In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

Rearrangements are two measurable real-valued functions that have equal measure of pre-images of upper level sets. In this thesis, I will investigate several matters and problems relating to rearrangements: the relationship between assumptions on the measure space and desirable properties of the set of rearrangements, and the validity of rearrangement inequalities; generalising the Mountain Pass Lemma over rearrangements; and applying topological degree theory to boundary value problems involving rearrangements. From suppositions on the measure space, such as the measure space having finite measure and no atoms, it can proved that the set of rearrangements is contractible and locally contractible. The Mountain Pass Lemma over rearrangements can be generalised, so instead of considering continuous paths from the closed unit interval to the set of rearrangements; it will consider the continuous functions from the closed unit disc into the set of rearrangements. Topological degree theory is used to associate admissible triples of functions, sets and points with integers. These methods will be applied to a boundary value problem involving rearrangements, where the domain is almost equal to the union of balls, which has been studied using variational methods, providing new multiplicity results. The minimum number of solutions to this boundary value problem is found to be related exponentially to the number of balls contained in the domain.

No description available.

O principal objetivo deste trabalho é apresentar a construção do grau topológico em dimensão finita e infinita. Veremos, também, algumas de suas propriedades e aplicações topológicas, como o clássico Teorema de ponto fixo de Brouwer. Seguindo o que fizeram Manàsevich e Mawhin no artigo ``Periodic Solutions for Nonlinear Systems with p-Laplacian-Like Operators}. Journal of Differential Equations, vol. 145, p. 367-393, 1998\'\', vamos provar a existência de soluções para um sistema diferencial não linear com condições de contorno, usando, entre outras ferramentas, o grau topológico. / The main purpose of this work is the construction of the topological degree in finite and infinite dimension. In addition, we will see some of its properties and topological applications. Following the approach of Mannàsevich and Mawhin in the paper ``Periodic Solutions for Nonlinear Systems with p-Laplacian-Like Operators. Journal of Differential Equations, vol. 145, p. 367-393, 1998\'\', we will prove the existence of solutions for a nonlinear differential system with boundary conditions, using, among other tools, the topological degree.

We will study the solution stability of a parametric quasi-variational inequality without the monotonicity assumption of operators. By using the degree theory and the natural map we show that under certain conditions, the solution map of the problem is lower semi-continuous with respect to parameters.

Neste trabalho, apresentamos condições suficientes para a existência e a unicidade de soluções periódicas para equações diferenciais funcionais com retardo e impulsos. Os resultados sobre existência estão ancorados num Teorema de Continuação de Jean Mawhin. Por outro lado, as condições que garantem a unicidade de soluções periódicas são condições do tipo Lipschitz / In this work, we present sufficient conditions for the existence and the uniqueness of periodic solutions for retarded functional differential equations with impulses. The results on the existence of periodic solutions are anchored by a Jean Mawhin continuation theorem. Moreover, the conditions that guarantee the uniqueness of the periodic solutions are Lipschitz type

Submitted by Luciana Sebin (lusebin@ufscar.br) on 2016-09-12T17:21:38Z No. of bitstreams: 1 TeseFMF.pdf: 3344288 bytes, checksum: 3bfb3caf40d3dcd756aa8697b58b5f19 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-12T17:29:02Z (GMT) No. of bitstreams: 1 TeseFMF.pdf: 3344288 bytes, checksum: 3bfb3caf40d3dcd756aa8697b58b5f19 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-12T17:44:44Z (GMT) No. of bitstreams: 1 TeseFMF.pdf: 3344288 bytes, checksum: 3bfb3caf40d3dcd756aa8697b58b5f19 (MD5) / Made available in DSpace on 2016-09-12T17:46:25Z (GMT). No. of bitstreams: 1 TeseFMF.pdf: 3344288 bytes, checksum: 3bfb3caf40d3dcd756aa8697b58b5f19 (MD5) Previous issue date: 2015-08-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The aim of this work is to present results about the existence of non-trivial solutions for some classes of resonant and superlinear eliptic systems employing topological methods. More specifcally, we use a-priori bounds on the eventual solutions of this problems and topological degree theory. / Neste trabalho apresentamos a existência de soluções não triviais para classes de sistemas elípticos ressonantes e superlineares. Tais sistemas são tratados via métodos topológicos. Encontramos estimativas a priori para possíveis soluções destes sistemas e utilizamos estas estimativas juntamente com a teoria do grau topológico para garantir a existência de soluções.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2018-06-29T19:43:37Z No. of bitstreams: 2 Tese - Kaye Oliveira da Silva - 2015.pdf: 3763230 bytes, checksum: 2a51ab65a386fdff2c014712b4f5a7fd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-03T15:21:01Z (GMT) No. of bitstreams: 2 Tese - Kaye Oliveira da Silva - 2015.pdf: 3763230 bytes, checksum: 2a51ab65a386fdff2c014712b4f5a7fd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-07-03T15:21:01Z (GMT). No. of bitstreams: 2 Tese - Kaye Oliveira da Silva - 2015.pdf: 3763230 bytes, checksum: 2a51ab65a386fdff2c014712b4f5a7fd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-07-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study two problems in partial differential equations. The first one is a nonlinear eigenvalue problem given by: ( 􀀀div( (jruj)ru) = f(x; u) em , u = 0 em @ , where the nonlinearity f is oscilatory. By using Orlicz-Sobolev spaces and techniques of minimization, degree theory, lower and upper solutions and regularization of solutions, we show that for each sufficiently big, there is a family of solutions, which is finite when f oscillates a finite number of times (with respect to the second variable) and it is infinite when f oscillates infinitely many times. On the second problem, we use the shooting method, to show that the problem: ( 􀀀(r (ju0(r)j)u0(r))0 = r f(u(r)); 0 < r < R; u(R) = u0(0) = 0; has for each sufficiently small, a family fukg1k =1 of solutions, where for each positive integer k, uk has exactly k roots in the interval (0;R). / Neste trabalho estudamos dois problemas de equações diferenciais parciais. O primeiro é um problema não linear de autovalores da forma: ( 􀀀div( (jruj)ru) = f(x; u) em , u = 0 em @ , cuja não linearidade f é oscilatória. Utilizando os espaços de Orlicz-Sobolev e técnicas de minimização, teoria do grau, sub e super soluções e regularização de soluções, mostramos que para cada suficientemente grande, existe uma família de soluções, que é finita no caso de f oscilar um número finito de vezes (com relação a segunda variável) e infinita no caso de f oscilar um número infinito de vezes. No segundo problema, usamos o método de shooting, para mostrar que o problema ( 􀀀(r (ju0(r)j)u0(r))0 = r f(u(r)); 0 < r < R; u(R) = u0(0) = 0; possui para cada > 0 suficientemente pequeno, uma família fukg1k =1 de soluções, onde para cada k inteiro positivo, uk tem exatamente k raízes no intervalo (0;R).

Neste trabalho, apresentamos condições suficientes para a existência e a unicidade de soluções periódicas para equações diferenciais funcionais com retardo e impulsos. Os resultados sobre existência estão ancorados num Teorema de Continuação de Jean Mawhin. Por outro lado, as condições que garantem a unicidade de soluções periódicas são condições do tipo Lipschitz / In this work, we present sufficient conditions for the existence and the uniqueness of periodic solutions for retarded functional differential equations with impulses. The results on the existence of periodic solutions are anchored by a Jean Mawhin continuation theorem. Moreover, the conditions that guarantee the uniqueness of the periodic solutions are Lipschitz type