FUNCTIONAL EQUATIONS

Author: RODRÍGUEZ LÓPEZ ROSANA.
Year: 2004.
University: SANTIAGO DE COMPOSTELA [www.usc.es].
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD DEM ATEMÁTICAS. UNIVERSIDAD DE SANTIAGO DE COMPOSTELA.

Summary: The differential equations have been used to model the mechanisms of evolution of many important dynamic processes in various fields of application. However, for many real phenomena, obtaining a suitable model requires states to take into account the heavy system, resulting in the functional differential equations. They provide a mathematical model for real systems in which the rate of change may depend on the influence of his previous statements and that lead to various kinds of equations involving ordinary differential equations, differential equations with delay, with maximum functional equations, equations integro-diferenciales, etc.. The phenomenon of retardation is present in many fields such as engineering, control theory, physics, chemistry, engineering, biology, medicine, economics, or information theory. Among the salient issues in the theory and application of functional differential equations include border problems and the search for solutions periodicals. Many evolutionary processes are characterized by the fact that at certain moments undergo an abrupt change due to disruptions lasting very small and, therefore, for practical purposes, negligible duration (perturbations snapshots), resulting from a mathematical point of view, the concept of momentum and impulse equations. The theory of differential system impulse emerges as an important area, and still under development, research on the new phenomena appearing (confluence of solutions, the phenomenon momentum, loss of autonomy ..), which had no place in the corresponding theory of ordinary differential equations classic. The mathematical modeling of real phenomena we are basically faced with two challenges: the complexity of the model and the uncertainty caused by the inability to differentiate itself events accurately and precisely. It is therefore necessary to have tools or mathematical concepts to describe the uncertainty present in a multitude of real phenomena: fuzzy sets, which are very important in fields such as robotics, artificial intelligence, or social sciences. From a mathematical point of view, metric spaces of fuzzy sets provide a mathematical framework suitable for various applications of fuzzy sets. This memory is organized into four distinct part, each of which is part of a following topics: 1-Differential Equations ordinary 2-Differential Equations functional 3-Differential Equations diffuse 4-Extension of differential equations to metric spaces general. First, we apply a generalization of the theorem of Punto Fijo to Fanach to partially ordered sets, with the aim of obtaining results of existence and uniqueness of solutions for ordinary differential equations with periodic boundary conditions. With regard to the functional equations, we approached the study of the problem of periodic boundary for certain functional equations and functional equations with pulses of the first order, discussing issues such as the existence, uniqueness, location and approximate solutions. For this, we use the theory of fixed points, the method of sub and sobresolucioles and iterative techniques monotonous. In addition, we get the Green's function for the expression of the solution of border newspaper associated with a functional differential equation of second order, in the particular case in which the functional unit is continuing to pieces. In the third part, we study equations arising in certain spaces more generally, such as differential equations diffuse, for which it is more difficult to respond to problems such as exists 8 ncia of 76c solution, or the marking of such solutions. Thus, we tried issues such as the existence of solution and the calculation of its linear expression for specific problems of the first order, the existence of a solution to the problem Cauchy on fuzzy equations of higher order, determining conditions for the existence of solutions extremales to diffuse quadratic equations using fixed-point theorems, the marking of the solutions to differential equations and integral diffuse, or the existence of functional equations solution to diffuse type neutral in Banach spaces. In conclusion, we spent the last part of this memory to define and explore a possible generalization to metric spaces abstract differential equations: what we call Dynamic Systems Metric. In this area, we developed a theory similar to the existing classical equations for ordinary defining concepts such as the primitive for a function in a metric space. In addition, test results are similar to theorem Picard-Lipschitz, there are conditions for the existence of approximate solutions, discusses the continuing dependence of the solution as regards the data are provided for comparison results, develop techniques approximation as Polygonal method of Euler, .. Finally, we extend this broader context as the concepts of functional equation, with or without pulse, or the hybrid equation.