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The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

Le sujet de cette thèse porte sur la résolution d'équations d'opérateurs dans l'algèbre B(H) des opérateurs linéaires bornés sur un espace de Hilbert H. Nous étudié celles qui sont associées aux dérivations généralisées. Mon sujet de thèse explore aussi des équations beaucoup plus générales comme celles du type AXB - XD = E ou AXB - CXD = E où A, B, C, D et E appartiennent à B(H). Plus précisément il s'agit de donner une description des solutions de ces équations pour E appartenant à une famille précise(autoadjoint, normal, rang un, rang fini, compact, couple de Fuglède Putnam) et pour des opérateurs A, B, C et D appartenant à des bonnes classes d'opérateurs ( celles qui interviennent dans les applications, notamment en physique) comme les opérateurs autoadjoints, les opérateurs normaux, sous normaux,... En dehors du cas où les spectres de A et B sont disjoints, il n'existe pas de méthode générale pour construire de manière effective l'ensemble des solutions de l'équation de Sylvester AX - XB = C à partir des opérateurs A, B et C. Un des objectifs de mon travail de thèse est de fournir une méthode constructive dans le cas où A, B et C appartiennent à des bonnes classes d'opérateurs. Une étude spectrale des solutions est également faite. A coté de cette étude qualitative, il y a aussi une étude quantitative.Il s'agit d'obtenir aussi des estimations précises de la norme d'opérateur(ou norme de Schatten) des solutions en fonction des normes des opérateurs correspondants aux données. Ceci nous a d'ailleurs conduit à des résultats concernant quelques inégalités intéressantes pour les dérivations généralisées, et enfin quelques résultats concernant les opérateurs dans un espace de Banach sont également donnés / The subject of this thesis focuses on the resolution of operator equationsin B(H) algebra of bounded linear operators on a Hilbert space. We studythose associated with generalized derivations. In this thesis, we also exploremore general equations such as the type AXB - XD = E or AXB -CXD = E where A, B, C, D and E belong to B(H). Specifically it is adescription of the solutions of these equations for E belongs in a precisefamily (Self-adjoint, normal, rank one, finite rank, compact, pair of FugledePutnam) and the operators A, B, C and D belonging to the good classesof operators (Those involved in applications , especially in physics) as theself-adjoint operators, normal operators, subnormal operators... Apart fromthe case where the spectra of A and B are disjoint, there is not any generalmethod for constructing effectively all solutions of the Sylvester equationAX - XB = C from the given operators A, B and C. One objective of thisthesis is to provide a constructive approach in when A, B and C belong toconventional families of operators. A spectral study of the solutions is alsostudied. Besides this qualitative study, there is also a quantitative study.It is also to obtain accurate estimates of the operator norm (or norm ofSchatten) of the solutions in terms of operator norms corresponding to data.This also led us to obtain results concerning some interesting inequalitiesfor generalized derivations, and finally some examples and properties ofoperators on a Banach space are also given

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.