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General properties of $$(p,q)$$-convex functions and $$(p,q)$$-monotone operators. (English) Zbl 0555.49007
Summary: The paper deals with general properties of a class of functions, $$(p,q)$$-convex functions, and a class of nonlinear operators, $$(p,q)$$-monotone operators, defined in a Hilbert space $$H$$. These classes extend, respectively, that of $$C^{1,1}$$ perturbations of convex functions and that of Lipschitz continuous perturbations of monotone operators. The aim is to tackle problems with nonconvex constraints.
In particular, theorems are given, concerning the minimization of certain functions which are not lower semicontinuous with respect to the weak topology of $$H$$. The class of $$(p,q)$$-convex functions was introduced in the note by E. De Giorgi, A. Marino and M. Tosques [Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Natur. VIII. Ser. 73, 6–14 (1982; Zbl 0521.49011)], where some of the results contained in the paper were announced.

##### MSC:
 49J27 Existence theories for problems in abstract spaces 46G05 Derivatives of functions in infinite-dimensional spaces 47H20 Semigroups of nonlinear operators 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 49J45 Methods involving semicontinuity and convergence; relaxation