A maximum theorem for generalized convex functions

Motivated by the Maximum Theorem for convex functions (in setting of linear spaces) and subadditive Abelian semigroups), we establish a class generalized functions, i.e., $f:X\to\R$ that satisfy inequality $f(x\circ y)\leq pf(x)+qf(y)$, where $\circ$ is binary operation on $X$ $p,q$ are positive constants. As an application, also obtain extension Karush--Kuhn--Tucker theorem this functions.