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The project of this book is concerned with various aspects of bifurcation theory and spectral theory of linear and nonlinear differential equations describing many problems in mathematical physics. First part of the present thesis deals with a nonlinear transport equation derived from a model introduced by Rotenberg, which describes the growth of a cell population. Second part, we introduce and study the essential approximate of pseudospectrum and condition pseudospectrum of closed, densely defined linear operators in the Banach space. We begin by the definition and we investigate the characterization, the stability and some properties of these pseudospectrum.