Modular Space Stability for Cubic Functional Equations in Nonlinear Material Modeling
Resumo
This paper investigates the stability of a generalized cubic functional equation of the form
\begin{align*}
(m-n)\Big[(m+n)^3 f\Big(\tfrac{nu + mv}{\,n+m\,}\Big) + (n-m)^3 f\Big(\tfrac{nu - mv}{\,n-m\,}\Big)\Big]\\
+(m+n)\Big[(m+n)^3 f\Big(\tfrac{mu + nv}{\,m+n\,}\Big) + (m-n)^3 f\Big(\tfrac{mu - nv}{\,m-n\,}\Big)\Big]\\
=~mn(m^2+n^2)\big[f(u+v)+f(u-v)\big] + 2\,(m^4-n^4)f(u),
\end{align*}
within the setting of modular normed spaces. Using the direct method of Hyers--Ulam and a suitably defined control function, we establish explicit stability bounds for approximately cubic mappings. In addition, by employing an operator constructed in the modular space without $\Delta_2$-conditions and applying the fixed point alternative, we obtain existence, uniqueness, and generalized Hyers--Ulam stability of the exact cubic solution.
An application to nonlinear constitutive modeling in continuum mechanics is presented to illustrate the physical relevance of the cubic equation. The cubic stress--strain relation, widely used in modeling polymers, biological tissues, and metals under finite deformation, fits naturally into the functional framework developed in this study. The stability results guarantee that experimentally observed or numerically computed approximate constitutive laws admit a unique exact cubic model in their vicinity, enhancing robustness in material characterization and computational simulations. The findings demonstrate both the theoretical depth and practical significance of stability analysis for higher-order functional equations in modern applied mathematics.
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Referências
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Aboutaib, I., Benzarouala, C., Brzdęk, J., Leśniak, Z. and Oubbi, L.,
{\it Ulam Stability of a General Linear Functional Equation in Modular Spaces},
Symmetry 14 (11), 2468, (2022).
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Aczel, J. and Dhombres, J.,
{\it Functional Equations in Several Variables},
Cambridge Univ. Press, (1989).
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Aoki, T.,
{\it On the stability of the linear transformation in Banach spaces},
J. Math. Soc. Japan 2, 64--66, (1950).
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Arunkumar, M., Murthy, S. and Ganapathy, G.,
{\it Stability of generalized $n$-dimensional cubic functional equation in fuzzy normed spaces},
Int. J. Pure Appl. Math. 77, 179--190, (2012).
\bibitem{MASK}
Arunkumar, M. and Karthikeyan, S.,
{\it Solution and intuitionistic fuzzy stability of $n$-dimensional quadratic functional equation: Direct and fixed point methods},
Int. J. Adv. Math. Sci. 2 (1), 21--33, (2014).
\bibitem{MAKR}
Arunkumar, M., Karthikeyan, S. and Ramamoorthi, S.,
{\it Generalized Ulam--Hyers stability of $n$-dimensional cubic functional equation in FNS and RNS: Various methods},
Middle-East J. Sci. Res. 24 (S2), 386--404, (2016).
\bibitem{Baak07}
Baak, C. and Moslenian, M. S.,
{\it On the stability of an orthogonally cubic functional equation},
Kyungpook Math. J. 47, 69--76, (2007).
\bibitem{bodagh}
Bodaghi, A.,
{\it Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations},
J. Intell. Fuzzy Syst. 30, 2309--2317, (2016).
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Czerwik, S.,
{\it Functional Equations and Inequalities in Several Variables},
World Scientific, (2002).
\bibitem{DeviRaniKumar23}
Devi, S., Rani, A. and Kumar, M., {\it Stability of functional equations in modular spaces and fuzzy normed spaces}, Int. J. Math. Its Appl. 11(3), 113--127, (2023).
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Fast, H.,
{\it Sur la convergence statistique},
Colloq. Math. 2, 241--244, (1951).
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{\it On statistical convergence},
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Ganapathy, G., Sakthi, R., Karthikeyan, S., Vijaya, N., Suresh, M. and Venkataramanan, K.,
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J. Math. Comput. Sci. 35 (1), 82--95, (2024).
\bibitem{g1} Gowri, S., Donganont, S., Karthick, S., Balaanandhan, R., Tamilvanan, K., {\it Hyers-Ulam stability of generalized quartic mapping in non-Archimedean $(n,\beta)$-normed spaces}, Eur. J. Pure Appl. Math. 18, 6699-6699 (2025).
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G\u{a}vru\c{t}a, P.,
{\it A generalization of the Hyers--Ulam--Rassias stability of approximately additive mappings},
J. Math. Anal. Appl. 184, 431--436, (1994).
\bibitem{hyers}
Hyers, D. H.,
{\it On the stability of the linear functional equation},
Proc. Natl. Acad. Sci. U.S.A. 27, 222--224, (1941).
\bibitem{KWJ}
Jun, K. W. and Kim, H. M.,
{\it The generalized Hyers--Ulam--Rassias stability of a cubic functional equation},
J. Math. Anal. Appl. 274, 867--878, (2002).
\bibitem{jungbook}
Jung, S. M.,
{\it Hyers--Ulam--Rassias Stability of Functional Equations in Mathematical Analysis},
Hadronic Press, (2001).
\bibitem{MJRMAES}
Rassias, M. J., Arunkumar, M. and Sathya, E.,
{\it Stability of a $k$-cubic functional equation in quasi-beta normed spaces: Direct and fixed point methods},
Brit. J. Math. Comput. Sci. 8 (5), 346--360, (2015).
\bibitem{Junkim03}
Jun, K. W. and Kim, H. M.,
{\it On the Hyers--Ulam--Rassias stability of a general cubic functional equation},
Math. Inequal. Appl. 6 (2), 289--302, (2003).
\bibitem{kara}
Karakus, S.,
{\it Statistical convergence on probabilistic normed spaces},
Math. Commun. 12, 11--23, (2007).
\bibitem{Karthikeyan}
Karthikeyan, S., Rassias, J. M., Arunkumar, M. and Sathya, E.,
{\it Generalized Ulam--Hyers stability of $(a,b;k>0)$-cubic functional equation in intuitionistic fuzzy normed spaces},
J. Anal., (2018).
\bibitem{KarthikeyanJMCS}
Karthikeyan, S., Park, C., Palani, P. and Kumar, T. R. K.,
{\it Stability of an additive--quartic functional equation in modular spaces},
J. Math. Comput. Sci. 26 (1), 22--40, (2022).
\bibitem{kolk}
Kolk, E.,
{\it The statistical convergence in Banach spaces},
Tartu Ul Toime. 928, 41--52, (1991).
\bibitem{murs1}
Mursaleen, M.,
{\it $\lambda$-statistical convergence},
Math. Slovaca 50, 111--115, (2000).
\bibitem{murs2}
Mursaleen, M. and Mohiuddine, S. A.,
{\it On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space},
J. Comput. Appl. Math. 233, 142--149, (2009).
\bibitem{Mursa1}
Mursaleen, M. and Ansari, K. J.,
{\it Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation},
Appl. Math. Inf. Sci. 7 (5), 1677--1684, (2013).
\bibitem{Mursa2}
Mursaleen, M. and Mohiuddine, S. A.,
{\it On stability of a cubic functional equation in intuitionistic fuzzy normed spaces},
Chaos Solitons Fractals 42, 2997--3005, (2009).
\bibitem{park}
Park, C. and Bodaghi, A.,
{\it Two multi-cubic functional equations and some results on the stability in modular spaces},
J. Inequal. Appl., Art. 6, (2020).
\bibitem{AN}
Najati, A. and Park, C.,
{\it On the stability of a cubic functional equation},
Acta Math. Sin. 24, 1953--1964, (2008).
\bibitem{JMRassis82}
Rassias, J. M.,
{\it On approximation of approximately linear mappings by linear mappings},
J. Funct. Anal. 46, 126--130, (1982).
\bibitem{JMRassis00}
Rassias, J. M.,
{\it Solution of the Ulam problem for cubic mappings},
An. Univ. Timi\c{s}oara Ser. Mat. Inform. 38, 121--132, (2000).
\bibitem{JMRassias01(1)}
Rassias, J. M.,
{\it Solution of the Ulam stability problem for the cubic mapping},
Glasnik Mat. 36 (56), 63--72, (2001).
\bibitem{JMRSK}
Rassias, J. M. and Karthikeyan, S.,
{\it Stability of additive--quadratic 3D functional equation in modular spaces by direct method},
Asian-Eur. J. Math. 15 (8), 2250145, (2022).
\bibitem{Rassias78}
Rassias, Th. M.,
{\it On the stability of the linear mapping in Banach spaces},
Proc. Amer. Math. Soc. 72, 297--300, (1978).
\bibitem{raviarunrassias}
Ravi, K., Arunkumar, M. and Rassias, J. M.,
{\it On the Ulam stability for the orthogonally general Euler--Lagrange type functional equation},
Int. J. Math. Sci. 3, 36--47, (2008).
\bibitem{SahaMondalChoudhury25}
Saha, P., Mondal, P. and Choudhury, B. S., \textit{ Stability of general quadratic Euler--Lagrange functional equations in modular spaces: a fixed point approach}, Ural Math. J. 11(1), 114--123, (2025).
\bibitem{SahaEtAl24}
Saha, P., Kayal, N. C., Choudhury, B. S., Dutta, S. and Mondal, S. P., \textit{ A fixed point approach to the stability of a quadratic functional equation in modular spaces without $\Delta_{2}$-conditions}, Tatra Mt. Math. Publ. 86, 47--64, (2024).
\bibitem{stein}
Steinhaus, H.,
{\it Sur la convergence ordinaire et la convergence asymptotique},
Colloq. Math. 2, 73--84, (1951).
\bibitem{s1}
Donganont, S.,
{\it An additive functional inequality in $C^{*}$-algebras},
Bol. Soc. Paran. Mat. 43, 1--14, (2025).
\bibitem{s2}
Donganont, S. and Park, C.,
{\it Combined system of additive functional equations in Banach algebras},
Open Math. 22, Art. ID 20230177, (2024).
\bibitem{Ulam}
Ulam, S. M.,
{\it Problems in Modern Mathematics},
Wiley, (1964).
Aboutaib, I., Benzarouala, C., Brzdęk, J., Leśniak, Z. and Oubbi, L.,
{\it Ulam Stability of a General Linear Functional Equation in Modular Spaces},
Symmetry 14 (11), 2468, (2022).
\bibitem{Aczel}
Aczel, J. and Dhombres, J.,
{\it Functional Equations in Several Variables},
Cambridge Univ. Press, (1989).
\bibitem{Aoki}
Aoki, T.,
{\it On the stability of the linear transformation in Banach spaces},
J. Math. Soc. Japan 2, 64--66, (1950).
\bibitem{MASM}
Arunkumar, M., Murthy, S. and Ganapathy, G.,
{\it Stability of generalized $n$-dimensional cubic functional equation in fuzzy normed spaces},
Int. J. Pure Appl. Math. 77, 179--190, (2012).
\bibitem{MASK}
Arunkumar, M. and Karthikeyan, S.,
{\it Solution and intuitionistic fuzzy stability of $n$-dimensional quadratic functional equation: Direct and fixed point methods},
Int. J. Adv. Math. Sci. 2 (1), 21--33, (2014).
\bibitem{MAKR}
Arunkumar, M., Karthikeyan, S. and Ramamoorthi, S.,
{\it Generalized Ulam--Hyers stability of $n$-dimensional cubic functional equation in FNS and RNS: Various methods},
Middle-East J. Sci. Res. 24 (S2), 386--404, (2016).
\bibitem{Baak07}
Baak, C. and Moslenian, M. S.,
{\it On the stability of an orthogonally cubic functional equation},
Kyungpook Math. J. 47, 69--76, (2007).
\bibitem{bodagh}
Bodaghi, A.,
{\it Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations},
J. Intell. Fuzzy Syst. 30, 2309--2317, (2016).
\bibitem{czerwik}
Czerwik, S.,
{\it Functional Equations and Inequalities in Several Variables},
World Scientific, (2002).
\bibitem{DeviRaniKumar23}
Devi, S., Rani, A. and Kumar, M., {\it Stability of functional equations in modular spaces and fuzzy normed spaces}, Int. J. Math. Its Appl. 11(3), 113--127, (2023).
\bibitem{Fast}
Fast, H.,
{\it Sur la convergence statistique},
Colloq. Math. 2, 241--244, (1951).
\bibitem{fridy}
Fridy, J. A.,
{\it On statistical convergence},
Anal. 5, 301--313, (1985).
\bibitem{Ganapathy}
Ganapathy, G., Sakthi, R., Karthikeyan, S., Vijaya, N., Suresh, M. and Venkataramanan, K.,
{\it Stabilities and instabilities of Euler--Lagrange cubic functional equation},
J. Math. Comput. Sci. 35 (1), 82--95, (2024).
\bibitem{g1} Gowri, S., Donganont, S., Karthick, S., Balaanandhan, R., Tamilvanan, K., {\it Hyers-Ulam stability of generalized quartic mapping in non-Archimedean $(n,\beta)$-normed spaces}, Eur. J. Pure Appl. Math. 18, 6699-6699 (2025).
\bibitem{g2} Gowri, S., Pushpalatha, A. P., Vijaya, N., Sreelatha Devi, V., Balamurugan, M., Ramachandran, A., Tamilvanan, K., {\it Ulam stability of quadratic mapping connected with homomorphisms and derivations in non-Archimedean Banach algebras}, Int. J. Anal. Appl. 23, 119-119 (2025).
\bibitem{g3} Gowri, S., Sudharsan, S., Banu Priya, V., Annadurai, S., Ganapathy, G., Vijayalakshmi, A., {\it Hyers-Ulam stability of $n$-dimensional additive functional equation in modular spaces using fixed point method}, Int. J. Anal. Appl. 23, 148-148 (2025).
\bibitem{g4} Gowri, S., Vijaya, N., Gayathri, S., Vijayalakshmi, P., Balamurugan, M., Karthikeyan, S., {\it Hyers-Ulam stability of quartic functional equation in IFN-spaces and 2-Banach spaces by classical methods}, Int. J. Anal. Appl. 23, 68-68 (2025).
\bibitem{gavruta94}
G\u{a}vru\c{t}a, P.,
{\it A generalization of the Hyers--Ulam--Rassias stability of approximately additive mappings},
J. Math. Anal. Appl. 184, 431--436, (1994).
\bibitem{hyers}
Hyers, D. H.,
{\it On the stability of the linear functional equation},
Proc. Natl. Acad. Sci. U.S.A. 27, 222--224, (1941).
\bibitem{KWJ}
Jun, K. W. and Kim, H. M.,
{\it The generalized Hyers--Ulam--Rassias stability of a cubic functional equation},
J. Math. Anal. Appl. 274, 867--878, (2002).
\bibitem{jungbook}
Jung, S. M.,
{\it Hyers--Ulam--Rassias Stability of Functional Equations in Mathematical Analysis},
Hadronic Press, (2001).
\bibitem{MJRMAES}
Rassias, M. J., Arunkumar, M. and Sathya, E.,
{\it Stability of a $k$-cubic functional equation in quasi-beta normed spaces: Direct and fixed point methods},
Brit. J. Math. Comput. Sci. 8 (5), 346--360, (2015).
\bibitem{Junkim03}
Jun, K. W. and Kim, H. M.,
{\it On the Hyers--Ulam--Rassias stability of a general cubic functional equation},
Math. Inequal. Appl. 6 (2), 289--302, (2003).
\bibitem{kara}
Karakus, S.,
{\it Statistical convergence on probabilistic normed spaces},
Math. Commun. 12, 11--23, (2007).
\bibitem{Karthikeyan}
Karthikeyan, S., Rassias, J. M., Arunkumar, M. and Sathya, E.,
{\it Generalized Ulam--Hyers stability of $(a,b;k>0)$-cubic functional equation in intuitionistic fuzzy normed spaces},
J. Anal., (2018).
\bibitem{KarthikeyanJMCS}
Karthikeyan, S., Park, C., Palani, P. and Kumar, T. R. K.,
{\it Stability of an additive--quartic functional equation in modular spaces},
J. Math. Comput. Sci. 26 (1), 22--40, (2022).
\bibitem{kolk}
Kolk, E.,
{\it The statistical convergence in Banach spaces},
Tartu Ul Toime. 928, 41--52, (1991).
\bibitem{murs1}
Mursaleen, M.,
{\it $\lambda$-statistical convergence},
Math. Slovaca 50, 111--115, (2000).
\bibitem{murs2}
Mursaleen, M. and Mohiuddine, S. A.,
{\it On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space},
J. Comput. Appl. Math. 233, 142--149, (2009).
\bibitem{Mursa1}
Mursaleen, M. and Ansari, K. J.,
{\it Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation},
Appl. Math. Inf. Sci. 7 (5), 1677--1684, (2013).
\bibitem{Mursa2}
Mursaleen, M. and Mohiuddine, S. A.,
{\it On stability of a cubic functional equation in intuitionistic fuzzy normed spaces},
Chaos Solitons Fractals 42, 2997--3005, (2009).
\bibitem{park}
Park, C. and Bodaghi, A.,
{\it Two multi-cubic functional equations and some results on the stability in modular spaces},
J. Inequal. Appl., Art. 6, (2020).
\bibitem{AN}
Najati, A. and Park, C.,
{\it On the stability of a cubic functional equation},
Acta Math. Sin. 24, 1953--1964, (2008).
\bibitem{JMRassis82}
Rassias, J. M.,
{\it On approximation of approximately linear mappings by linear mappings},
J. Funct. Anal. 46, 126--130, (1982).
\bibitem{JMRassis00}
Rassias, J. M.,
{\it Solution of the Ulam problem for cubic mappings},
An. Univ. Timi\c{s}oara Ser. Mat. Inform. 38, 121--132, (2000).
\bibitem{JMRassias01(1)}
Rassias, J. M.,
{\it Solution of the Ulam stability problem for the cubic mapping},
Glasnik Mat. 36 (56), 63--72, (2001).
\bibitem{JMRSK}
Rassias, J. M. and Karthikeyan, S.,
{\it Stability of additive--quadratic 3D functional equation in modular spaces by direct method},
Asian-Eur. J. Math. 15 (8), 2250145, (2022).
\bibitem{Rassias78}
Rassias, Th. M.,
{\it On the stability of the linear mapping in Banach spaces},
Proc. Amer. Math. Soc. 72, 297--300, (1978).
\bibitem{raviarunrassias}
Ravi, K., Arunkumar, M. and Rassias, J. M.,
{\it On the Ulam stability for the orthogonally general Euler--Lagrange type functional equation},
Int. J. Math. Sci. 3, 36--47, (2008).
\bibitem{SahaMondalChoudhury25}
Saha, P., Mondal, P. and Choudhury, B. S., \textit{ Stability of general quadratic Euler--Lagrange functional equations in modular spaces: a fixed point approach}, Ural Math. J. 11(1), 114--123, (2025).
\bibitem{SahaEtAl24}
Saha, P., Kayal, N. C., Choudhury, B. S., Dutta, S. and Mondal, S. P., \textit{ A fixed point approach to the stability of a quadratic functional equation in modular spaces without $\Delta_{2}$-conditions}, Tatra Mt. Math. Publ. 86, 47--64, (2024).
\bibitem{stein}
Steinhaus, H.,
{\it Sur la convergence ordinaire et la convergence asymptotique},
Colloq. Math. 2, 73--84, (1951).
\bibitem{s1}
Donganont, S.,
{\it An additive functional inequality in $C^{*}$-algebras},
Bol. Soc. Paran. Mat. 43, 1--14, (2025).
\bibitem{s2}
Donganont, S. and Park, C.,
{\it Combined system of additive functional equations in Banach algebras},
Open Math. 22, Art. ID 20230177, (2024).
\bibitem{Ulam}
Ulam, S. M.,
{\it Problems in Modern Mathematics},
Wiley, (1964).
Publicado
2026-02-21
Seção
Special Issue: Non-Linear Analysis and Applied Mathematics
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