spaces. Appl. Math. Lett. 24, 1795–1798 (2011)

    Article  MathSciNet  Google Scholar 

  5. A. Amini-harandini, Fixed point theory for generalized quasicontraction maps in vector modular spaces. Comp. Math. Appl. 61, 1891–1897 (2011)

    Article  MathSciNet  Google Scholar 

  6. H. Aydi, H.K. Nashine, B. Samet, H. Yazidi, Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 74(17), 6814–6825 (2011)

    Article  MathSciNet  Google Scholar 

  7. H. Aydi, Common fixed point results for mappings satisfying (\(\psi \),\(\phi \))-weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 12(2), 53–64 (2012)

    MathSciNet  MATH  Google Scholar 

  8. W. Sintunavarat, P. Kumam, Weak condition for generalized multi-valued \((f,\alpha, \beta )\)-weak contraction mappings. Appl. Math. Lett. 24, 460–465 (2011)

    Article  MathSciNet  Google Scholar 

  9. W. Sintunavarat, Y.J. Cho, P. Kumam, Common fixed point theorems for \(c\)-distance in ordered cone metric spaces. Comput. Math. Appl. 62, 1969–1978 (2011)

    Article  MathSciNet  Google Scholar 

  10. C. Mongkolkeha, P. Kumam, Fixed point theorems for generalized asymptotic pointwise \(\rho \)-contraction mappings involving orbits in modular function spaces. Appl. Math. Lett. https://doi.org/10.1016/j.aml.2011.11.027 (2012) (in press)

    Article  MathSciNet  Google Scholar 

  11. M. Geraghty, On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)

    Article  MathSciNet  Google Scholar 

  12. A. Amini-Harandi, H. Emani, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, 2238–2242 (2010)

    Article  MathSciNet  Google Scholar 

  13. M. Eshaghi Gordji, M. Ramezani, Y.J. Cho, S. Pirbavafa, A generalization of Geraghty’s theorem in partially ordered metric spaces and application to ordinary differential equations. Fixed point Theory Appl. 2012, 74 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. H. Nakano, in Modulared Semi-Ordered Linear Spaces, vol. 1 (Tokyo Math. Book Ser.) (Maruzen Co., Tokyo, 1950)

    Google Scholar 

  16. H. Nakano, in Topology and Linear Topological Spaces, vol. 3 (Tokyo Math. Book Ser.) (Maruzen Co., Tokyo, 1951)

    Google Scholar 

  17. P. Kumam, Some geometric properties and fixed point theorem in modular spaces, in Fixed Point Theorem and Its Applications, ed. by J. Garcia Falset, L. Fuster, B. Sims (Yokohama Publishers, Yokohama, 2004), pp. 173–188

    Google Scholar 

  18. P. Kumam, Fixed point theorems for nonexpansive mappings in modular spaces. Archivum Mathematicum (BRONO) 40(4), 345–353 (2004)

    MathSciNet  MATH  Google Scholar 

  19. P. Kumam, On uniform opial condition and uniform Kadec-Klee property in modular spaces. J. Interdisc. Math. 8(3), 377–385 (2005)

    Article  MathSciNet  Google Scholar 

  20. P. Kumam, On nonsquare and Jordan-Von Neumann constants of modular spaces. SE Asian Bull. Math. 30(1), 67–77 (2006)

    MathSciNet  MATH  Google Scholar 

  21. C. Mongkolkeha, P. Kumam, Common fixed point for generalized weak contraction mapping in Modular spaces. Scientiae Mathematicae Japonicae, online, e-2012, 117–127

    Google Scholar 

  22. Y.J. Cho, R. Saadati, G. Sadeghi, Quasi-contractive mappings in modular metric spaces. J. Appl. Math. 2012, Article ID 907951, 5 p (2012)

    Google Scholar 

  23. P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed-point theorems for multivalued mappings in modular metric spaces. Abstr. Appl. Anal. 2012, Article ID 503504, 14 p (2012)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. M.R. Taskovic, A generalization of Banach’s contraction principle. Nouvell serie tome. 23(37), 179–191 (1978)

    MathSciNet  MATH  Google Scholar 

  27. L.J. Ćirić, A generalization of Banachs contraction principle. Proc. Am. Math. Soc. 45, 267273 (1974)

    Article  MathSciNet  Google Scholar 

  28. S. Czerwik, Contraction mappings in \(b\)-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)

    MathSciNet  MATH  Google Scholar 

  29. T. Suzuki, Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. App. 253(2), 440458 (2001)

    MathSciNet  Google Scholar 

  30. H.L. Guang, Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl 332, 14681476 (2007)

    MathSciNet  Google Scholar 

  31. B. Samet, C. Vetro, P. Vetro, Fixed point theorems for \(\alpha \)-\(\psi \)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)

    Article  MathSciNet  Google Scholar 

  32. 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(3), 128–139 (2015)

    Article  MathSciNet  Google Scholar 

  33. D. Gopal, M. Abbas, D.K. Patel, C. Vetro, Fixed points of \(\alpha \)-type \(F\)-contractive mappings with anapplication to nonlinear fractional differential equation. Acta Math. Scientia 36B(3), 1–14 (2016)

    MATH  Google Scholar 

  34. N. Hussain, A. Latif, I. Iqbal, Fixed point results for generalized \(F\)-contractions in modular metric and fuzzy metric spaces. Fixed Point Theory Appl. 2015, 158 (2015)

    Article  MathSciNet  Google Scholar 

  35. A. Padcharoen, D. Gopal, P. Chaipunya, P. Kumam, Fixed point and periodic point results \(\alpha \)-type \(F\)-contractions in modular metric spaces. Fixed Point Theory Appl. 2016 12 p (2016)

    Google Scholar 

  36. D. Wardowski, Fixed points of new type of contractive mappings in complete metric space. Fixed Point Theory Appl. (2012). https://doi.org/10.1186/1687-1812-2012-94

    Article  MathSciNet  MATH  Google Scholar 

  37. A.N.N. Abdou, M.A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. (1), 163 (2013)

    Article  MathSciNet  Google Scholar 

  38. J. Jachymski, The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 1(136), 1359–1373 (2008)

    MathSciNet  MATH  Google Scholar 

  39. A.C. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132(5), 1435–1443 (2004)

    Article  MathSciNet  Google Scholar 

  40. G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs (CRC Press, New York, 2011)

    MATH  Google Scholar 

  41. R. Johnsonbaugh, Discrete Mathematics (Prentice Hall, New York, 1997)

    MATH  Google Scholar 

  42. M.R. Alfuraidan, On monotone Ćirić quasi-contraction principle for mappings with a graph. Fixed Point Theory Appl. 2015, 93 11 p (2015)

    Google Scholar 

  43. A. Padcharoen, P. Kumam, D. Gopal, Coincidence and periodic point results in a modular metric space endowed with a graph and applications. Creat. Math. Inform. 26(1), 95–104 (2017)

    MathSciNet  MATH  Google Scholar 

  44. G.S. Jeong, B.E. Rhoades, Maps for which \(F(T) = F(T^n)\). Fixed Point Theory Appl. 6, 87–131 (2005)

    Google Scholar 

  45. C. Chifu, G. Petrusel, Generalized contractions in metric space endowed with a graph. Fixed Point Theory Appl. 2012, 161, 9 p (2012)

    Google Scholar 

  46. W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011, Article ID 637958, 14 p

    Google Scholar 

  47. P. Sumalai, P. Kumam, Y.J. Cho, A. Padcharoen, The \((CLR_g)\)-property for coincidence point theorems and Fredholm integral equations in modular metric spaces. Eur. J. Pure Appl. Math. 10(2), 238–254 (2017)

    MathSciNet  MATH  Google Scholar 

  48. P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, P. Kumam, Erratum to “fixed-point theorems for multivalued mappings in modular metric spaces”. Abstr. Appl. Anal. 2012, 2 (2012)

    MATH  Google Scholar 

  49. P. Chaipunya, P. Kumam, An observation on set-valued contraction mappings in modular metric spaces. Thai J. Math. 13(1), 9–17 (2015)

    MathSciNet  MATH  Google Scholar 

  50. S.B. Nadler Jr., Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)

    Article  MathSciNet  Google Scholar 

  51. M. Sgroi, C. Vetro, Multi-valued \(F\)-contractions and the solution of certain functional and integral equations. Filomat 27, 1259–1268 (2013)

    Article  MathSciNet  Google Scholar 

  52. D. Jain, A. Padcharoen, P. Kumam, D. Gopal, A new approach to study fixed point of multivalued mappings in modular metric spaces and applications. Mathematics 4, 51 (2016). https://doi.org/10.3390/math4030051

    Article  MATH  Google Scholar 

  53. S. Moradi, M. Mohammadi Anjedani, E. Analoei, On existence and uniqueness of solutions of a nonlinear Volterra-Fredholm integral equation. Int. J. Nonlinear Anal. Appl. 6, 62–68 (2015)

    MATH  Google Scholar 

  54. M. Cosentino, P. Vetro, Fixed point results for \(F\)-contractive mappings of Hardy-Rogers-type. Filomat 28(4), 715–722 (2014)

    Article  MathSciNet  Google Scholar 

  55. H. Piri, P. Kumam, Some fixed point theorems concerning \(F\)-contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210, 11 p (2014)

    Google Scholar 

  56. B.E. Rhoades, Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)

    Article  MathSciNet  Google Scholar 

  57. V.V. Chistyakov, Fixed points of modular contractive maps. Doklady Math. 86(1), 515–518 (2012)

    Article  MathSciNet  Google Scholar 

  58. V.V. Chistyakov, Modular contractions and their application, in Models, Algorithms, and Technologies for Network Analysis, volume 32 of Springer Proceedings in Mathematics & Statistics, ed. by B. Goldengorin, V.A. Kalyagin, P.M. Pardalos (Springer, New York, 2013), pp. 65–92

    Google Scholar 

  59. O. Acar, G. Durmaz, G. Minak, Generalized multivalued \(F\)-contractions on complete metric spaces. Bull. Iranian Math. Soc. 40, 1469–1478 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Poom Kumam .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Padcharoen, A., Chaipunya, P., Kumam, P. (2018). Existence Theory on Modular Metric Spaces. In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_2

Download citation

Publish with us

Policies and ethics