We analyze existence and uniqueness of solutions for perturbations of a Jensen functional inequality in several variables. Applications in connection with asymptotic behaviors of isomorphisms, derivations and $n$-Jordan derivations on Banach algebras are also provided. The results of this paper correct and improve the main results of [12, 16, 22, 23] and improve the corresponding results in [2, 9, 27], but under weaker assumptions.
Hamid KHODAEI
. FUNCTIONAL INEQUALITIES, ISOMORPHISMS AND DERIVATIONS ON BANACH ALGEBRAS[J]. Acta mathematica scientia, Series B, 2025
, 45(3)
: 855
-866
.
DOI: 10.1007/s10473-025-0307-3
[1] Alinejad A, Khodaei H, Rostami M. $n$-Derivations and functional inequalities with applications. Math Inequal Appl, 2020, 23: 1343-1360
[2] An J S. On an additive functional inequality in normed modules over a C*-algebra. J Korea Soc Math Educ Ser B Pure Appl Math, 2008, 15: 393-400
[3] Badora R. On approximate ring homomorphisms. J Math Anal Appl, 2002, 276: 589-597
[4] Badora R. On approximate derivations. Math Inequal Appl, 2006, 9: 167-173
[5] Badora R, Przebieracz B. On approximate group homomorphisms. J Math Anal Appl, 2018, 462: 505-520
[6] Brzdęk J, Fǒsner A. Remarks on the stability of Lie homomorphisms. J Math Anal Appl, 2013, 400: 585-596
[7] Brzdęk J, Fǒsner A, Leśniak Z. A note on asymptotically approximate generalized Lie derivations. J Fixed Point Theory Appl, 2020, 22: Paper No 40
[8] Brzdęk J, Popa D, Raşa I, Xu B.Ulam Stability of Operators. Oxford: Elsevier, 2018
[9] Cho Y J, Saadati R, Yang Y O, Kenari H M. A fixed point technique for approximate a functional inequality in normed modules over C*-algebras. Filomat, 2016, 30: 1691-1696
[10] Fischer P, Muszély Gy. On some new generalizations of the functional equation of Cauchy. Canadian Math Bull, 1967, 10: 197-205
[11] Ger R. On a characteritation of strictly convex spaces. Atti Accad Sci Torino Cl Sci Fiz Mat Natur, 1993, 127: 131-138
[12] Ghaleh S G, Ghasemi K. Stability of $n$-Jordan *-derivations in C*-algebras and JC*-algebras. Taiwanese J Math, 2012, 16: 1791-1802
[13] Gilányi A. Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequat Math, 2001, 62: 303-309
[14] Gilányi A. On a problem by K Nikodem. Math Inequal Appl, 2002, 5: 707-710
[15] Herstein I N. Lie and Jordan structures in simple, associative rings. Bull Amer Math Soc, 1961, 67: 517-531
[16] Jamalzadeh J, Ghasemi K, Ghaffary S. $n$-Jordan *-derivations in Fréchet locally C*-algebras. Int J Nonlinear Anal Appl, 2022, 13: 555-562
[17] Khodaei H. Asymptotic behavior of $n$-Jordan homomorphisms. Mediterr J Math, 2020, 17: Paper No 143
[18] Khodaei H, Khodabakhsh R, Gordji M E. Fixed point, Lie *-homomorphisms and Lie *-derivations on Lie C*-algebras. Fixed Point Theory, 2013, 14: 387-400
[19] Kim H M, Chang I S. Approximate linear derivations and functional inequalities with applications. Appl Math Lett, 2012, 25: 830-836
[20] Kim H M, Chang I S. Asymptotic behavior of generalized *-derivations on C*-algebras with applications. J Math Phys, 2015, 56: Art ID 041708
[21] Lanski C. Generalized derivations and $n$th power maps in rings. Commun Algebra, 2007, 35: 3660-3672
[22] Lee J R, Shin D Y. Isomorphisms and derivations in C*-algebras. Acta Math Sci, 2011, 31B: 309-320
[23] Lee S J, Park C, Shin D Y. An additive functional inequality. Korean J Math, 2014, 22: 317-323
[24] Maksa G, Volkmann P. Characterization of group homomorphisms having values in an inner product space. Publ Math Debrecen, 2000, 56: 197-200
[25] Park C. Bi-additive $s$-functional inequalities and quasi-*-multipliers on Banach algebras. Bull Braz Math Soc, 2019, 50: 561-574
[26] Park C. Biderivations and bihomomorphisms in Banach algebras. Filomat, 2019, 33: 2317-2328
[27] Park C, Lee J R. Comment on ''Functional inequalities associated with Jordan-von Neumann type additive functional equations''. J Inequal Appl, 2012, 2012: Paper No 47
[28] Park C, Rassias Th M.Additive functional equations and partial multipliers in C*-algebras. Rev Real Acad Cienc Exactas Fis Nat Ser A-Mat, 2019, 113: 2175-2188
[29] Park C, Rassias Th M. Homomorphisms and derivations in proper JCQ*-triples. J Math Anal Appl, 2008, 337: 1404-1414
[30] Rădulescu M, Rădulescu S. On the Maksa-Volkmann functional inequality $|f(x+y)|\ge|f(x)+f(y)|$ when the range of $f$ is a space of functions. Carpathian J Math, 2014, 30: 253-256
[31] Rätz J. On inequalities associated with the Jordan-von Neumann functional equation. Aequat Math, 2003, 66: 191-200
[32] Rostami M, Alinejad A, Khodaei H. On $n$-Jordan derivations in the sense of Herstein. Rev Real Acad Cienc Exactas Fis Nat Ser A-Mat, 2023, 117: Paper No 85
