Multiplicative set--valued equation related to certain set--valued mappings and its stability on Banach algebras

Year 2025, Volume: 8 Issue: 4, 187 - 199, 15.12.2025

Wuttichai Suriyacharoen Wutiphol Sintunavarat

https://doi.org/10.33205/cma.1715909

Abstract

Let B be a Banach algebra, and let Ccb(B) denote the set of all closed convex bounded subsets of B. Assume that ⪰ is a partial order defined on Ccb(B), and define D := {A ∈ Ccb(B) : A ≻ 0}, where 0 denotes the zero element of Ccb(B). Furthermore, suppose that for every A,B ∈ D, the set A ⊗ B also belongs to D, where A ⊗ B means the closure of the product set AB. In this paper, general solutions F : D → D of the multiplicative set-valued functional equation
F(X ⊗ Y ) = F(X) ⊗ F(Y )
for all X, Y ∈ D are determined. These solutions are closely involved with some set-valued mappings. Moreover, its stability is also proved on Banach algebras. The results not only generalize classical findings in functional equations but also open avenues for further exploration in nonlinear analysis and set-valued operator theory.

Keywords

Banach algebras , Hausdorff metric , stability , multiplicative set-valued equations , set-valued mappings

References

• J. A. Baker: The stability of the cosine equation, Proc. Am. Math. Soc., 80 (1980), 411–416.
• A. R. Baias, B. Mo¸snegu¸tu and D. Popa: Set-valued solutions of a generalized quadratic functional equation, Results Math., 73 (2018), Article ID: 129.
• H.-Y. Chu, A. Kim and S. K. Yoo: On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett., 37 (2014), 7–14.
• D. R. Henney: Quadratic set-valued functions, Arkiv För Matematik, 4 (1962), 377–378.
• S. Y. Jang, C. Park and Y. Cho: Hyers-Ulam stability of a generalized additive set-valued functional equation, Appl. Math. Lett., 23 (2010), 1312–1316.
• H. A. Kenary, H. Rezaei, Y. Gheisari and C. Park: On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., 2012 (2012), Article ID: 81.
• J. R. Lee, C. Park, D. Y. Shin and S. Yum: Set-valued quadratic functional equations, Results Math., 72 (2017), 665–677.
• G. Lu, C. Park: Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett., 24 (2011), 1312–1316.
• K. Nikodem: On quadratic set-valued functions, Publ. Math. Debr., 30 (1984), 297–301.
• K. Nikodem: On Jensen’s functional equation for set-valued functions, Rad. Mat., 3 (1987), 23–33.
• K. Nikodem: Set-valued solutions of the Pexider functional equation, Funkc. Ekvacioj, 31 (1988), 227–231.
• C. Park, D. O’Regan and R. Saadati: Stability of some set-valued functional equations, Appl. Math. Lett., 24 (2011), 1910–1914.
• C. Park, Y. Sungsik, J. R. Lee and D. Y. Shin: Set-Valued Additive Functional Equations, Constr. Math. Anal., 2 (2019), 89–97.
• C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Springer, Berlin (1977).
• G. Debreu: Integration of correspondences, Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability II, Berkeley (1966), 351–372.