Francesca Papalini

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.

On the Existence of Three Nontrivial Solutions for Periodic Problems Driven by the Scalar p-Laplacian

Abstract We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential function. First we establish an alternative minimax expression for the first nonzero eigenvalue for the negative periodic scalar p-Laplacian and then using it we prove the existence of three nontrivial solutions, two of which have constant sign. Our approach is variational, based on the nonsmooth critical point theory.

Some representations of midconvex set-valued functions

SummaryIn this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.

Existence theorems for nonlinear evolution inclusions

SummaryIn this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u′∈−∂−f(u)+G(u), u(O)=x0, where ∂−f is the Fréchet subdifferential of a functionf defined on an open subset Ω of a real separable Hilbert space H, taking its values in R ∪ {+∞} and G is a multifunction from C([0, T], Ω) into the nonempty subsets of L2([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x′∈−∂−f(x)+F(t, x), x(0)=x0, where F:[0, T]×Ω→(H) is a multifunction satisfying some regularity assumptions.

On the continuity of midpoint hull convex set-valued functions

SummaryThe concept of “hull convexity” (“midpoint hull convexity”) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of “convexity” (“midpoint convexity”).The main result is a sufficient condition for a “midpoint hull convex” set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).

Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators

In this paper we study the optimal control of a class of nonlinear finitedimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the “relaxed problem”. We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Carathéodory’s theorem for convex sets in R and the fourth uses lower semicontinuity arguments and Γ-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the “closure” of the original one.

On the solution set of a vector duffing equation

In this paper we consider a vector Duffing equation with periodic boundary conditions. First we prove an existence result assuming on f(t,x) Caratheodory type conditions. Then by imposing also a monotonicity assumption we show that the solution set is acyclic.

Caratterizzazione della semicontinuità inferiore della frontiera di una multifunzione in spazi normati

In this note we obtain for a multifunctionG defined in a topological spaceT and taking as its values «bounded» and «convex» subsets with non empty interior in a Banach spaceX (dimX<∞), the following result:G continuous int0εT ⇔ ∂G lower semicontinuous int0εT. This theorem contains the results stated by M.D.P. Monteiro Marques in [4] and by D. Averna—T. Cardinali in [1].