Existence of solutions of the dynamic Cauchy problem in Banach spaces

• Autorzy: Cichoń M. , Kubiaczyk I. , Sikorska-Nowak A. , Yantir A.
• Języki publikacji: EN

Abstrakty

EN

In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2012
• Tom: Vol. 45, nr 3
• Strony: 561--573
• Opis fizyczny: Bibliogr. 36 poz.

Twórcy

autor

Cichoń M.

autor

Kubiaczyk I.

autor

Sikorska-Nowak A.

autor

Yantir A.

• Faculty Of Mathematics And Computer Science Adam Mickiewicz University Ul. Umultowska 87 61-614 Poznań, Poland,

Bibliografia

• [1] R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), 3–22.
• [2] R. P. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (2001), 535–557.
• [3] R. P. Agarwal, D. O’Regan, Nonlinear boundary value problems on time scales, Nonlinear Anal. 44 (2001), 527–535.
• [4] R. P. Agarwal, D. O’Regan, Difference equations in Banach spaces, J. Austral. Math. Soc. ser. A 64 (1998), 277–284.
• [5] R. P. Agarwal, D. O’Regan, A fixed point approach for nonlinear discrete boundary value problems, Comput. Math. Appl. 36 (1998), 115–121.
• [6] R. P. Agarwal, D. O’Regan, Existence principle for continuous and discrete equations on infinite intervals in Banach spaces, Math. Nachr. 207 (1999), 5–19.
• [7] R. P. Agarwal, D. O’Regan, S. H. Saker, Properties of bounded solutions of nonlinear dynamic equations on time scales, Canad. Appl. Math. Quart. 14 (2006), 1–10.
• [8] A. Ambrosetti, Un teorema di esistenza por le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349–361.
• [9] D. R. Anderson, L. M. Moats, q-dominant and q-recessive matrix solutions for linear quantum systems, Electron J. Qual. Theory Differ. Equ. 11 (2007), 1–29.
• [10] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990.
• [11] B. Aulbach, L. Neidhard, Integration on Measure Chains, in: Aulbach, B., et al., editor, Proceedings of the 6th International Congress on Difference Equations and Applications (Augsburg, Germany, 2001), 239–252. Chapman & Hall/CRC, Boca Raton, 2004.
• [12] Jia Baoguoa, L. Erbe, A. Peterson, Oscillation of a family of q-difference equations, Appl. Math. Lett. 22 (2008), 871–875.
• [13] J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Dekker, New York-Basel, 1980.
• [14] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, 2001.
• [15] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
• [16] A. Cabada, D. R. Vivero, Criterions for absolute continuity on time scales, J. Difference Equ. Appl. 11 (2005), 1013–1028.
• [17] A. Cellina, On existence of solutions of ordinary differential equations in Banach spaces, Func. Ekvac. 14 (1971), 129–136.
• [18] M. Cichoń, On solutions of differential equations in Banach spaces, Nonlinear Anal. 60 (2005), 651–667.
• [19] M. Cichoń, A note on Peano’s theorem on time scales, (to appear).
• [20] M. Cichoń, I. Kubiaczyk, A. Sikorska-Nowak and A. Yantir, Weak solutions for the dynamic Cauchy problem in Banach spaces, Nonlinear Anal. 71 (2009), 2936–2943.
• [21] M. Dawidowski, I. Kubiaczyk, J. Morchało, A discrete boundary value problem in Banach spaces, Glasnik Mat. 36 (2001), 233–239.
• [22] K. Deimling, Ordinary Differential Equations in Banach Spaces, LNM 596, Springer, Berlin, 1977.
• [23] R. Dragoni, J. W. Macki, P. Nistri, P. Zecca, Solution Sets of Differential Equations in Abstract Spaces, Longmann, 1996.
• [24] L. Erbe, A. Peterson, Green’s functions and comparison theorems for differential equations on measure chains, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 6 (1999), 121–137.
• [25] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), 107–127.
• [26] C. Gonzalez, A. Jimenez-Meloda, Set-contractive mappings and difference equations in Banach spaces, Comp. Math. Appl. 45 (2003), 1235–1243.
• [27] S. Hilger, Ein Masskettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988.
• [28] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.
• [29] V. Kac, P. Chueng, Quantum Calculus, Springer, Berlin, 2002.
• [30] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Akademic Publishers, Dordrecht, 1996.
• [31] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607–614.
• [32] I. Kubiaczyk, P. Majcher, On some continuous and discrete equations in Banach spaces on unbounded intervals, Appl. Math. Comput. 136 (2003), 463–473.
• [33] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985–999.
• [34] S. Szufla, Measure of noncompactness and ordinary differential equations in Banach spaces, Bull. Acad. Polish Sci. Math. 19 (1971), 831–835.
• [35] M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York, 2000.
• [36] C. C. Tisdell, A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008), 3504–3524.