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# Existence Theorems: Weak Convergence and Growth Conditions

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

## Abstract

Let A be a subset of the tx-space Rn+1 and let A(t) denote its sections, that is, A(t) = [xR n | (t, x) ∈ A]. For every (t, x) ∈ A let Q(t, x) be a given subset of the z-space R n , x = (x1,…x n ), z = (z1,…,z n ). Let Mo be the set Mo = [(t,x, z) | (t, x) ∈ A, zQ(t, x)] ⊂ R1+2n and let F o (t, x, z) be a given real valued function defined on Mo. Let B be a given subset of the t1x1t2x2-space R2n+2, and let g(t 1 , x 1 , t 2 , x2) be a real valued function defined on B. Let Ω = {x} denote a nonempty collection of AC functions x(t) = (x1,…, x n ), t1t ≤ t2, such that
$$x\left( t \right) \in A\left( t \right),{\text{ }}x'\left( t \right) \in Q\left( {t,x\left( t \right)} \right)for{\text{ }}t \in \left[ {{{t}_{1}},{{t}_{2}}} \right]\left( {a.e.} \right),e\left[ x \right] = \left( {{{t}_{1}},x\left( {{{t}_{1}}} \right),{{t}_{2}},x\left( {{{t}_{2}}} \right)} \right) \in B,{\text{ }}{{F}_{0}}(\cdot ,x\left( \cdot \right),x'\left( \cdot \right)) \in {{L}_{1}}\left[ {{{t}_{1}},{{t}_{2}}} \right]$$
(11.1.1)

## Keywords

Weak Convergence Existence Theorem Positive Semidefinite Absolute Minimum Extended Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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