Algunas Aplicaciones de Métodos de Espacios de Hilbert a Ecuaciones Diferenciales Semilineales con Espectro Discreto

Abstract

In this dissertation we expose some ideas that allow us to face problems related to semi-linear differential equations through Hilbert spaces theory and functional Analisis. In the chapters 2 and 3 we present a brief summary of some concepts and results in relation to certain Hilbert spaces and the spectral theory of operators that is useful in the statement of differential equations in the so mentioned spaces. At the first part of the chapter 4 we present the general form of the semi-linear differential equations which are the main focusing of this dissertation, and the way in which these are related with problems of operators in Hilbert spaces. At the end of the chapter 4, we develop as particular case of the semi-linear equations the armonic oscilator and the linear resonance phenomena for this problem; we exhibit solutions and we show the relations that exist between the spectrum of the differential operator and the existence and uniqueness of solutions to this problem. Finally, in the chapter 6 we introduce a generalization of the previous result to equations that involve self-adjoint differential operator over any Hilbert space and no-linearity more general.