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

Ireneusz Kubiaczyk; Aneta Sikorska-Nowak

doi:10.7151/dmdico.1108

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

Ireneusz Kubiaczyk, Aneta Sikorska-Nowak

Source

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2009 > 29 > 1 > 113-126

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.

Identifiers

journal ISSN :	1509-9407
DOI	10.7151/dmdico.1108

Authors

Ireneusz Kubiaczyk

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

Aneta Sikorska-Nowak

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

Keywords

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

Additional information

Publication languages: English

Data set: DML

Publisher

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra

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Bibliography

[1] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Result Math. 35 (1999), 3-22.
[2] R.P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4) (2001), 535-557.
[3] R.P. Agarwal and D. O'Regan, Nonlinear boundary value problems on time scales, Nonlin. Anal. TMA 44 (2001), 527-535.

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Ireneusz Kubiaczyk

Aneta Sikorska-Nowak

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Bibliography

[1] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Result Math. 35 (1999), 3-22.

[2] R.P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4) (2001), 535-557.

[3] R.P. Agarwal and D. O'Regan, Nonlinear boundary value problems on time scales, Nonlin. Anal. TMA 44 (2001), 527-535.

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