A system of functional equations and its stability using fixed point method

Mehdi  Dehghanian; Yamin  Sayyari; Mana Donganont; Choonkil Park

doi:10.5269/bspm.77695

Mehdi Dehghanian Sirjan University of Technology
Yamin Sayyari Sirjan University of Technology
Mana Donganont University of Phayao
Choonkil Park Hanyang University

Abstract

This paper introduces and analyzes the concept of \((g,h)\)-derivations in complex Banach algebras, extending the classical notion of \(g\)-derivations. We consider a nonlinear system of three functional equations that models approximate \((g,h)\)-derivations and examine its Hyers-Ulam stability. Using a fixed point framework in generalized metric spaces, we derive the existence, uniqueness, and error bounds for the corresponding exact solutions. The results not only unify and extend previous stability results for derivations and homomorphisms but also offer a novel analytical method for treating operator equations with asymmetric structure.

Downloads

Download data is not yet available.

Author Biography

Mehdi Dehghanian, Sirjan University of Technology

d

References

M. S. Abdo, S. T. M. Thabet and B. Ahmad, The existence and Ulam-Hyers stability results for Ψ-Hilfer fractional integrodifferential equations, J. Pseudo-Differ. Oper. Appl. 11 (2020), 1757–1780.

M. Ahmad and A. Zada, Ulam’s stability results of conformable neutral fractional differential equations Ψ-Hilfer fractional integrodifferential equations, Bol. Soc. Parana. Mat. (3) 41 (2023), 1–13.

L. Cadariu and V. Radu, Fixed points and generalized stability for functional equations in abstract spaces, J. Math. Inequal. 3(3) (2009), 463–473.

M. Dehghanian, Y. Sayyari and C. Park, Hadamard homomorphisms and Hadamard derivations on Banach algebras, Misk. Math. Notes 24(1) (2023), 129–137.

J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126(74) (1968), 305–309.

A. Ebadian, I. Nikoufar, Th. M. Rassias and N. Ghobadipour, Stability of generalized derivations on Hilbert C-modules associated to a pexiderized Cuachy-Jensen type functional equation, Acta Math. Sci. 32 (2012), 1226–1238.

M. Eshaghi Gordji and N. Ghobadipour, Nearly generalized Jordan derivations, Math. Slovaca 61(1) (2011), 55–62.

F. Haddouchi, Existence and Ulam-Hyers stability results for a class of fractional integro-differential equations involving nonlocal fractional integro-differential boundary conditions, Bol. Soc. Parana. Mat. (3) 42 (2024), 1–19.

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.

G. Lu and C. Park, Hyers-Ulam stability of general Jensen-type mappings in Banach algebras, Results Math. 66 (2014), 87–98.

D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572.

M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations on C∗-algebras, Proc. Amer. Math. Soc. 134 (2006), 3319–3327.

M. Mirzavaziri and M. S. Moslehian, σ-Derivations in Banach algebras, Bull. Iran. Math. Soc. 32 (2007), 65–78.

S. Paokanta, M. Dehghanian, C. Park and Y. Sayyari, A system of additive functional equations in complex Banach algebras, Demonstr. Math. 56 (2023), Article ID 20220165.

C. Park, Homomorphisms between Poisson JC∗-algebras, Bull. Braz. Math. Soc. 36(1) (2005), 79–97.

Y. Sayyari, M. Dehghanian, C. Park and J. Lee, Stability of hyper homomorphisms and hyper derivations in complex Banach algebras, AIMS Math. 7(6) (2022), 10700–10710.

S. M. Ulam, A Collection of the Mathematical Problem, Interscience, New York, 1960.

Z. Zamani, B. Yousefi and H. Azadi Kenary, Fuzzy Hyers-Ulam-Rassias stability for generalized additive functional equations, Bol. Soc. Parana. Mat. (3) 40 (2022), 1–14.