Implicit functions and differential equations in general analysis
1927; American Mathematical Society; Volume: 29; Issue: 3 Linguagem: Inglês
10.1090/s0002-9947-1927-1501403-4
ISSN1088-6850
Autores Tópico(s)Numerical methods for differential equations
ResumoThe chief purpose of this paper is to discuss some special cases of the implicit function theorems obtained by Hildebrandt and Graves in the paper entitled Implicit functions and their differentials in general analysis.tIn particular I wish to discuss a generalization of the notion of differential equation, combining ideas due to Hahn and to Carathéodory.§ The equations are of the form (1) f(r, £ g(r,r',y(r'),x)dr',y(r),x^ = 0.Here x may represent both initial values and parameters, and the functions /, g, and y are supposed to be bounded and measurable in r and r'.Imbedding and existence theorems for equations of this form are obtained in Parts VI and VII.As indicated by the paper of Hahn just referred to, such theorems find important applications in the calculus of variations.Special theorems relating to linear equations are found in Part VIII.Many other special cases of our general theory have appeared in the Uterature from other writers.|| In some of these cases it naturally occurs that the hypotheses of the writers must be strengthened in order to make their theories fit under our general theory.On the other hand, we should expect that only a limited part of any special theory will flow directly from
Referência(s)