Abstract
In this article, we obtain a characterizations and representations of set-valued solutions defined on an Abelian group G with values in a Hausdorff topological vector space of the following generalized bi-quadratic functional equation:
for some nonnegative real numbers a, b, c, and d.
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Baias, A.R., Moşneguţu, B., Popa, D.: Set-valued solutions of a generalized quadratic functional equation. Results Math. 73(4), 129 (2018)
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht (1993)
Benoist, J., Popovici, N.: Generalized convex set-valued maps. J. Math. Anal. Appl. 288(1), 161–166 (2003)
Breckner, W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4(1), 109–127 (1997)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (2006)
Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore (2002)
Henney, D.R.: Quadratic set-valued functions. Ark. Mat. 27(4), 377–378 (1961)
Inoan, D., Popa, D.: On selections of generalized convex set-valued maps. Aequ. Math. 88(3), 267–276 (2014)
Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, Berlin (2009)
Kurepa, S.: On the quadratic functional. Publ. Inst. Math. Acad. Serbe Sci. 13, 57–72 (1959)
Lee, J.R., Park, C., Shin, D.Y., Yun, S.: Set-valued quadratic functional equations. Results Math. 72(1–2), 665–677 (2017)
Nikodem, K.: On quadratic set-valued functions. Publ. Math. Debr. 30, 297–301 (1983)
Nikodem, K.: On Jensen’s functional equation for set-valued functions. Radovi Math. 3, 23–33 (1987)
Nikodem, K.: K-convex and K-concave set-valued functions. Zeszyty Nauk. Politech. Łódz. Mat. 559 (Rozprawy Nauk 114), Łódź (1989), 1–75
Nikodem, K., Popa, D.: On single valuedness of some classes of set-valued maps. Banach J. Math. Anal. 3(1), 44–51 (2009)
Park, W.G., Bae, J.H.: On a bi-quadratic functional equation and its stability. Nonlinear Anal. Theory Methods Appl. 62(4), 643–654 (2005)
Popa, D.: Set-valued solutions for an equation of Jensen type. Rev. d’Anal. Numér. Théor. de l’Approx. 28(1), 73–77 (1999)
Popa, D., Vornicescu, N.: Locally compact set-valued solutions for the general linear equation. Aequ. Math. 67, 205–215 (2004)
Rådström, H.: An embedding theorem for space of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952)
Rådström, H.: One-parameter semigroups of subsets of a real linear space. Ark. Mat. 4, 87–97 (1960)
Sahoo, P.K., Kannappan, P.: Introduction to Functional Equations. CRC Press, Boca Raton (2011)
Sikorska, J.: On a method of solving some functional equations for set-valued functions. Set Valued. Var. Anal. 27(1), 295–304 (2019)
Szczawińska, J.: On some families of set-valued functions. Aequ. Math. 78, 157–166 (2009)
Szczawińska, J.: On some equation for set-valued functions. Aequ. Math. 85, 421–428 (2013)
Urbański, R.: A generalization of the Minkowski–Rådström–Hörmander theorem. Bull. Pol. Acad. Sci. Math. 24, 709–715 (1976)
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EL-Fassi, Ii., El-Hady, ES. & Nikodem, K. On Set-Valued Solutions of a Generalized Bi-quadratic Functional Equation. Results Math 75, 89 (2020). https://doi.org/10.1007/s00025-020-01225-0
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DOI: https://doi.org/10.1007/s00025-020-01225-0