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Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach

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Abstract

We investigate the Hyers–Ulam–Rassias stability property of a quadratic functional equation. The analysis is done in the context of modular spaces. The type of stability considered here is very general in character which has been considered in various domains of mathematics. The speciality of the functional equation considered here is that it has a geometrical background behind its introduction. We approach the problem by applying a fixed point method for which a version of the contraction mapping principle in modular spaces is utilized. Also the results in this paper are established without using some familiar conditions on modular spaces.

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References

  1. K. W. Jun and H. M. Kim, “Solution of Hyers–Ulam stability problem for generalized Pappus equation,” J. Math. Anal. Appl. 299, 100–112 (2004).

    Article  MathSciNet  Google Scholar 

  2. S. M. Jung, “Hyers–Ulam stability of linear differential equations of first order. II,” App. Math. Lett. 19, 854–858 (2006).

    Article  MathSciNet  Google Scholar 

  3. L. Cadariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” J. Ineq. Pure and Applied Math. 4, 1–15 (2003).

    MathSciNet  MATH  Google Scholar 

  4. Y. Dong, “On approximate isometries and application to stability of a function,” J. Math. Anal. Appl. 426, 125–137 (2015).

    Article  MathSciNet  Google Scholar 

  5. L. Cadariu and V. Radu, “Fixed points and stability for functional equations in probabilistic metric and random normed spaces,” Fixed Point Theory and Applications, Article ID 589143 (2009).

    MATH  Google Scholar 

  6. J. Chmielinski and J. Tabor, “On approximate solutons of the pexider equation,” Aequ. Math. 46, 143–163 (1993).

    Article  Google Scholar 

  7. A. Grabiec, “The generalized Hyers–Ulam stability of a class of functional equations,” Publ. Math. Debrecen 48, 217–235 (1996).

    MathSciNet  MATH  Google Scholar 

  8. H. M. Kim and M. Y. Kim, “Generalized stability of Euler Lagrange quadratic functional equation,” Abstract and Applied Analysis, Article ID 219435 2012 (2012).

    Google Scholar 

  9. Th. M. Rassias, “On the stability of functional equations in Banach spaces,” J. Math. Anal. Appl 251, 264–284 (2000).

    Article  MathSciNet  Google Scholar 

  10. P. Saha, T. K. Samanta, P. Mondal, B. S. Choudhury, and M. D. L. Sen, “Applying fixed point techniques to stability problems in intuitionistic fuzzy Banach spaces,” Mathematics 974 (2020).

    Google Scholar 

  11. S. M. Ulam, Problems in Modern Mathematics, (Science Editions, Wiley, New York, 1964), Chap. VI.

    MATH  Google Scholar 

  12. D. H. Hyers, “On the stability of the linear functional equation,” Proc. Nat. Acad. Sci. 27, 222–224 (1941).

    Article  MathSciNet  Google Scholar 

  13. Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proc. Amer. Math. Soci. 72, 297–300 (1978).

    Article  MathSciNet  Google Scholar 

  14. P. M. Gruber, “Stability of isometries,” Trans. Amer. Soc. 245, 263–277 (1978).

    Article  MathSciNet  Google Scholar 

  15. N. C. Kayal, T. K. Samanta, P. Saha, and B. S. Choudhury, “A Hyers–Ulam–Rassias stability result for functional equations in intuitionistic fuzzy Banach spaces,” Iranian Journal of Fuzzy Systems 13, 87–96 (2016).

    MathSciNet  MATH  Google Scholar 

  16. A. K. Mirmostafaee and M. S. Moslehian, “Stability of additive mappings in non-Archimedean fuzzy normed spaces,” Fuzzy Sets and Systems 160, 1643–1652 (2009).

    Article  MathSciNet  Google Scholar 

  17. P. Saha, T. K. Samanta, P. Mondal, and B. S. Choudhury, “Stability of two variable pexiderized quadratic functional equation in intuitionistic fuzzy Banach spaces,” Proyecciones J. Math. 3, 447–467 (2019).

    Article  MathSciNet  Google Scholar 

  18. T. K. Samanta, P. Mondal, and N. C. Kayal, “The generalized Hyers–Ulam–Rassias stability of a quadratic functional equation in fuzzy Banach spaces,” Ann. Fuzzy Math. Inform. 6, 59–68 (2013).

    MathSciNet  MATH  Google Scholar 

  19. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. (Springer-Verlag, Berlin, 1983), Vol. 1034.

    Book  Google Scholar 

  20. H. Nakano, Modular Semi-Ordered Spaces (Maruzen C. Ltd., Tokyo (Japan), 1950).

    MATH  Google Scholar 

  21. M. A. Khamsi and W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces (Springer, Heidelberg–New York, 2015).

    Book  Google Scholar 

  22. W. M. Kozlowski, Modular Function Spaces in Series of Monographs and Textbooks in Pure and Applied Mathematics (Marcel Dekker, New York, 1988), Vol. 122.

    Google Scholar 

  23. A. A. N. Abdou and M. A. Khamsi, “Fixed point theorems in modular vector spaces,” J. Nonlinear Sci. Appl. 10, 4046–4057 (2017).

    Article  MathSciNet  Google Scholar 

  24. M. B. Ghaemi, M. Choubin, G. Sadeghi, and M. Gordji, “A fixed point approach to stability of quintic functional equations in modular spaces,” Kyungpook Math. J. 55, 313–326 (2015).

    Article  MathSciNet  Google Scholar 

  25. M. A. Khamsi, “Quasicontraction mappings in modular spaces without \(\Delta_2\)-conditions,” Fixed Point Theory and Applications, Article ID 916187 (2008).

    MathSciNet  MATH  Google Scholar 

  26. C. Kim and S. W. Park, “A Fixed point approach to the stability of additive-quadratic functional equations in modular spaces,” J. Chung. Math. Soc. 18, 321–330 (2015).

    Article  Google Scholar 

  27. H. Kim and Y. Hong, “Approximate Cauchy–Jensen type mappings in modular spaces,” Far East J. Math. Sci. 7, 1319–1336 (2017).

    Google Scholar 

  28. K. Wongkum, P. Chaipunya, and P. Kumam, “On the generalized Ulam–Hyers–Rassias stability of quadratic mappings in modular spaces without \(\Delta_2\)-conditions,” J. Function Spaces, 1–6 (2015).

    Article  Google Scholar 

  29. K. Wongkum, P. Kumama, Y. Chob, P. Thounthonge, and P. Chaipunyaa, “On the generalized Hyers–Ulam–Rassias stability for quartic functional equation in modular spaces,” J. Nonlinear Sci. Appl. 10, 1399–1406 (2017).

    Article  MathSciNet  Google Scholar 

  30. T. Z. Xu and J. M. Rassias, “Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces,” Advances in Difference Equations 1, 1399–1406 (2012).

    MathSciNet  MATH  Google Scholar 

  31. F. Crivelli, “Absolute values, valuations and completion,” Algebra in Positive Characteristic (2008).

    Google Scholar 

  32. M. A. Khamsi, W. M. Kozlowski, and S. Reich, “Fixed point theory in modular function spaces,” Nonlinear Anal. 14, 935–953 (1990).

    Article  MathSciNet  Google Scholar 

  33. G. Z. Eskandani and J. M. Rassias, “Stability of general A-cubic functional equations in modular spaces,” RACSAM, 425–435 (2018).

    Article  MathSciNet  Google Scholar 

  34. G. Sadeghi, “A fixed point approach to stability of functional equations in modular spaces,” Bull. Malays. Math. Sci. Soc. 37, 333–344 (2014).

    MathSciNet  MATH  Google Scholar 

  35. S. S. Kim, J. M. Rassias, and S. H. Kim, “A fixed point approach to the stability of a nonic functional equation in modular spaces,” WSEAS Transactions on Mathematics 17, 130–136 (2018).

    Google Scholar 

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Authors and Affiliations

  1. Indian Institute of Engineering Science and Technology, Shibpur, 711103, Howrah, India

    P. Saha & B. S. Choudhury

  2. Bijoy Krishna Girls’ College, Howrah, 711101, Howrah, India

    Pratap Mondal

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  1. P. Saha
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  2. Pratap Mondal
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  3. B. S. Choudhury
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Correspondence to P. Saha.

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Saha, P., Mondal, P. & Choudhury, B.S. Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach. Math Notes 109, 262–269 (2021). https://doi.org/10.1134/S0001434621010302

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