Faria, Teresa, 1958-Caetano, Diogo Loureiro2018-08-292018-08-2920182018http://hdl.handle.net/10451/34646Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de Ciências, 2018In this dissertation, we provide a proof of a Principle of Linearized Stability for a class of autonomous differential equations with infinite delay. This is done via techniques from functional analysis, namely duality theory for semigroups of bounded linear operators, following the approach of O. Diekmann and M. Gyllenberg in [12]. First, we make a detailed study of some aspects of the theory of strongly continuous semigroups (also called C0 semigroups) of linear operators in Banach spaces. In particular, we prove the classical theorem of Hille-Yosida, characterizing infinitesimal generators of C0 semigroups, and define the adjoint of a strongly continuous semigroup. Since the adjoint semigroup is not necessarily strongly continuous, we study whether it can be restricted to some subdomain where strong continuity holds. This is the starting point for the sun-star calculus, of which we make use throughout the remaining chapters. We introduce some elements of the sun-star theory for linear operators and give meaning to an abstract integral equation, for which we prove existence, uniqueness, continuation and regularity of solutions. We then consider, in a suitable (weighted) space of continuous functions on (-∞;0] that vanish at -∞, an initial value problem for a differential equation with infinite delay and prove an equivalence result between solutions of such equation and the solution semigroup of an abstract integral equation. After that, we study the characteristic equation of the linearized problem, and prove that the roots of this equation are precisely the eigenvalues of the infinitesimal generator of the solution semigroup of the linear equation. Moreover, we show that, on a fixed half-space, there are only finitely many such roots. Consequently, the spectral projection of the resolvent operator induces a decomposition of the phase space as the direct sum of two invariant subspaces - one with finite dimension, and the other where the semigroup is exponentially stable -, to which we can apply a theorem by Desch and Schappacher. As a result, we obtain a proof of the Principle of Linearized Stability, generalizing for this case the well-known result for ordinary and finite-delay differential equations.engEquações diferenciaisAtraso infinitoEstabilidade linearTeoria de semigruposTeoria sun-starTeses de mestrado - 2018master thesis201988933