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 theory 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 fi rst chapter we present some preliminaries on the spectral theory, most of the materials 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 characterizes 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 materials follow from Khalil Ezzinbi [5]; Alain Bensoussen, Guiseppe Da Prato, Michel C. Delphour, Sanjoy K. Mitter [2].