Semigroups Of Linear Operators And Application To Differential Equations

Abstract

This work concerns one of the most important tools to solve well-posed problems in the the- ory of evolution equations (e.g di usion equation, wave equations, ...) and in the theory of stochastic process, namely the semigroups of linear operators with application to di erential equations. A semigroup of linear operator on a Banach space X is a continuous operator valued function T : [0, +∞) → B(X, X) such that T (t + s) = T (t)T (s) for every t, s ≥ 0 and T (0) = I . The fact that every non zero continuous complex function that satis es f (s+t) = f (s)f (t) for every t, s ≥ 0 has the form f (t) = exp(at), and that f is determined by the number a = f (0), T (t)x − x ; motivates the association to (T (t)t≥0 ) of an operator A de ned by Ax := lim t→0+ t x ∈ D(A) and called the in nitesimal generator of (T (t))t≥0 . Furthemore the study of the converse is of essential interest in the line of Hille-Yosida. We divide this work into three chapters: In the rst chapter we present some preliminaries on the spectral theory, most of the ma- terials follow from A. D. Andrew and W. L. Green[1]; C.E. Chidume [4], G. Barbatis; E.B. Davies and J.A. Erdos[3]; Erwin Kreyszig [5]; Khalil Ezzinbi [5]. In the second chapter we present the generation and representation of semigroups of linear operators and provide Hille-Yosida theorem which caracterizes the in nitesimal generator of a class of Continuous semigroup; essentially most of the materials follow from A. Pazy [7]; Khalil Ezzinbi [5]. Lastly we present the Abstract Cauchy problem as application, essentially most of the ma- terials follow from Khalil Ezzinbi [5]; Alain Bensoussen, Guiseppe Da Prato, Michel C. Delphour, Sanjoy K. Mitter [2].