Set valued pexiderized quadratic functional equation

Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation

$$\begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$

for unknown functions \(f_1,f_2,f_3,f_4:X\rightarrow cc(Y)\). This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper.