On the stability of the generalized quadratic set-valued functional equation (Q2792450)

Let \(X\) be a real vector space and \(Y\) be a Banach space. Let \(C_{c} (Y)\) and \(C_{cb} (Y)\) be the set of all closed convex and closed bounded convex subsets of \(Y\), respectively. For \(A, B \in C_{c} (Y)\), \(A\oplus B\) is defined as \(A\oplus B = \overline{A+B}\), where \(\overline{A+B}\) denotes the closure of \(A+B\). In this paper, the authors consider the following generalized quadratic set-valued functional equation NEWLINE\[NEWLINE(4-n) f \bigg ( \sum_{i=1}^n x_i \bigg ) \oplus \sum_{i=1}^n f \bigg ( \sum_{j=1}^n \theta (i, j) \, x_j \bigg ) = \sum_{i=1}^n f(x_i) , NEWLINE\]NEWLINE where \(n \geq 2\) is an integer and \(\theta : \mathbb{N} \times \mathbb{N} \to \{-1, +1\}\) is given by \(\theta(i,j)=1\) if \(i\neq j\) or \(\theta (i,j) = -1\) if \(i=j\), and prove some results concerning the Hyers-Ulam stability for this set-valued functional equation.