On cyclic α(·)-monotone multifunctions

Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let ${\displaystyle \Gamma :X\to 2^{\Phi }}$ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), ${\displaystyle \Gamma \left(x\right)={\partial }^{-\alpha }\_\{\Phi \}f{\mid }_x}$.