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1	A construção do grau topológico e sua aplicação a um sistema diferencial não linear com condições de contorno / The construction of the topological degree and its application to a nonlinear differential system with boundary conditions Peixoto, Adriano Leandro da Costa 06 May 2014 (has links) 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. Brouwer degree Degree theory Grau de Brouwer Grau de Leray-Schauder Leray-Schauder degree Teoria do grau
2	積分微分方程的數值解吳舜堂, WU, SHUN-TANG Unknown Date (has links) 本論文是以探討積分微分方程數值解的問題為主。此文中吾人皆先對問題本身做分析，討論其存在解，然後再用有限元素法，對連續性的問題做分解，使其變為一非線性的方程組。而後藉由同倫（HOMOTOPY）法來解此非線性方程組。最後吾人可得到當區間分割得愈小，真實解與數值解的誤差會愈小。也就是吾人所用之方法，為一個收斂的方法。本文共分兩部分，第一部分中，吾人討論一維的微分積分方程在有限區間的問題。於此部分中，我們分了６個章節。第一節中，給了關於此問題的簡單介紹，並給序一些必需的假設。第二節中，吾人可得到在第一節的假設下，假如原問題有真實解的話，那麼此真實解絕對值的極大值（SUPREMUM）必不大於某個大於零的常數。第三節中，吾人討論原方程的存在解，而證此存在解是經由LERAY-SCHAUDER DEGREE 定理得來的。且在更強的條件下，會有存在唯一解。更而證明假如原來問題中函數不滿足所給予的假設，那麼可經由修正（MODIFIED）原來的問題，也可得到原問題存在有解。第四節中，對原來的方程，經由變分法（VARIATIONAL ）的方法，把它變成一非線性的方程組，而在某些條件下，吾人亦可得到此方程組有解。第五節中，吾人討論此非線性方程組的數值解。並可得知，當區間分割的愈小，此數值解會更趨近實實的解。第六節中，吾人給予平滑的多項式子空間來逼近真實解，結果可得到假如每個區間以（ｋ＋１）個點的LAGRANGE多項式來做內插（INTERPOLATION ），可知其收斂速度為O(Hk (big O),h 是分割區間的最大距離。第二部分中，吾人所討論的是二維以上的積分微分方程在有界區域的問題，於此部分中討論的與第一部分中類似，探討其存在，數值解等等問題。最後吾人並給予一些例子，來加以印證我們所得到的結果。積分微分方程數值解有限元素法同倫法變分法 LAGRANGE多項式 HOMOTOPY-METHOD VARIATIONAL-METHOD LERAY-SCHAUDER-DEGREE
3	Resultados de existência para alguns problemas não lineares com valores na fronteira de equações diferenciais / Existence results for some nonlinear problems of boundary value differential equations. Santos, Dionicio Pastor Dallos 26 May 2017 (has links) O principal objetivo deste trabalho é estudar a existência de soluções para alguns problemas de valores de contorno de equações diferenciais ordinárias não lineares em dimensão finita e infinita. Todos os sistemas considerados nesta investigação são transformados em equações funcionais nas quais o objetivo é encontrar um ponto fixo de um oportuno operador definido em um espaço de funções (que depende do problema estudado). Para isso, faremos uso do grau de Leray-Schauder e de um conceito de grau topológico, devido a R. Nussbaum, para perturbações não compactas da identidade em espaços de Banach. / The main purpose of this work is to study the existence of solutions to some boundary value problems for nonlinear ordinary differential equations in finite and infinite dimension. All systems considered in this research are transformed into functional equations in which the objective is to find a fixed point of a suitable operator defined in a space of functions (which depends on the studied problem). To do this, we use the Leray-Schauder degree and a concept of topological degree due to R. Nussbaum for non-compact perturbations of identity in Banach spaces. Boundary value problem Compact operators Fixed point problems Grau de Leray-Schauder Kuratowskis measure of non-compactness Leray-Schauder degree Medida de Kuratowski de não compacidade Operadores compactos Problemas de ponto fixo Problemas de valor na fronteira
4	Resultados de existência para alguns problemas não lineares com valores na fronteira de equações diferenciais / Existence results for some nonlinear problems of boundary value differential equations. Dionicio Pastor Dallos Santos 26 May 2017 (has links) O principal objetivo deste trabalho é estudar a existência de soluções para alguns problemas de valores de contorno de equações diferenciais ordinárias não lineares em dimensão finita e infinita. Todos os sistemas considerados nesta investigação são transformados em equações funcionais nas quais o objetivo é encontrar um ponto fixo de um oportuno operador definido em um espaço de funções (que depende do problema estudado). Para isso, faremos uso do grau de Leray-Schauder e de um conceito de grau topológico, devido a R. Nussbaum, para perturbações não compactas da identidade em espaços de Banach. / The main purpose of this work is to study the existence of solutions to some boundary value problems for nonlinear ordinary differential equations in finite and infinite dimension. All systems considered in this research are transformed into functional equations in which the objective is to find a fixed point of a suitable operator defined in a space of functions (which depends on the studied problem). To do this, we use the Leray-Schauder degree and a concept of topological degree due to R. Nussbaum for non-compact perturbations of identity in Banach spaces. Grau de Leray-Schauder Medida de Kuratowski de não compacidade Operadores compactos Problemas de ponto fixo Problemas de valor na fronteira Boundary value problem Compact operators Fixed point problems Kuratowskis measure of non-compactness Leray-Schauder degree
5	The degree theory and the index of a critical point for mappings of the type (<em>S</em><sub>+</sub>) Oinas, J. (Janne) 31 May 2007 (has links) Abstract The dissertation considers a degree theory and the index of a critical point of demi-continuous, everywhere defined mappings of the monotone type. A topological degree is derived for mappings from a Banach space to its dual space. The mappings satisfy the condition (S+), and it is shown that the derived degree has the classical properties of a degree function. A formula for the calculation of the index of a critical point of a mapping A : X→X* satisfying the condition (S+) is derived without the separability of X and the boundedness of A. For the calculation of the index, we need an everywhere defined linear mapping A' : X→X* that approximates A in a certain set. As in the earlier results, A' is quasi-monotone, but our situation differs from the earlier results because A' does not have to be the Frechet or Gateaux derivative of A at the critical point. The theorem for the calculation of the index requires a construction of a compact operator T = (A' + Γ)-1Γ with the aid of linear mappings Γ : X→X and A'. In earlier results, Γ is compact, but here it need only be quasi-monotone. Two counter-examples show that certain assumptions are essential for the calculation of the index of a critical point. Brouwer degree Browder degree Leray-Schauder theory Skrypnik degree degree theory index of a critical point index of a singular point mapping degree mappings of monotone type nonlinear analysis nonlinear functional analysis rotation of vector field the Brouwer-Hopf theory the Browder theory the Leray-Schauder degree the Skrypnik theory topological degree

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