This paper offers first- and higher-order necessary conditions for the local disjointness of a finite system of sets that
are nonlinear inverse images of convex sets. The proof is based on the characterizations of α-admissible and α-tangent variations to nonlinear inverse images of convex sets and a necessary condition for the local disjointness in terms
of these variations. As an application, the results are used to obtain first- and higher-order necessary conditions of optimality
in constrained optimization problems.
We consider means that are simultaneously of the form ψ-1(ψ(x)+ψ(y)/2) and -1((x)+(y)-(x+y /2)), where ϕ and ψ are continuous strictly monotone functions. We solve the corresponding functional equation
assuming that one of the functions is in C1. The result obtained enables us to improve that of J. Matkowski.
is considered, where 0 < p < 1 is a fixed parameter and f: I → R is an unknown function. The equivalence of this and Jensen’s functional equation is completely characterized in terms of
the algebraic properties of the parameter p. As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.