Sobre algumas aplicações da análise funcional na teoria de equações integrais

Abstract

Functional Analysis studies finite and infinite spaces composed of functions, vectors, matrices, sequences and other more complicated mathematical objects Mathematical Analysis, Function Theory and Set Theory, Algebra and Geometry come together, giving a deeper and more complete idea about functional dependence. As a proper branch of Mathematics, Functional Analysis emerged in the late eighteenth and early nineteenth century. The first works on Functional Analysis belonged to the Italian mathematician Volterra1, the French mathematician Poincaré2 and the German mathematician Hilbert3 . Metric spaces were introduced by the French mathematician Fréchet4 at the beginning of the 20th century, normed spaces appeared in 1922 thanks to the Polish mathematician Banach5 and the American Wiener6. Equations that contain the unknown under an integral sign are called integral equations. They have wide application, as most problems that arise in physics, mechanics, and other technical sciences boil down to integral equations. Note that different types of differential equations boil down to integral equations. In the study of integral equations, the first question that arises is about the existence of a solution. For certain classes of functional equations, the question about the existence of a solution is intrinsically linked to the investigation of the performance of the respective operator in different functional spaces. In this work, I made, in a pedagogical and didactic way, a bibliographical review concerning some topics on the theory of operators and its application in integral equations. In a simple way, I approached some applications of Functional Analysis in the Theory of Integral Equations, having studied the theory of functionals and continuous and linear operators, the compact operators, asking questions about the existence and uniqueness of the solution of integral equations. I also discussed the method of iterative approximations and resolvent construction. I have rigorously demonstrated some theorems such as a theorem analogous to the Caccioppoli 7-Banach theorem about the existence and uniqueness of the fixed point for a contraction application, Hausdorff's theorem5 about the compacticity of a set, the theorem about compacticity in space of continuous functions and theorems analogous to the Kolmogorov9-Riesz10 theorem on compacticity in spaces of summable functions with a variable weight. As a methodology I used the Theory of Integral Equations, Functional Analysis and Mathematical Analysis.