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## Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2009 | 29 | 1 | 113-126
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
113-126
Opis fizyczny
Daty
wydano
2009
otrzymano
2009-08-03
Twórcy
autor
• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
autor
• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
Bibliografia
• [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.
• [4] R.P. Agarwal and D. O'Regan, Difference equations in Banach spaces, J. Austral. Math. Soc. (A) 64 (1998), 277-284.
• [5] R.P. Agarwal and D. O'Regan, A fixed point approach for nonlinear discrete boundary value problems, Comp. Math. Appl. 36 (1998), 115-121.
• [6] R.P. Agarwal, D. O'Regan and S.H. Saker, Properties of bounded solutions of nonlinear dynamic equations on time scales, Can. Appl. Math. Q. 14 (1) (2006), 1-10.
• [7] E. Akin-Bohner, M. Bohner and F. Akin, Pachpate inequalities on time scale, J. Inequal. Pure and Appl. Math. 6 (1) Art. 6, (2005).
• [8] A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Univ. Padova 39 (1967), 349-361.
• [9] G. Aslim and G.Sh. Guseinov, Weak semiring, ω-semirings, and measures, Bull. Allahabad Math. Soc. 14 (1999), 1-20.
• [10] B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990.
• [11] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Dekker, New York and Basel, 1980.
• [12] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkäuser, 2001.
• [13] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkäuser, Boston, 2003.
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• [15] M. Cichoń, On solutions of differential equations in Banach spaces, Nonlin. Anal. TMA 60 (2005), 651-667.
• [16] M. Dawidowski, I. Kubiaczyk and J. Morchało, A discrete boundary value problem in Banach spaces, Glasnik Mat. 36 (2001), 233-239.
• [17] R. Dragoni, J.W. Macki, P. Nistri and P. Zecca, Solution Sets of Differential Equations in Abstract Spaces, Longmann, 1996.
• [18] L. Erbe and A. Peterson, Green's functions and comparison theorems for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems 6 (1) (1999), 121-137.
• [19] G.Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), 107-127.
• [20] C. Gonzalez and A. Jimenez-Meloda, Set-contractive mappings and difference equations in Banach spaces, Comp. Math. Appl. 45 (2003), 1235-1243.
• [21] S. Hilger, Ein Maßkettenkalkül mit Anvendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität at Würzburg, 1988.
• [22] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56.
• [23] B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Akademic Publishers, Dordrecht, 1996.
• [24] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614.
• [25] B.N. Sadovskii, Limit-compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144.
• [26] S. Szufla, Measure of noncompanctness and ordinary differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 19 (1971), 831-835.
Typ dokumentu
Bibliografia
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