The existence of at least two solutions for nonlinear equations close to semilinear equations at resonance is obtained by the degree theory methods. The same equations have no solutions if one slightly changes the right-hand side. The abstract result is applied to boundary value problems with specific nonlinearities.
On décrit une formule de trace [S] pour les résonances, qui est valable en toute dimension et pour les perturbations à longue portée du Laplacien. On établit une nouvelle application à l’éxistence de nombreuses résonances pour des opérateurs de Schrödinger semi-classiques.
We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii)...
We consider convex versions of the strong approximation property and the weak bounded approximation property and develop a unified approach to their treatment introducing the inner and outer Λ-bounded approximation properties for a pair consisting of an operator ideal and a space ideal. We characterize this type of properties in a general setting and, using the isometric DFJP-factorization of operator ideals, provide a range of examples for this characterization, eventually answering a question...
There is a one parameter family of bilinear Hilbert transforms. Recently, some progress has been made to prove Lp estimates for these operators uniformly in the parameter. In the current article we present some of these techniques in a simplified model...
We provide a local as well as a semilocal convergence analysis for Newton's method using unifying hypotheses on twice Fréchet-differentiable operators in a Banach space setting. Our approach extends the applicability of Newton's method. Numerical examples are also provided.
Let be a -semigroup with unbounded generator . We prove that has generically a very irregular behaviour for as .