On the Rockafellar theorem for Φγ(.,.)-monotone multifunctions

Let X be an arbitrary set, and gamma : X x X --> R any function. Let Phi be a family of real-valued functions defined on X. Let Gamma : X --> 2(Phi) be a cyclic Phi(gamma(., .)) -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f : X --> R such that Gamma is contained in the Phi gamma((.,.))-subdifferential of f, Gamma(x) subset of partial derivative(Phi)(gamma(.,.))f\x.