Irene Benedetti

Nonlocal semilinear evolution equations without strong compactness: theory and applications

A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integro-differential equation arising in age-structure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation.MSC:34G25, 34B10, 34B15, 47H04, 28B20, 34H05.

An approximation solvability method for nonlocal differential problems in Hilbert spaces

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integro-differential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.

On Noncompact Fractional Order Differential Inclusions with Generalized Boundary Condition and Impulses in a Banach Space

We provide existence results for a fractional differential inclusion with nonlocal conditions and impulses in a reflexive Banach space. We apply a technique based on weak topology to avoid any kind of compactness assumption on the nonlinear term. As an example we consider a problem in population dynamic described by an integro-partial-differential inclusion.

BVP for Caratheodory inclusions in Hilbert spaces: sharp existence conditions and applications

Abstract This article concerns an existence result for Floquet boundary value problems associated to semilinear differential inclusions with Carathéodory right hand side in a Hilbert space. We apply a continuation principle and we require a sharp (i.e., localized on the boundary) transversality condition. We give an application to a nonlinear partial differential inclusion with periodic conditions.

Relative topological degree and variational inequalities

We give a new definition of the relative topological degree for multimaps compositions of approximable multimaps and continuous map. We apply this notion to prove an existence result for variational inequalities of Stampacchia’s type in finite dimensional vector spaces. Finally we obtain the same results also for multimaps compositions of selectionable multimaps and continuous map.

On generalized boundary value problems for a class of fractional differential inclusions

Abstract We prove existence of mild solutions to a class of semilinear fractional differential inclusions with non local conditions in a reflexive Banach space. We are able to avoid any kind of compactness assumptions both on the nonlinear term and on the semigroup generated by the linear part. We apply the obtained theoretical results to two diffusion models described by parabolic partial integro-differential inclusions.

A further generalization of midpoint convexity of multimaps towards common fixed point Theorems and applications

We furtherly generalize midpoint convexity for multivalued maps and derive Fixed Point Theorems and Common Fixed Point Theorems without requiring strong compactness. As an application we obtain some Best Approximation results, and minimax and variational inequalities.

Existence results for nonlinear variational inequalities via topological methods

In this paper we find existence results for nonlinear variational inequalities involving a multivalued map. Both the cases of a lower semicontinuous multimap and an upper semicontinuous one are considered. We solve the problem using a linearization device and a suitable continuation principle.