Generic properties of functional equations

Abstract

A PROPERTY is said to be generic on a Baire space if the subset on which it is true is residual, that is its complement is a set of Baire first category. The study of generic properties of differential equations was started by W. Orlicz [l] who showed that the subset allf for which the Cauchy problem for the equation y’ = f(t, y) has not uniqueness is of first category in the space of all continuous and boundedf with values in R”, equipped with a natural metric. An analogous result for hyperbolic equations was proved by A. Alexiewicz and W. Orlicz [2]. More recently A. Lasota and J. Yorke [3] studied properties concerning existence, uniqueness and continuous dependence of solutions of the Cauchy problem for ordinary differential equations in a Banach space. Similar problems for functional differential equations y’ = f(t, y,) in a Banach space have been discussed in [4]. In [5] ([6]) it has been shown that the convergence of successive approximations for ordinary differential equations in a Banach space (for a class of hyperbolic partial differential equations) is a generic property in the corresponding spaces of continuous functions. (For the R” case see [7].) Generic properties of fured points for nonlinear nonexpansive mappings in a Banach space have been investigated in [8,9]. Other problems in the same spirit of the aforementioned ones have been treated in [l&12]. In this note we shall present some contributions to the theory of generic properties for functional equations and functional differential equations. Section 1 contains results concerning generic properties of existence uniqueness, continuous dependence of solutions and convergence of successive approximations for abstract functional equations. In Sections 2 and 3 we establish, by using similar arguments, generic properties of existence, uniqueness, continuous dependence of solutions and convergence of successive approximations for a class of functional integral equations. In Section 4 the results of Section 1 are used to obtain generic properties of nonexpansive mappings in a Banach space. The question how those maps, for which the generic properties under consideration fail to be true, are scattered is discussed in Section 5.