The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 30 (2010), p. 121-140 / Harvested from The Polish Digital Mathematics Library

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Résumé

We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

Publié le : 2010-01-01

Zbl 1201.35082

EUDML-ID : urn:eudml:doc:271181

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Adrian Karpowicz. The existence of Carathéodory solutions of hyperbolic functional differential equations. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 30 (2010) pp. 121-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1115/

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