Results Math 72 (2017), 665–677
2017 Springer International Publishing
published online April 4, 2017
Results in Mathematics
Set-Valued Quadratic Functional Equations
Jung Rye Lee, Choonkil Park, Dong Yun Shin, and Sungsik Yun
Abstract. In this paper, we introduce set-valued quadratic functional
equations and prove the Hyers–Ulam stability of the set-valued quadratic
functional equations by using the ﬁxed point method.
Mathematics Subject Classiﬁcation. 47H10, 54C60, 39B52, 47H04, 91B44.
Keywords. Hyers–Ulam stability, set-valued quadratic functional equa-
tion, ﬁxed point.
1. Introduction and Preliminaries
Set-valued functions in Banach spaces have been developed in the last decades.
The pioneering paper by Aumann  and Debreu  were inspired by prob-
lems arising in Control Theory and Mathematical Economics. We can refer
to the papers by Arrow and Debreu , McKenzie , the momographs by
Hindenbrand , Aubin and Frankowska , Castaing and Valadier , Klein
and Thompson  and the survey by Hess .
The stability problem of functional equations originated from a question
of Ulam  concerning the stability of group homomorphisms. Hyers gave
a ﬁrst aﬃrmative partial answer to the question of Ulam for Banach spaces.
Hyers’ Theorem was generalized by Aoki  for additive mappings and by
Rassias  for linear mappings by considering an unbounded Cauchy diﬀer-
ence. A generalization of the Rassias theorem was obtained by G˘avruta 
by replacing the unbounded Cauchy diﬀerence by a general control function
in the spirit of Rassias’ approach
The functional equation f(x + y)+f (x − y)=2f(x)+2f(y) is called
a quadratic functional equation. In particular, every solution of the quadratic
functional equation is said to be a quadratic mapping. A Hyers–Ulam sta-
bility problem for the quadratic functional equation was proved by Skof