Download PDF - Integro-differential equations on time scales with Henstock-Kurzweil delta integralsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Integro-differential equations on time scales with Henstock-Kurzweil delta integrals

Authors 1

Affiliations

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

Abstract

In this paper we prove existence theorems for integro - differential equations xΔ(t)=f(t,x(t),∫₀tk(t,s,x(s))Δs), t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale.

Keywords

ENG
integro-differential equations, nonlinear Volterra integral equation, time scales, Henstock-Kurzweil delta integral, HL delta integral, Banach space, fixed point, measure of noncompactness, Carathéodory solutions

Bibliography

  1. [1] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Result Math. 35 (1999), 3-22.
  2. [2] R.P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales, a survey, Math. Inequal. Appl. 4 (4) (2001), 535-557.
  3. [3] R.P. Agarwal and D. O'Regan, Difference equations in Banach spaces, J. Austral. Math. Soc. (A) 64 (1998), 277-284. doi: 10.1017/S1446788700001762
  4. [4] E. Akin-Bohner, M. Bohner and F. Akin, Pachpate inequalities on time scale, J. Inequal. Pure and Appl. Math. 6 (1) Art. 6, 2005.
  5. [5] A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Univ. Padova 39 (1967), 349-361.
  6. [6] B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990.
  7. [7] 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.
  8. [8] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkäuser, 2001.
  9. [9] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkäuser, Boston, 2003. doi: 10.1007/978-0-8176-8230-9
  10. [10] A. Cabada and D.R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; applications of the calculus of Δ-antiderivatives, Math. Comp. Modell. 43 (2006) 194-207. doi: 10.1016/j.mcm.2005.09.028
  11. [11] S.S. Cao, The Henstock integral for Banach valued functions, SEA Bull. Math. 16 (1992), 36-40.
  12. [12] M. Cichoń, On integrals of vector-valued functions on time scales, Commun. Math. Anal. 11 (1) (2011), 94-110.
  13. [13] M. Cichoń, I. Kubiaczyk, A. Sikorska-Nowak and A. Yantir, Weak solutions for the dynamic Cauchy problem in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 71 (2009), 2936-2943. doi: 10.1016/j.na.2009.01.175
  14. [14] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991, MR 92h:45001.
  15. [15] 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.
  16. [16] D. Franco, Green's functions and comparison results for impulsive integro-differential equations, Nonlin. Anal. Th. Meth. Appl. 47 (2001), 5723-5728. doi: 10.1016/S0362-546X(01)00674-5
  17. [17] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996.
  18. [18] D. Guo, Initial value problems for second-order integro-differential equations in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 37 (1999), 289-300. doi: 10.1016/S0362-546X(98)00047-9
  19. [19] D. Guo and X. Liu, Extremal solutions of nonlinear impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 177 (1993), 538-553. doi: 10.1006/jmaa.1993.1276
  20. [20] G.Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5
  21. [21] R. Henstock, The General Theory of Integration, Oxford Math. Monographs, Clarendon Press, Oxford, 1991.
  22. [22] S. Hilger, Ein Maβkettenkalkül mit Anvendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität at Würzburg, Germany, 1988.
  23. [23] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56.
  24. [24] B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Akademic Publishers, Dordrecht, 1996.
  25. [25] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2001.
  26. [26] V.B. Kolmanovskii, E. Castellanos-Velasco and J.A. Torres-Munoz, A survey: stability and boundedness of Volterra difference equations, Nonlin. Anal. Th. Meth. Appl. 53 (2003), 669-681. doi: 10.1016/S0362-546X(02)00272-9
  27. [27] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614.
  28. [27a] I. Kubiaczyk and A. Sikorska-Nowak, Existence of solutions of the dynamic Cauchy problem of infinite time scale intervals, Discuss. Math. Differ. Incl. 29 (2009), 113-126. doi: 10.7151/dmdico.1108
  29. [28] V. Lakshmikantham, S. Sivasundaram and B. Kaymarkcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
  30. [29] P.Y. Lee, Lanzhou Lectures on Henstock Integration, Ser. Real Anal. 2, World Sci., Singapore, 1989.
  31. [30] L. Liu, C.Wu and F. Guo, A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces, J. Math. Anal. Appl. 275 (2002), 369-385. doi: 10.1016/S0022-247X(02)00366-9
  32. [31] X. Liu and D. Guo, Initial value problems for first order impulsive integro-differential equations in Banach spaces, Comm. Appl. Nonlinear Anal. 2 (1995), 65-83.
  33. [32] H. Lu, Extremal solutions of nonlinear first order impulsive integro-differential equations in Banach spaces, Indian J. Pure Appl. Math. 30 (1999), 1181-1197.
  34. [33] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 4 (1980), 985-999. doi: 10.1016/0362-546X(80)90010-3
  35. [34] A. Peterson and B. Thompson, Henstock-Kurzweil delta and nabla integrals, J. Math. Anal. Appl. 323 (2006), 162-178. doi: 10.1016/j.jmaa.2005.10.025
  36. [35] B.N. Sadovskii, Limit-compact and condesing operators, Russian Math. Surveys 27 (1972), 86-144. doi: 10.1070/RM1972v027n01ABEH001364
  37. [36] S. Schwabik, Generalized Ordinary Differential Equations, Word Scientic, Singapore, 1992.
  38. [37] A.P. Solodov, On condition of differentiability almost everywhere for absolutely continuous Banach-valued function, Moscow Univ. Math. Bull. 54 (1999), 29-32.
  39. [38] G. Song, Initial value problems for systems of integrodifferential equations in Banach spaces, J. Math. Anal. Appl. 264 (2001), 68-75. doi: 10.1006/jmaa.2001.7630
  40. [39] V. Spedding, Taming Nature's Numbers, New Scientist, July 19, 2003, 28-31.
  41. [40] S. Szufla, Measure of noncompactness and ordinary differential equations in Banach spaces, Bull. Acad. Poland Sci. Math. 19 (1971), 831-835.
  42. [41] C.C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlin. Anal. Th. Meth. Appl. 68 (2008), 3504-3524. doi: 10.1016/j.na.2007.03.043
  43. [42] Y. Xing, M. Han and G. Zheng, Initial value problem for first-order integro-differential equation of Volterra type on time scales, Nonlin. Anal. Th. Meth. Appl. 60 (2005), 429-442.
  44. [43] Y. Xing, W. Ding and M. Han, Periodic boundary value problems of integro-differential equation of Volterra type on time scales, Nonlin. Anal. Th. Meth. Appl. 68 (2008), 127-138. doi: 10.1016/j.na.2006.10.036
  45. [44] H. Xu and J.J. Nieto, Extremal solutions of a class of nonlinear integro-differential equations in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), 2605-2614. doi: 10.1090/S0002-9939-97-04149-X
  46. [45] S. Zhang, The unique existence of periodic solutions of linear Volterra difference equations, J. Math. Anal. Appl. 193 (1995), 419-430. doi: 10.1006/jmaa.1995.1244
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2011/31/1
Export to PBN, Opens in new tab
Pages:
71-90
Main language of publication
English
Received
2010-04-20
Published
2011

DOI: 10.7151/dmdico.1128

Exact and natural sciences