Stability of generalized cubic set-valued functional equations (Q2794921)

Let \(X\) be a real vector space, \(A \subset X\) be a cone with zero and \(Y\) be a Banach space. Let \(C_{b} (Y)\) and \(C_{cb} (Y)\) be the set of all closed bounded and closed bounded convex subsets of \(Y\), respectively. For \(A, B \in C_{b} (Y)\), \(A\oplus B\) is defined as \(\overline{A+B}\), where \(\overline{A+B}\) denotes the closure of the set \(A+B\). For an integer \(a \geq 2\), let \(A_a := (a-1) ( a^2 - 1)\), \(B_a := (a+1)(a^2-1) \) and NEWLINE\[NEWLINEH_f (x, y) := h \bigg ( f(ax+y) \oplus f(x-ay) \oplus B_a f(y), \, a^2 f(x+y) \oplus a f(x-y) \oplus A_a f(x)\bigg ),NEWLINE\]NEWLINE where \(h ( \cdot, \cdot )\) denotes the Hausdorff distance.NEWLINENEWLINEIn this paper, the author proves some stability results for the set-valued functional equation NEWLINE\[NEWLINEf(ax+y)\oplus f(x-ay) \oplus B_a f(y) = a^2 f(x+y) \oplus a f(x-y) \oplus A_a f(x) .NEWLINE\]NEWLINE The following is one of the main results of this paper: Let \(\phi : X \times X \to [0, \infty )\) be a function such that \(\tilde{\phi} (x,y) := \sum_{j=0}^{\infty} {1 \over {a^{3j}}} \, \phi (a^j x , a^j y ) < \infty \) for all \(x, y \in X\) and an integer \(a \geq 2\). Suppose that \(f : X \to (C_{cb}(Y), h)\) is a mapping with \(f(0)=0\) satisfying \(H_f (x, y) \leq \phi (x, y) \) for all \(x, y \in X\). Then there exists a unique cubic set-valued mapping \(g : X \to (C_{cb} (Y), h)\) such that \(h(f(x), g(x) ) \leq {1 \over {a^3 }} \, \tilde{\phi} (x, 0)\) for all \(x \in X\).