Existence Theorems: Problems of Slow Growth

  • Lamberto Cesari
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 17)

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.

Keywords

Manifold 

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Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • Lamberto Cesari
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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