Abstract
Using the Mazur–Orlicz theorem and some results about the Fréchet functional equation, we consider functional equations related to generalized monomials of degree n. From these considerations, we give some results on existence of single-valuedness and selections for convex-valued maps satisfying functional inclusions. Also, the Diaz–Margolis fixed point alternative is applied to solve the stability problem for set-valued generalized monomials of degree n. Finally, several particular cases are discussed and some applications are given.
Similar content being viewed by others
References
Arrow, K.J., Debereu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)
Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)
Brzdȩk, J.: On approximately additive functions. J. Math. Anal. Appl. 381, 299–307 (2011)
Brzdȩk, J., Piszczek, M.: Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability. J. Fixed Point Theory Appl. 19, 2441–2448 (2017)
Brzdȩk, J., Piszczek, M.: Selections of set-valued maps satisfying some inclusions and the Hyers-Ulam stability, in: Handbook of Functional Equations, in: Springer Optim. Appl., vol. 96, Springer, New York, pp. 83-100 (2014)
Brzdȩk, J., Piszczek, M.: Ulam stability of some functional inclusions for multi-valued mappings. Filomat 31, 5489–5495 (2017)
Brzdȩk, J., Popa, D., Xu, B.: Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal. 74, 324–330 (2011)
Cascales, T., Rodrigeuz, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)
Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)
Cho, I.G., Kang, D., Koh, H.: Stability problems of quintic mappings in quasi-\(\beta \)-normed spaces, J. Inequal. Appl. 2010 Art. ID 368981 (2010)
Czerwik, S.: Functional Equations and Inequalities in Several Variables, World Scientific London, (2002)
Debreu, G.: Integration of correspondences. In: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, Part I, 351-372 (1966)
Diaz, J., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74, 305–309 (1968)
Djoković, D.Z̆.: A representation theorem for \((X_1-1)(X_2-1)\cdots (X_n-1)\) and its applications, Ann. Pol. Math. 22, 189-198 (1969)
Fréchet, M.: Un definition fonctionnelle des polynômes. Nouv. Ann. de Math. 9, 145–162 (1909)
Gajda, Z., Ger, R.: Subadditive multifunctions and Hyers-Ulam stability. Numer. Math. 80, 281–291 (1987)
Khodaei, H.: Selections of generalized convex set-valued functions satisfying some inclusions. J. Math. Anal. Appl. 474, 1104–1115 (2019)
Khodaei, H., El-Fassi, Iz., Hayati, B.: On selections of set-valued Euler-Lagrange inclusions with applications, Acta Math. Sci. 40 1-11 (2020)
Khodaei, H., Rassias, Th.M.: Set-valued dynamics related to generalized Euler-Lagrange functional equations, J. Fixed Point Theory Appl. 20 Art. 32 (2018)
Lu, G., Park, C.: Hyers-Ulam stability of additive set-valued functional equations. Appl. Math. Lett. 24, 1312–1316 (2011)
Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der polynomischen Operationen, I., II., Studia Math. 5 (1934) 50-68, 179-189 (in German)
Narasimman, P., Rassias, J.M., Ravi, K.: \(n\)-dimensional quintic and sextic functional equations and their stabilities in Felbin type spaces. Georgian Math. J. 23, 121–137 (2016)
Nikodem, K.: \(K\)-Convex and \(K\)-Concave Set-Valued Functions, Zeszyty Naukowe, Politech. Łódź. Mat. 559, Łódź, (1989)
Nikodem, K., Popa, D.: On selections of general linear inclusions. Publ. Math. Debrecen 75, 239–249 (2009)
Park, C.: Fixed point method for set-valued functional equation. J. Fixed Point Theory Appl. 19, 2297–2308 (2017)
Park, C., O’Regan, D., Saadati, R.: Stabiltiy of some set-valued functional equations. Appl. Math. Lett. 24, 1910–1914 (2011)
Piszczek, M.: On selections of set-valued inclusions in a single variable with applications to several variables. Results Math. 64, 1–12 (2013)
Piszczek, M.: The properties of functional inclusions and Hyers-Ulam stability. Aequat. Math. 85, 111–118 (2013)
Popa, D.: A stability result for a general linear inclusion. Nonlinear Funct. Anal. Appl. 3, 405–414 (2004)
Popa, D.: Functional inclusions on square-symmetric groupoids and Hyers-Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004)
Rådström, H.: An embedding theorem for space of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)
Sahoo, P.K.: A generalized cubic functional equation, Acta Math. Sin. (Engl. Ser.) 21, 1159-1166 (2005)
Smajdor, A.: Additive selections of superadditive set-valued functions. Aequat. Math. 39, 121–128 (1990)
Székelyhidi, L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific, Singapore (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Miahi, M., Mirzaee, F. & Khodaei, H. On convex-valued G-m-monomials with applications in stability theory. RACSAM 115, 76 (2021). https://doi.org/10.1007/s13398-021-01022-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-021-01022-6