More on stability of two functional equations

Longfa Sun; Yunbai Dong

doi:10.37863/umzh.v75i6.7121

Longfa Sun School of Mathematics and Physics, North China Electric Power University, Baoding, China
Yunbai Dong Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China

DOI: https://doi.org/10.37863/umzh.v75i6.7121

Keywords: Ulam stability problem, uniform convexity, Banach spaces

Abstract

UDC 517.5

We prove the generalized stability of the functional equations $\|f(x+y)\|=\|f(x)+f(y)\|$ and $\|f(x-y)\|= \|f(x)-f(y)\|$ in $p$-uniformly convex spaces with $p\geq 1.$

References

J. Aczél, J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, Cambridge (1989). DOI: https://doi.org/10.1017/CBO9781139086578

M. R. Abdollahpour, R. Aghayaria, M. Th. Rassiasb, Hyers–Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. and Appl., 437, 605–612 (2016). DOI: https://doi.org/10.1016/j.jmaa.2016.01.024

D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, 223–237 (1951). DOI: https://doi.org/10.1090/S0002-9904-1951-09511-7

L. Cheng, Q. Fang, S. Luo, L. Sun, On non-surjective coarse isometries between Banach spaces, Quaest. Math., 42, № 3, 347–362 (2019). DOI: https://doi.org/10.2989/16073606.2018.1448900

S. Czerwik, Functional equations and inequalities in several variables, World Sci. Publ. Co., New Jersey etc. (2002). DOI: https://doi.org/10.1142/4875

J. Brzdek, D. Popa, B. Xu, On approximate solutions of the linear functional equation of higher order, J. Math. Anal. and Appl., 373, 680–689 (2011). DOI: https://doi.org/10.1016/j.jmaa.2010.08.028

Y. Dong, Generalized stability of two functional equations, Aequationes Math., 86, 269–277 (2013). DOI: https://doi.org/10.1007/s00010-012-0180-8

P. Fischer, G. Muszély, On some new generalizations of the functional equation of Cauchy, Canad. Math. Bull., 10, 197–205 (1967). DOI: https://doi.org/10.4153/CMB-1967-018-x

R. Ger, A Pexider-type equation in normed linear space, Österreich. Akad. Wiss. Math.-Natur. K1. Sitzungsber. II, 206, 291–303 (1997).

O. Hanner, On uniformly convexity of $L_p$ and $ell_p$, Ark. Mat., 3, 239–244 (1956). DOI: https://doi.org/10.1007/BF02589410

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27, 222–224 (1941). DOI: https://doi.org/10.1073/pnas.27.4.222

D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequationes Math., 44, 125–153 (1992). DOI: https://doi.org/10.1007/BF01830975

Y. Jin, C. Park, M. Th. Rassias, Hom-derivations in $C^ast$-ternary algebras, Acta Math. Sin. (Engl. Ser.), 36, № 9, 1025–1038 (2020). DOI: https://doi.org/10.1007/s10114-020-9323-3

S. M. Jung, D. Popa, M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim., 59, 165–171 (2014). DOI: https://doi.org/10.1007/s10898-013-0083-9

S. M. Jung, Hyers–Ulam–Rassias stability of functional equations in nonlinear analysis, Springer Optim. and Appl., vol. 48, Springer, New York (2011). DOI: https://doi.org/10.1007/978-1-4419-9637-4

S. M. Jung, K. Lee, M. Th. Rassias, S. Yang, Approximation properties of solutions of a mean value-type functional inequality, II, Mathematics, 8, (2020); DOI: 10.3390/math8081299. DOI: https://doi.org/10.3390/math8081299

S. M. Jung, P. K. Sahoo, On the stability of a mean value type functional equation, Demonstr. Math., 33, 793–796 (2000). DOI: https://doi.org/10.1515/dema-2000-0411

P. Kannappan, Functional equations and inequalities with applications, Springer, New York (2009). DOI: https://doi.org/10.1007/978-0-387-89492-8

J. Lindenstrauss, A. Szankowski, Non linear perturbations of isometries, Astérisque, 131, 357–371 (1985).

T. Miura, S. Miyajima, S. Takahasi, A characterization of Hyers–Ulam stability of first order linear differential operators, J. Math. Anal. and Appl., 286, 136–146 (2003). DOI: https://doi.org/10.1016/S0022-247X(03)00458-X

Th. M. Rassias, On the stability of the linear mapping in Banach space, Proc. Amer. Math. Soc., 72, 297–300 (1978). DOI: https://doi.org/10.1090/S0002-9939-1978-0507327-1

Th. M. Rassias, On the stability of functional equations in Banach space, J. Math. Anal. and Appl., 251, 264–284 (2000). DOI: https://doi.org/10.1006/jmaa.2000.7046

P. K. Sahoo, P. Kannappan, Introduction to functional equations, Taylor Francis Group (2011). DOI: https://doi.org/10.1201/b10722

J. Sikorska, Stability of the preservation of the equality of distance, J. Math. Anal. and Appl., 311, 209–217 (2005). DOI: https://doi.org/10.1016/j.jmaa.2005.02.039

F. Skof, On the functional equation $|f(x+y)-f(x)|=|f(y)|$, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 127, 229–237 (1993).

J. Tabor, Stability of the Fischer–Musz'{e}ly functional equation, Publ. Math., 62, 205–211 (2003). DOI: https://doi.org/10.5486/PMD.2003.2725

S. M. Ulam, A collection of mathematical problems, Intersci. Publ., New York (1968).