Existence Theorems: Weak Convergence and Growth Conditions

Abstract

Let A be a subset of the tx-space Rⁿ⁺¹ and let A(t) denote its sections, that is, A(t) = [x ∈ Rⁿ| (t, x) ∈ A]. For every (t, x) ∈ A let Q(t, x) be a given subset of the z-space Rⁿ, x = (x¹,…xⁿ), z = (z¹,…,zⁿ). Let M_o be the set M_o = [(t,x, z) | (t, x) ∈ A, z ∈ Q(t, x)] ⊂ R¹⁺²ⁿ and let F _o (t, x, z) be a given real valued function defined on M_o. Let B be a given subset of the t₁ x₁ t₂ x₂-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,…, xⁿ), t₁ ≤ t ≤ 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)