Dynamic equations (…) on time scales

• Autorzy: Sikorska-Nowak A.
• Języki publikacji: EN

Abstrakty

EN

In this paper we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem (…), where (…) denotes a mth order (…) - derivative,T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence (…) E -a Banach space and f is a continuous function or satis .es 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. As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2011
• Tom: Vol. 44, nr 2
• Strony: 317--333
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Sikorska-Nowak A.

• Faculty of Mathematics and Computer Science Adam Mickiewicz University 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, S.H.Saker, Properties of bounded solutions of nonlinear dynamic equations on time scales, Canad. Appl. Math. Quart. 14 (2006), 1–10.
• [7] E.Akin-Bohner, M.Bohner, 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. Mat. Univ. Padova 39 (1967), 349–361.
• [9] B.Aulbach, S.Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990.
• [10] J.Banaś, K.Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Appl. Math. 60, Dekker, New York and Basel, 1980.
• [11] M.Bohner, A.Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkäuser, 2001.
• [12] M.Bohner, A.Peterson, Advances in Dynamic Equations on Time Scales, Birkäuser, Boston, 2003.
• [13] A.Cabada, D.R.Vivero, Exression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; Applications of the calculus of Δ-aniderivatives, Math. Comput. Modelling 43 (2006), 194–207.
• [14] M.Cichoń, I.Kubiaczyk, A.Sikorska-Nowak, A.Yantir, Weak solutions for the dynamic Cauchy problem in Banach spaces, Nonlinear Anal. 71 (2009), 2936–2943.
• [15] L.Erbe, A.Peterson, Green’s functions and comparison theorems for differential equations on measure chains, Dyn. Contin. Discrete Impuls. Syst. 6 (1999), 121–137.
• [16] G.Sh.Guseinov, Integration on time scales, J.Math. Anal. Appl. 285 (2003), 107–127.
• [17] C.Gonzalez, A.Jimenez–Meloda, Set-contractive mappings and difference equations in Banach spaces, Comp. Math. Appl. 45 (2003), 1235–1243.
• [18] S.Hilger, Ein Maßkettenkalkül mit Anvendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988.
• [19] S.Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.
• [20] V.Kac, P.Cheung, Quantum Calculus, Springer, New York, 2001.
• [21] B.Kaymakcalan, V.Lakshmikantham, S.Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Akademic Publishers, Dordrecht, 1996.
• [22] I.Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607–614.
• [23] I.Kubiaczyk, A.Sikorska–Nowak, The set of pseudo-solutions of the differential equation x(m)=f(t,x) in Banach spaces, Publ. Math. Debrecen 68 (2006), 297–308.
• [24] B.N.Sadovskii, Limit-compact and condesing operators, Russian Math. Surveys 27 (1972), 86–144.
• [25] A.Sikorska-Nowak, The existence theory for the differential equation x(m)=f(t,x) in Banach spaces and Henstock–Kurzweil integral, Demonstratio Math. 40 (2007), 115–124.
• [26] V.Spedding, Taming Nature’s Numbers, New Scientist, July 19, 2003, 28–31.
• [27] S.Szufla, On the differential equation x(m)=f(t,x) in Banach spaces, Funkcial. Ekvac. 41 (1998), 101–105.
• [28] S.Szufla, Measure of noncompanctness and ordinary differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 19 (1971), 831–835.
• [29] S.Szufla, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 30 (1982), 507–515.
• [30] A.Szukala, On the application of some measures of noncompactness to existence theorem for an m-th order differential equation, Demonstratio Math. 34 (2001), 819–824.