Let G be a subset of the dual of a real Banach space X and F ⊂ G. Then F is a James boundary of G if each w∗ -continuous linear functional on X attains its supremum over G on an element of the set F. We ask whether a norm bounded subset of X which is countably compact for the topology generated by F is ne- cessary sequentially compact for the topology generated by G. The main content of our work is a positive solution to this problem. As a corollary we obtain James characterization of weakly compact subsets of a real Banach space. Due to the Eberlein-Šmuljan theorem a positive solution to the so called boundary problem is shown as a special case of the affirmative answer to the question raised above. The question is further discussed for a case of Banach spaces defined over the complex field. In this case we cannot use the old definition of the James boun- dary but by a "natural" way it is possible to redefine the term James boundary and then we are able to answer our question positively again. 1

In this thesis we present an extensive characterization of weak* sequentially precompact subsets of the dual of a sequentially order complete M-space with an order unit. This central part of the thesis generalizes results due to H.H. Schaefer and X.D. Zhang showing that small weak* compact subsets of the dual of a space of bounded measurable real-valued functions (continuous real-valued functions on a compact quasi-Stonian space) are weakly compact. Moreover, while the proofs of Schaefer and Zhang use measure theoretical arguments, the arguments presented here are purely elementary and are based on the well-known result, that the space l1 has the Schur property. Finally some applications are given. For example, we investigate compact or sequentially precompact subsets, which consist of order-weakly compact operators, in the space of continuous linear operators defined on a sequentially order complete Riesz space with values in a Banach space provided with the strong operator topology: as an immediate consequence of the results, we can easily deduce extended versions of the Vitali-Hahn-Saks theorem for vector measures. For this we need a generalization of the Yosida-Hewitt decomposition theorem, which is proved here with other techniques like the factorization of an order-weakly compact operator through a Banach lattice with order continuous norm.

Dans cette thèse nous regardons plusieurs modèles de populations structurés s’écrivant à l’aide d’équations de transport. Le caractère bien posé ainsi que la positivité des solutions sont montrés de manière systématique au sens des sémiologues dans un cadre L1. Un premier travail est consacré à un système de type proie prédateur structuré en âge. Une étude de stabilité des équilibres nous permet de formuler explicitement un seuil un seuil d’extinction ainsi qu’in seuil pouvant amener à l’explosion des populations. On obtient numériquement la possibilité d’un cycle limite ainsi que la convergence vers un équilibre de coexistence des populations. Dans un cas particulier, ce modèle se réécrit comme un système différentiel à retard. A l’aide de fonctionnelle de Lyapunov, on montre la stabilité globale de cet équilibre sous certaines conditions. On étudie également 2 modèles structuré en taille, issus de la dynamique cellulaire. L’un est composé de deux équations de transport où la cellule peut être soit prolifèrent soit quiescente ; et le deuxième est une équation de type transport/ diffusion avec des conditions aux bords FELLER. On vérifie à chaque fois l’irréductibilité du semi groupe puis des arguments de faibles capacité L1 nous donne l’existence d’un « gap spectral » sous certaines conditions. On démontre ainsi dans certains cas la croissance exponentielle asynchrone du semi groupe / This thesis is dedicated to some structured populations models described with transport or transport-diffusion equations. The well-posedness, in the semigroupes setting in L1 and the positivity of the solutions are systematically shown. A first work is dedicated to an age-structured predator/prey system. A stability study of the equilibria allow us to give explicit formulations of an extinction threshold and an threshold which can lead to explosion of solutions. We numerically obtain the possibility to get a limit cycle and the convergence to a coexistence equilibrium of the populations. In a specific case, this model rewrites as a delay differential system. Using Lyapunov functional, we show the global stability of this equilibrium under some assumptions. We also study two size-structured models that come from cellular dynamics. The first one consists on two transport equations, where the cell can either proliferate or be quiescent, and the second one is a transport-diffusion equation with Feller boundary conditions. The irreducibility of the semigroup governing this latter model is always satisfied using the Hopf maximum principle. However, the irreducibility for the first model is true only under a necessary and sufficient condition that we give. We also show for these two models, using some weak compactness arguments in L1, the existence of a `spectral gap' (essential type strictly less than the type) ensuring the asynchronous exponential growth of the semigroup.