Neste trabalho estudaremos a existência de soluções fracas, pseudo quase periódicas e periódicas, para uma classe de sistemas não autônomo do tipo neutro com retardamento não limitado modelados na forma \' d SUP. dt\' (u(t) + F(t, ut)) = A(t)u(t) + G(t, \'u IND.t\' ), t \'PERTENCE A\' (0, a), \'u IND. 0\' = \'varphi\' \'PERTENCE A\' B, onde {A(t)} ´e uma família de operadores lineares fechados, com um dom´?nio comum D =D(A(t)), a história ut : (-\'INFINITO\'1, 0] \'SETA\' X, \'u IND. t\'(THETA) = u(t+\'THETA\'), pertence a um espaço de fase abstrato B definido axiomaticamente e F,G : [0, a] × B \'SETA\' X são funções apropriadas. Para obter alguns de nossos resultados, precisaremos usar as propriedades da família de operadores de evolução (U(t, s))\'t > OU=\'s, para o sistema u? (t) - A(t)u(t) = 0, t \'Pertencer A\' (0, a), \'u IND.0\' = \'phi\', onde U(t, s) ´e uma fam´?lia de operadores lineares limitados em X / In this work we study the existence of mild, pseudo almost-periodic and periodic solution, concepts introduced be later for a class of abstract neutral functional systems with unbounded delay in the form \'d SUP dt\' (u(t) + F(t, \'u IND.t\')) = A(t)u(t) + G(t, \'u IND. t\'), t IT BELONGS\' (0, a), \'u IND.0\' = \'varphi\' \'IT BELONGS\' , where is a family of closed linear operator in a Banach space X, with a common domain D = D(A(t)), t \'IT BELONGS\' R, densely defined in X; the history \'u IND. t\' : (-\'THE infinite\', 0] \' ARROW\' X, ut(\'THETA\') = x(t+\'THETA\'), belongs to some abstract phase space B defined axiomatically and F,G : I ×B \'ARROW\' X are appropriate functions and I is a bounded or unbounded interval in R. To establish some of our results, we will use the properties of a systems of evolution (U(t, s))\' t IND. > OR =\'s, for a system in the form u? (t) - A(t)u(t) = 0, t \'IT BELONGS\' (0, a), \'u IND.0\' = \'PHI\', where (U(t, s))\'t IND. > 0R =\'s is a family of bounded linear operators on X

We study the problem on the exact estimation of the value of the best mean-square approximation on the period to an arbitrary complex-valued periodic function by trigonometric polynomials of degree not exceeding a given number in terms of its nonclassical L2 -modulus of continuity generated by the finite-difference operator of order 2m-1, m>= 2, with alternating constant coefficients equal to 1 in absolute value. / Исследуется задача о точной оценке величины наилучшего среднеквадратического приближения на периоде произвольной комплекснозначной периодической функции тригонометрическими полиномами порядка не выше заданного через ее неклассический L2 -модуль непрерывности, порожденный конечно-разностным оператором порядка 2m-1, m>= 2, с постоянными знакочередующимися коэффициентами, равными по модулю единице.

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Neste trabalho, estudamos equações diferenciais em espaços de Banach. Duas questões são abordadas: a robustez da estabilidade assintótica, e a aproximação de soluções de sistemas periódicos por wavelets. Observa-se que a estabilidade exponencial do sistema x = A(t)x é qualitativamente preservada pelo sistema perturbado x=A(t)x+B(t)x se B(t) for integralmente pequeno. Consequentemente, tal propriedade é preservada por uma perturbação B(wt)x para w suficientemente grande, mesmo se B(t) pertence a uma classe mais geral de funções do que as funções quase-periódicas, aqui apresentada. Além disso, estudamos o efeito de aproximações de uma função periódica f (t) por wavelets periódicas na solução de um sistema periódico x = Ax+ f (t). Conclui-se que as soluções do problema inicial podem inclusive ser aproximadas utilizando a wavelet base não-periódica. / In this work, we study differential equations in Banach spaces. Two questions were considered: the robustness of the asymptotic stability, and the approximation of solutions of periodic systems by wavelets. It is observed that the exponential stability of the system x = A(t)x is qualitatively preserved by the perturbed system x = A(t)x+B(t)x if B(t) is integrally small. As a consequence, this property is preserved by a perturbation B(wt) for w sufficiently large, even if B(t) is in a class of functions which is more general than almost-periodic functions, presented here. Furthermore, we study the effect of approximating a periodic function f (t) by periodic wavelets in the solution of a periodic system x = Ax+ f (t). It is concluded that the solutions of the initial problem can even be approximated using the non-periodic base wavelet.

Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter-dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton's method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton's method commonly used on large-scale problems. Frequently, these variations are more robust than Newton's method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large-scale case, yet notably more robust than Newton's method in practice. The normal flow method is a generalization of Newton's method for solving the system F:R^m→R^n, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large-scale case. These are developed in direct analogy with inexact-Newton, globalized inexact-Newton, and trust-region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is treated in two parts. The first is to find sufficient conditions for a function so that its Fourier series distributions become common and uniform, as well as an approach to Fejér’s Theorem, an interesting and useful result of no Fourier Series study. A second part of the application of the Fourier Series, Weyl equidistribution theorem. A problem that lies at the frontier of Dynamic Systems with a Theory of Numbers. The same refers to the distribution of irrational numbers in the range [0, 1). / Este trabalho é tratado em duas partes. A primeira consiste em encontrar condições suficientes sobre uma dada função para que sua expansão em Série de Fourier convirja pontualmente e uniformemente, como também uma abordagem ao Teorema de Fejér, resultado interessante e útil no estudo de Séries de Fourier. A segunda parte uma aplicação provenientes das Séries de Fourier, o Teorema de equidistribuição de Weyl. Um problema que se encontra na fronteira dos Sistemas Dinâmicos com a Teoria dos Números. O mesmo refere-se à distribuição de números irracionais no intervalo [0, 1).

Cette thèse porte sur les équations d’évolution et s’articule autour de trois parties. Dans la première partie, on se propose de se concentrer sur le critère oscillatoire de certaines équations différentielles. Des résultats classiques sur les fonctions presque-périodiques sont rassemblés dans le premier chapitre. Le deuxième chapitre de cette thèse a pour objectif de prouver l’existence d’une solution presque-périodique de Besicovitch d’une équation différentielle de second ordre sur un espace de Hilbert. L’approche utilisée se base sur un formalisme variationnel. La deuxième partie de cette thèse traite le comportement asymptotique des problèmes de Cauchy dans le cas non autonome. Les semi-groupes et les familles d’évolution étant les outils principaux utilisés dans cette partie, le troisième chapitre introduit des résultats importants de cette théorie, notamment ceux permettant de caractériser la stabilité des semigroupes et des familles d’évolution périodiques. Dans le quatrième chapitre de cette contribution, on prouve, en utilisant une approche basée sur les semigroupes, un résultat liant la bornitude de solutions de problèmes de Cauchy périodiques et la stabilité exponentielle uniforme des familles d’évolution issues de ces problèmes. Dans une troisième partie, on focalise l’attention sur quelques résultats sur la dichotomie exponentielle comme une propriété liée au comportement asymptotique des systèmes différentiels. Quelques résultats connus sont, par suite, réunis au cinquième chapitre qui introduit brièvement la notion de dichotomie exponentielle. Dans un dernier chapitre, une caractérisation de la dichotomie exponentielle d’une famille d’évolution en termes de bornitude des solutions de problèmes de Cauchy opératoriels correspondants sera démontrée. / This PhD thesis deals with the evolution equations and is organized in three parts. The first part is devoted to the almost periodic solutions of certain differential equations. Classic results on the almost periodic functions are collected in the first chapter. The second chapter of this thesis aims to prove the existence of an almost-periodic solution of Besicovitch of a second-order differential equation on Hilbert space. The used approach is based on a variational formalism. In the second part of this thesis, we study the asymptotic behavior of Cauchy problems in the non-autonomous case. We give in the third chapter important results on semigroups and evolution families, namely, those allowing to characterize the stability of semigroups and periodic evolution families. We prove in the fourth chapter sufficient conditions for the uniform exponential stability of a strongly continuous, q-periodic evolution family acting on a complex Banach space. The last part in this work focuses the attention on some results on the exponential dichotomy as a property for the asymptotic behavior of the differential systems. Some well-known results are given in the fifth chapter which introduces briefly the concept of the exponential dichotomy. A characterization of the exponential dichotomy for evolution family in terms of boundedness of the solutions to periodic operatorial Cauchy problems will be established.