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Title

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

Authors 1, 1

Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland

Abstract

In the paper, we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem xΔ(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.

Keywords

ENG
Cauchy dynamic problem, Banach space, measure of noncompactness, Carathéodory's type solutions, time scales, fixed point

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Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2009/29/1
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Pages:
113-126
Main language of publication
English
Received
2009-08-03
Published
2009

DOI: 10.7151/dmdico.1108

Exact and natural sciences