Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

Ireneusz Kubiaczyk ; Aneta Sikorska-Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 29 (2009), p. 113-126 / Harvested from The Polish Digital Mathematics Library

Résumé

In the paper, we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problem $x^Δ(t) = f(t,x(t))$, t ∈ T, x(0) = x₀, where T denotes an unbounded time scale (a nonempty closed subset of R and such that there exists a sequence (xₙ) in T and xₙ → ∞) and f is continuous or satisfies Carathéodory’s conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result. The results presented in the paper are new not only for Banach valued functions, but also for real-valued functions.

Détail

Publié le : 2009-01-01

Zbl 1198.34197

EUDML-ID : urn:eudml:doc:271140

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