Let X be a real Banach space and G1, G2 two nonempty, open and bounded subsets of X such that 0 ∈ G2 and G2 ⊂ G1. The problem (∗) T x + Cx = 0 is considered, where T : X ⊃ D(T) → X is an accretive or monotone operator with 0 ∈ D(T) and T(0) = 0, while C : X ⊃ D(C) → X can be, e.g., one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type (S+) with T positively homogeneous of degree one; (d) quasi-bounded and satisfies a generalized (S+)-condition w.r.t. the operator T, while T is positively homogeneous of degree one. Solutions are sought for the problem (∗) lying in the set D(T + C) ∩ (G1 \ G2). Nontrivial solutions of (∗) exist even when C(0) = 0. The degree theories of Leray and Schauder, Browder, and Skrypnik as well as the degree theory by Kartsatos and Skrypnik for densely defined operators T, C are used. The last three degree theories do not assume any compactness conditions on the operator C. The excision and additivity properties of these degree theories are employed, and the main results are significant extensions or generalizations of previous results by Krasnoselskii, Guo, Ding and Kartsatos involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator T. Moreover, a new degree theory developed by Kartsatos and Skrypnik has been used to prove a similar result for operators of type T + C, where T : X ⊃ D(T) → 2 X∗ is a multi-valued maximal monotone operator, with 0 ∈ D(T) and 0 ∈ T(0), and C : X ⊃ D(C) → X∗ is a densely defined quasi-bounded and finitely continuous operator of type (S˜+). The problem of existence of nonzero solutions for T x + Cx + Gx 3 0 is also considered. Here, T is maximal monotone, C is bounded demicontinuous of type (S+), and G is of class (P). Eigenvalue and invariance of domain results have also been established for the sum L + T + C : G ∩ D(L) → 2 X∗ , where G ⊂ X is open and bounded, L : X ⊃ D(L) → X∗ densely defined linear maximal monotone, T : X → 2X∗ bounded maximal monotone, and C : G → X∗ bounded demicontinuous of type (S+) w. r. t. D(L).

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

Les jeux stochastiques à somme nulle possèdent une structure récursive qui s'exprime dans leur opérateur de programmation dynamique, appelé opérateur de Shapley. Ce dernier permet d'étudier le comportement asymptotique de la moyenne des paiements par unité de temps. En particulier, le paiement moyen existe et ne dépend pas de l'état initial si l'équation ergodique - une équation non-linéaire aux valeurs propres faisant intervenir l'opérateur de Shapley - admet une solution. Comprendre sous quelles conditions cette équation admet une solution est un problème central de la théorie de Perron-Frobenius non-linéaire, et constitue le principal thème d'étude de cette thèse. Diverses classes connues d'opérateur de Shapley peuvent être caractérisées par des propriétés basées entièrement sur la relation d'ordre ou la structure métrique de l'espace. Nous étendons tout d'abord cette caractérisation aux opérateurs de Shapley "sans paiements", qui proviennent de jeux sans paiements instantanés. Pour cela, nous établissons une expression sous forme minimax des fonctions homogènes de degré un et non-expansives par rapport à une norme faible de Minkowski. Nous nous intéressons ensuite au problème de savoir si l'équation ergodique a une solution pour toute perturbation additive des paiements, problème qui étend la notion d'ergodicité des chaînes de Markov. Quand les paiements sont bornés, cette propriété d'"ergodicité" est caractérisée par l'unicité, à une constante additive près, du point fixe d'un opérateur de Shapley sans paiement. Nous donnons une solution combinatoire s'exprimant au moyen d'hypergraphes à ce problème, ainsi qu'à des problèmes voisins d'existence de points fixes. Puis, nous en déduisons des résultats de complexité. En utilisant la théorie des opérateurs accrétifs, nous généralisons ensuite la condition d'hypergraphes à tous types d'opérateurs de Shapley, y compris ceux provenant de jeux dont les paiements ne sont pas bornés. Dans un troisième temps, nous considérons le problème de l'unicité, à une constante additive près, du vecteur propre. Nous montrons d'abord que l'unicité a lieu pour une perturbation générique des paiements. Puis, dans le cadre des jeux à information parfaite avec un nombre fini d'actions, nous précisons la nature géométrique de l'ensemble des perturbations où se produit l'unicité. Nous en déduisons un schéma de perturbations qui permet de résoudre les instances dégénérées pour l'itération sur les politiques. / Zero-sum stochastic games have a recursive structure encompassed in their dynamic programming operator, so-called Shapley operator. The latter is a useful tool to study the asymptotic behavior of the average payoff per time unit. Particularly, the mean payoff exists and is independent of the initial state as soon as the ergodic equation - a nonlinear eigenvalue equation involving the Shapley operator - has a solution. The solvability of the latter equation in finite dimension is a central question in nonlinear Perron-Frobenius theory, and the main focus of the present thesis. Several known classes of Shapley operators can be characterized by properties based entirely on the order structure or the metric structure of the space. We first extend this characterization to "payment-free" Shapley operators, that is, operators arising from games without stage payments. This is derived from a general minimax formula for functions homogeneous of degree one and nonexpansive with respect to a given weak Minkowski norm. Next, we address the problem of the solvability of the ergodic equation for all additive perturbations of the payment function. This problem extends the notion of ergodicity for finite Markov chains. With bounded payment function, this "ergodicity" property is characterized by the uniqueness, up to the addition by a constant, of the fixed point of a payment-free Shapley operator. We give a combinatorial solution in terms of hypergraphs to this problem, as well as other related problems of fixed-point existence, and we infer complexity results. Then, we use the theory of accretive operators to generalize the hypergraph condition to all Shapley operators, including ones for which the payment function is not bounded. Finally, we consider the problem of uniqueness, up to the addition by a constant, of the nonlinear eigenvector. We first show that uniqueness holds for a generic additive perturbation of the payments. Then, in the framework of perfect information and finite action spaces, we provide an additional geometric description of the perturbations for which uniqueness occurs. As an application, we obtain a perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.