Volume 5, Issue 3

The aim of this paper consists of introducing on a locally compact and σ-compact metric space a notion of set convergence, which generalizes the Hausdorff convergence, the local Hausdorff convergence and the Kuratowski convergence. We analyze the connections beetwen the three new notions: and. in particular, we prove a compactness result. As a first application of this convergence we give, on a sequence of sets, a condition which assures the lower semicontinuity of the Hausdorff measure with respect to this new convergence and we show that this condition is satisfied by any minimizing sequence of Mumford-Shah functional.

We consider the functional F : H 1 0 (B(0,1)) → ℝ where α > 0,p > 0, 1 < γ ≤ 2, and B(0,1) is the unit ball in ℝ 2 . We prove that for any p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 no maximizer of F(u) on the unit ball in H 1 0 is radially symmetric provided that cu is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the Hénon equation with Dirichlet boundary conditions.

Large amplitude solutions of asymptotically positively homogeneous perturbations of hamiltonians systems at resonance can be unbounded, either in the past, or in the future. We present conditions for boundedness or unboundedness, generalizing in particular the results obtained by Alonso and Ortega [1] for scalar second order equations with asymmetric nonlinearities.

Global bifurcation in boundary value problems for systems of ordinary differential equations is studied. Rotation numbers are associated with solutions and are shown to be invariant along bifurcating continua. This invariance is then used to analyze the global structure of the bifurcating continua, and to demonstrate the existence of infinitely many solutions to some boundary value problems.

Using Mawhin’s continuation theorem, sufficient conditions are obtained for the existence of positive periodic solutions for periodic delay-Lotka-Volterra systems.

We prove a fixed point theorem for continuous mappings which satisfy a compression-expansion condition on the boundary of a N-dimensional cell of ℝ N . Our work is motivated by a previous theorem of Andres, Gaudenzi and Zanolin (1990) dealing with the existence of periodic solutions for dissipative-repulsive systems and is also related to some recent results by Zgliczyński and Gidea (2004) concerning covering relations for Markov partitions. Applications are given to the study of the iterates of a continuous map of ℝ N . In this case we obtain some theorems on the existence of periodic points and chaotic-like dynamics.