on the form of the functional $F$ , except that for each $t$ in some interval $[t_{0},$. $Â¥infty)$ and $y$ in a suitably restricted family of $n-1$ times differentiable ...
Funkcialaj Ekvacioj, 31 (1988) 161-178
Efficient Application of the Schauder-Tychonoff Theorem to Functional Perturbations of $x^{(n)}=0$ By
William F. TRENCH (Trinity University, U. S. A.)
1. Introduction We consider the scalar functional equation
(1.1)
$y^{(n)}=F(t;y)$
$(n¥geq 2)$
as a perturbation of $x^{(n)}=0$ . Our main theorem requires no specific assumptions on the form of the functional , except that for each in some interval and in a suitably restricted family of $n-1$ times differentiable functions, $F(t;y)$ at one or more (possibly infinitely , is determined by the values of it is sufficiently general Consequently, . , interval many) points in some [ $F$
$[t_{0},$
$t$
$¥infty)$
$y$
$ y,¥ldots$
$a$
$y^{(n-1)}$
$¥infty)$
to be applicable to ordinary differential equations, equations with deviating arguments, and integro-differential equations. Our objective is to use the Schauder-Tychonoff theorem to establish sufficient such that on some interval [ , conditions for (1.1) to have a solution $t_{0}$
$¥mathcal{Y}¥mathrm{o}$
(1.2) where
$y_{0}^{(,)}(t)=p^{(r)}(t)+o(t^{m-r})$
$p$
is a given polynomial and
(1.2)
$m$
,
$¥infty)$
$0¥leq r¥leq n-1$ ,
is an integer, with
$0¥leq m¥leq¥deg p¥leq n-1$ .
Neither this problem nor the application of the Schauder-Tychonoff theorem to it is new; nevertheless, we believe that our method of applying the theorem has some advantages over the standard approach, in which the integrability conditions imposed on virtually always imply that there are constants and $M$ and a positive function $W$ such that $F$
