Submitted by Laura Rezzieri Gambera null (laura.rgambera@gmail.com) on 2016-04-14T18:58:22Z No. of bitstreams: 1 Dissertação de mestrado - Laura R Gambera.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-18T13:27:01Z (GMT) No. of bitstreams: 1 gambera_lr_me_sjrp.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) / Made available in DSpace on 2016-04-18T13:27:01Z (GMT). No. of bitstreams: 1 gambera_lr_me_sjrp.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) Previous issue date: 2016-03-29 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho estudamos a existência de solução fraca quase automórfica para equações diferenciais abstratas de segunda ordem descritas na forma x’’(t) = Ax(t) + f(t, x(t)), t real, onde x(t) pertence a X para todo t real, X é um espaço de Banach, A : D(A) C X -> X é o gerador infinitesimal de uma família cosseno fortemente contínua de operadores lineares limitados em X e f : R x X -> X é uma função apropriada. / In this work we study the existence of an almost automorphic mild solution to second order abstract differential equations given by x’’(t) = Ax(t) + f(t, x(t)), t real, where x(t) lies in X for all t real, X is a Banach space, A : D(A) C X ->X is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on X and f : R x X -> X is an appropriate function. / FAPESP: 2013/22813-3

Dans cette thèse, nous affinons l'étude des fonctions presque automorphes sur time scales introduites dans la littérature par Lizama et Mesquita, nous explorons de nouvelles propriétés de ces fonctions et appliquons les résultats à étudier l'existence et l'unicité de solution presque automorphe d'une nouvelle classe d'équations dynamiques sur time scales. Puis nous introduisons la notion de fonction presque automorphe de classe Cn, nous investiguons les propriétés fondamentales de ces fonctions et utilisons les résultats pour établir l'existence, l'unicité et la stabilité globale et exponentielle de solution presque automorphe de classe C1 d'un système d'équations dynamiques avec délai variable fini modélisant un réseau de neurones. Ensuite nous présentons le concept de fonctions asymptotiquement presque automorphes de classe Cn. Nous démontrons quasiment toutes les propriétés de ces fonctions, lesquelles nous permettent, sous des hypothèses convenables, d'établir, d'une part, que l'unique solution d'un problème avec condition initiale est asymptotiquement presque automorphe de classe C1, et d'autre part, l'existence et l'unicité de solution asymptotiquement presque automorphe pour une équation intégro-dynamque avec conditon initiale non locale sur time scales. Enfin, en utilisant la notion de semi-groupe sur time scales de Hamza et Oraby, nous généralisons les résultats de Lizama et Mesquita en dimension infinie, c'est-à-dire, nous étudions l'existence et l'unicité des solutions presque automorphes pour des équations dynamiques semi linéaires abstraites sur time scales. / In this thesis, we refine the notion of almost automorphic functions on time scales introduced in the literature by Lizama and Mesquita, we explore some new properties of such functions and apply the results to study the existence and uniqueness of almost automorphic solution for a new class of dynamic equations on time scales. Then we introduce the concept of almost automorphic functions of order n on time scales, we investigate the fundamental properties of these functions and we use the findings to establish the existence and uniqueness and the global stability of almost automorphic solution of one to a first order dynamical equation with finite time varying delay. Then we present the concept of asymptotically almost automorphic functions of order n on time scales. We study the properties of these functions and we use the results to prove, under suitable hypothesis, that the unique solution to a problem with initial condition is asymptotically almost automorphic of order one at the one hand, and the existence and uniqueness of asymptotically almost automorphic solution for an integro-dynamic equation with nonlocal initial conditon on time scales in other hand. Finally, using the concept of semigroup on time scales introduced by Hamza and Oraby, we generalize the results in Lizama and Mesquita's paper for abstract Banach spaces, that is, we study the existence and uniqueness of almost automorphic solution for semilinear abstract dynamic equations on time scales.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we first develop a brief theoretical approach of semigroups of bounded linear operators, culminating on Hille-Yosida Theorem. Then we used the extrapolation theory to study su cient conditions to obtain existence and uniqueness of Almost Automorphic and Pseudo-Almost Automorphic mild solutions, through the Banach's Fixed Point Theorem for the semilinear evolution equation x(t) = Ax(t) + f(t; x(t)); t E R, where A : D(A) X ! X is a Hille-Yosida operator of negative type and not necessary dense domain on the Banach space X. / Neste trabalho, desenvolvemos inicialmente uma breve abordagem te orica dos semigrupos de operadores lineares limitados, culminando no Teorema de Hille-Yosida. Em seguida, usamos a teoria de extrapolação a fim de estudar condições suficientes para obtermos a existência e a unicidade de soluções brandas Quase Automórficas e Pseudo-quase Automórficas, por meio do Teorema do Ponto Fixo de Banach, para a equação de evolução semilinear x(t) = Ax(t) + f(t; x(t)); t E R, onde A : D(A) X ! X é um operador de Hille-Yosida de tipo negativo e dom ínio não necessariamente denso, definido no espaço de Banach X.