Partial second-order subdifferentials of -prox-regular functions

Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2  functions. The class of prox-regular functions covers all convex functions, lower C2  functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regular functions have been developed in Banach and Hilbert spaces. In this paper, the parametric prox-regular functions are defined using limiting subdifferentials. Also, a partial second-order subdifferential is defined here for extended real valued functions of two variables corresponding to its variables through coderivatives of first-order partial subdifferential mappings. Then, relations between maximal monotonicity of  the partial first-order subdifferentials of these functions  and the positive-semidefiniteness of  the coderivatives of partial first order subdifferential mapping are investigated. Finally, we present necessary and sufficient conditions for ∂ -prox-regular functions to be convex based on positive-semidefiniteness of the partial second-order subdifferentials mappings.