In this work, using the basic tools of functional analysis, we obtain a technique that allows us to obtain important results, related to quadratic equations in two variables that represent a natural number and differential equations. We show the possible ways to write an even number that ends in six, as the sum of two odd numbers and we establish conditions for said odd numbers to be prime, also making use of a suitable linear functional F : R 3 → R we obtain representations of natural numbers of the form (10A + 9), A ∈ N in order to obtain positive integer solutions of the equation quadratic (10x + 9)(10y + 9) = P where P is a natural number given that it ends with one. And finally, we show with three examples the use of the proposed technique to solve some ordinary and partial linear differential equations.We believe that the third corollary of our first result of this investigation can help to demonstrate the strong Goldbach conjecture.

In this work, we give one generalization of the concept and the linear teoria of aplications pfatoráveis for the multilinear case. We supply two defiuitions; based in definition 2.2 we arrive to get some results. Following the ideas of the Pietsch, and based in definition 3.9 it foresaw generalization of some definitions and theorems of the linear ideais for the multilinear case we try to prove the equivalence of the two definitions.iii Resumo Neste trabalho, damos uma generalização do conceito e da teoría das aplicações lineares p-fatoráveis para o caso multilinear. Fornecemos duas definições; baseadas na definição 2.2 chegamos a obter alguns resultados. Seguindo a ideas do Pietsch, e baseada na definição 3. 9 previa generalização de algumas definições e teoremas dos ideais lineares para o caso multilinear tentamos provar a equivalência das duas definições. i v Agradecimentos Á Deus por darme a força de não ter perdido as esperanzas de terminar este doutoramento. Ao profesor Mário C. Matos, pela orientação sobre todos os aspectos deste trabalho, por sua paciência e vontade com que me orientou.Aos profesores e funcionários do IMECC.Aos meus colegas Miguel Y glesias, Merlyng Zavaleta e Alberto Santana pela ajuda oferecida para que este trabalho chegue a seu termino.A minha família, pelo apoio em todos os momentos mais difícies.Á Jey, pela paciência, carinho e alegría com que me brindou nestes anos.Á CAPES pelo apoio financiero.

In this article, given a number P ∈ N that ends in one and assuming that there are integer solutions (A; B) ∈ N × N for the equations P = (10x + 9)(10y + 9) or P = (10x + 7)(10y + 3) or P = (10x + 1)(10y + 1), the straight line was used passing through the center of gravity of the triangle bounded by the vertices (A; A), (B; A), (A; B). Considering A ≥ 25, we manage to divide the domain of the curve P = (10x + 9)(10y + 9) into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation P = (10x + 9)(10y + 9) is found. Similar process is done when P = (10x + 1)(10y + 1), in case P is of the form P = (10x + 7)(10y + 3) or P = (10x + 3)(10y + 7). These curves are different and to obtain a process similar to the one carried out previously, we proceeded according to Observation 2.2. Our results allow minimizing the number of operations to perform when our problem requires to be implemented computationally.Furthermore, we obtain some conditions to find the solution of the equations:

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