Massimo Furi

About well-posed optimization problems for functionals in metric spaces

A necessary and sufficient condition of correctness of extremal problems for lower semicontinuous functionals defined in metric spaces is given.

On the solvability of nonlinear operator equations in normed spaces

SuntoSi studia una classe di applicazioni continue negli spazi normati e si dimostrano, per tali applicazioni, proprietà analoghe a quelle della teoria del grado di Leray- Schauder. Si danno esempi di problemi ai limiti per equazioni differenziali ordinarie che possono essere agevolmente trattati usando la teoria sviluppata nel presente lavoro.

Contributions to the spectral theory for nonlinear operators in Banach spaces

SuntoSi introduce una definizione di spettro σ(f) per applicazioni continue definite in uno spazio di Banach. Tale definizione coincide con quella classica nel caso in cui f sia lineare e continua. Alcuni dei risultati più noti della teoria spettrale lineare vengono estesi al caso non lineare. In particolare si dimostra che σ(f) è chiuso e che la sua frontiera è contenuta nello spettro puntuale approssimato σπ(f).

Handbook of topological fixed point theory

Preface. I. Homological Methods in Fixed Point Theory. 1. Coincidence theory. 2. On the Lefschetz fixed point theorem. 3. Linearizations for maps of nilmanifolds and solvmanifolds. 4. Homotopy minimal periods. 5. Perodic points and braid theory. 6. Fixed point theory of multivalued weighted maps. 7. Fixed point theory for homogeneous spaces - a brief survey. II. Equivariant Fixed Point Theory. 8. A note on equivariant fixed point theory. 9. Equivariant degree. 10. Bifurcations of solutions of SO (2)-symmetric nonlinear problems with variational structure. III. Nielsen Theory. 11. Nielsen root theory. 12. More about Nielsen theories and their applications. 13. Algebraic techniques for calculating the Nielsen number on hyperbolic surfaces. 14. Fibre techniques in Nielsen theory calculations. 15. Wecken theorem for fixed and periodic points. 16. A primer of Nielsen fixed point theory. 17. Nielsen fixed point theory on surfaces. 18. Relative Nielsen theory. IV. Applications. 19. Applicable fixed point principles. 20. The fixed point index of the Poincare translation. 21. On the existence of equilibria and fixed points of maps under constraints. 22. Topological fixed point theory and nonlinear differential equations. 23. Fixed point results based on the Wazeski method.

A nonlinear spectral approach to surjectivity in Banach spaces

Nx) -f(x) = Y9 GEE. (4 We are interested in finding conditions on f and h which ensure the solvability of Eq. (e) for all y E E. In other words we look for the surjectivity of the map A f defined by (A -f)(x) = x -f(x). Throughout this paper we will always assume that f is a quasibounded map (see Granas [2]). That is, there exist A, B > 0 such that Ilf(x)ll G A + B II * II f or all x E E. The quasinorm off, written j f I, is the infimum of all B > 0 such that the above inequality holds for some A 3 0, i.e.,

On the concept of orientability for Fredholm maps between real Banach manifolds

In [ A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree , Ann. Sci. Math. Quebec 22 (1998), 131–148] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.

A characterization of well-posed minimum problems in a complete metric space

Given a functionalI:X →R, defined on a complete metric space (X,d), we give a necessary and sufficient condition for the minimum problem forI onX to be well posed or well posed in the generalized sense.

Second order differential equations on manifolds and forced oscillations

These notes are a brief introductory course to second order dierential equations on manifolds and to some problems regarding forced oscillations of motion equations of constrained mechanical systems. The intention is to give a comprehensive exposition to the mathematicians, mainly analysts, that are not particularly familiar with the formalism of dierential geometry. The material is divided into ve

A MULTIDIMENSIONAL VERSION OF ROLLE'S THEOREM

In this paper we obtain for functions f: Rn > RP a version of Rolles Theorem which we hope the readers will find useful and interesting for the following reasons. Three fundamental results from Calculus: namely Rolles Theorem, the Mean Value Theorem and the Cauchy Generalized Mean Value Theorem can be easily derived from it. The version has intuitive geometrical applications and the proof is very simple. Teachers may find it appropriate to incorporate our result in a course on Multivariable Calculus, since it provides an example of how certain one-dimensional theorems can be rephrased in higher dimensional spaces, and it shows that by expanding our mathematical horizon we frequently gain in organization and unity. Professional mathematicians are all familiar with these facts, but students will surely derive from them a motivation to learn more. The basic idea of our result is to assume a certain behavior of f on the boundary dR of a n-dimensional region R (in the real line this behavior reduces to the familiar condition f(a) = f(b)) to obtain information on the derivative of f at an interior point of R. Of particular relevance to the result is the Mean Value Theorem of Sanderson [10] for a function v: [a, b] RP. We extend his theorem