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Analytical spaces --- Banach spaces --- Sequences (Mathematics) --- Series --- Banach, Espaces de --- Suites (Mathématiques) --- Séries (Mathématiques) --- Sequence (Mathematics) --- 517.984 --- Algebra --- Mathematics --- Processes, Infinite --- Mathematical sequences --- Numerical sequences --- Functions of complex variables --- Generalized spaces --- Topology --- Spectral theory of linear operators --- 517.984 Spectral theory of linear operators --- Suites (Mathématiques) --- Séries (Mathématiques) --- Analyse fonctionnelle --- Functional analysis --- Functional analysis. --- Espaces de banach --- Suites et series

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299 G(t), and to obtain the corresponding properties of its Laplace transform (called the resolvent of - A) R(p) = (A + pl)-l , whose existence is linked with the spectrum of A. The functional space framework used will be, for simplicity, a Banach space(3). To summarise, we wish to extend definition (2) for bounded operators A, i.e. G(t) = exp( - tA) , to unbounded operators A over X, where X is now a Banach space. Plan of the Chapter We shall see in this chapter that this enterprise is possible, that it gives us in addition to what is demanded above, some supplementary information in a number of areas: - a new 'explicit' expression of the solution; - the regularity of the solution taking into account some conditions on the given data (u , u1,f etc ... ) with the notion of a strong solution; o - asymptotic properties of the solutions. In order to treat these problems we go through the following stages: in § 1, we shall study the principal properties of operators of semigroups {G(t)} acting in the space X, particularly the existence of an upper exponential bound (in t) of the norm of G(t). In §2, we shall study the functions u E X for which t --+ G(t)u is differentiable.

517.9 --- 517.5 --- 517.4 --- 51-7 --- 51-7 Mathematical studies and methods in other sciences. Scientific mathematics. Actuarial mathematics. Biometrics. Econometrics etc. --- Mathematical studies and methods in other sciences. Scientific mathematics. Actuarial mathematics. Biometrics. Econometrics etc. --- 517.4 Functional determinants. Integral transforms. Operational calculus --- Functional determinants. Integral transforms. Operational calculus --- 517.5 Theory of functions --- Theory of functions --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Mathematical analysis. --- Numerical analysis. --- 517.984 --- 517.984 Spectral theory of linear operators --- Spectral theory of linear operators --- #KVIV:BB --- 519.6 --- 681.3 *G18 --- 681.3*G19 --- 681.3*G19 Integral equations: Fredholm equations; integro-differential equations; Volterra equations (Numerical analysis) --- Integral equations: Fredholm equations; integro-differential equations; Volterra equations (Numerical analysis) --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- 519.63 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- Mathematical analysis --- Numerical analysis --- Analyse mathématique --- Analyse numérique --- Partial differential equations. --- Partial Differential Equations. --- Numerical Analysis. --- Partial differential equations --- Chemometrics. --- Computational intelligence. --- Applied mathematics. --- Engineering mathematics. --- Mathematical physics. --- Math. Applications in Chemistry. --- Computational Intelligence. --- Mathematical and Computational Engineering. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Engineering --- Engineering analysis --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Chemistry, Analytic --- Analytical chemistry --- Chemistry --- Mathematics --- Measurement --- Statistical methods --- System theory. --- Calculus of variations. --- Systems Theory, Control. --- Calculus of Variations and Optimal Control; Optimization. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Systems, Theory of --- Systems science --- Science --- Philosophy --- Mechanics. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Analysis (Mathematics). --- Analysis. --- 517.1 Mathematical analysis