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 (2009)

  • Volume: 29, Issue: 1, page 113-126
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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Ireneusz Kubiaczyk, and Aneta Sikorska-Nowak. "Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 113-126. <http://eudml.org/doc/271140>.

@article{IreneuszKubiaczyk2009,
abstract = {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.},
author = {Ireneusz Kubiaczyk, Aneta Sikorska-Nowak},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Cauchy dynamic problem; Banach space; measure of noncompactness; Carathéodory's type solutions; time scales; fixed point; Carathéodory type solutions},
language = {eng},
number = {1},
pages = {113-126},
title = {Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals},
url = {http://eudml.org/doc/271140},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Ireneusz Kubiaczyk
AU - Aneta Sikorska-Nowak
TI - Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2009
VL - 29
IS - 1
SP - 113
EP - 126
AB - 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.
LA - eng
KW - Cauchy dynamic problem; Banach space; measure of noncompactness; Carathéodory's type solutions; time scales; fixed point; Carathéodory type solutions
UR - http://eudml.org/doc/271140
ER -

References

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