Existence Theorems: Problems of Slow Growth

Abstract

We discuss here existence theorems for the usual integrals $I\left[ x \right] = {\text{ }}\int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {{{F}_{0}}(t,x(t),x'(t))dt} {\text{ }}$ as in Section 11.1, but where F₀(t, x, x′) does not satisfy any of the growth conditions we have considered in Chapters 11, 12, 13. Well known problems are of this kind (cf. Section 3.12). There are a number of methods to cope with these problems; we mention here one based on their reduction to equivalent “parametric problems” (Sections 14.1–2). In Section 14.3 we state a number of existence theorems for the usual integrals I[x], and in Section 14.4 we present many examples of problems for which the existence theorems in Section 14.3 hold.