# 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

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topIreneusz 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 -

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