On $N$-parameter families and interpolation problems for nonlinear ordinary differential equations

Abstract:

Let $y = ({y_0}, \ldots ,{y_{N - 1}})$. This paper is concerned with the existence of solutions of a system of ordinary differential equations $y’ = g(t,y)$ satisfying interpolation conditions $(^\ast ) {y_0}({t_j}) = {c_j}$ for $j = 1, \ldots$, N and ${t_1} < \cdots < {t_N}$. It is shown that, under suitable conditions, the assumption of uniqueness for all such problems and of “local” solvability (i.e., for ${t_1}, \ldots ,{t_N}$ on small intervals) implies the existence for arbitrary ${t_1}, \ldots ,{t_N}$ and ${c_1}, \ldots ,{c_N}$. A result of Lasota and Opial shows that, in the case of a second order equation for ${y_0}$, the assumption of uniqueness suffices, but it will remain undecided if the assumption of “local” solvability can be omitted in general. More general interpolation conditions involving N points, allowing coincidences, are also considered. Part I contains the statement of the principal results for interpolation problems and those proofs depending on the theory of differential equations. Actually, the main theorems are consequences of results in Part II dealing with “N-parameter families” and “N-parameter families with pseudoderivatives.” A useful lemma states that if F is a family of continuous functions $\{ {y^0}(t)\}$ on an open interval (a, b), then F is an N-parameter family (i.e., contains a unique solution of the interpolation conditions $(^\ast )$ for arbitrary ${t_1} < \cdots < {t_N}$ on (a, b) and ${c_1}, \ldots ,{c_N}$) if and only if (i) ${y^0},{z^0} \in F$ implies ${y^0} - {z^0} \equiv 0$ or ${y^0} - {z^0}$ has at most N zeros; (ii) the set $\Omega \equiv \{ ({t_1}, \ldots ,{t_N},{y^0}({t_1}), \ldots ,{y^0}({t_N})):a < {t_1} < \cdots < {t_N} < b$ and ${y^0} \in F\}$ is open in ${R^{2N}}$; (iii) ${y^1},{y^2}, \ldots , \in F$ and the inequalities ${y^n}(t) \leqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots$ or ${y^n}(t) \geqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots$ on an interval $[\alpha ,\beta ] \subset (a,b)$ imply that either ${y^0}(t) = \lim {y^n}(t)$ exists on (a, b) and ${y^0} \in F$ or $\lim |{y^n}(t)| = \infty$ on a dense set of (a, b); and finally, (iv) the set $S(t) = \{ {y^0}(t):{y^0} \in F\}$ is not bounded from above or below for $a < t < b$. The notion of pseudoderivatives permits generalizations to interpolation problems involving some coincident points.