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

Aneta Sikorska-Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)

  • Volume: 31, Issue: 1, page 71-90
  • ISSN: 1509-9407

Abstract

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In this paper we prove existence theorems for integro - differential equations x Δ ( t ) = f ( t , x ( t ) , ∫ ₀ t k ( t , s , x ( s ) ) Δ s ) , 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.

How to cite

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Aneta Sikorska-Nowak. "Integro-differential equations on time scales with Henstock-Kurzweil delta integrals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.1 (2011): 71-90. <http://eudml.org/doc/271148>.

@article{AnetaSikorska2011,
abstract = {In this paper we prove existence theorems for integro - differential equations $x^Δ (t) = f(t,x(t),∫₀^t k(t,s,x(s))Δs)$, 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.},
author = {Aneta Sikorska-Nowak},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {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; time scale; Henstock-Lebesgue delta integral},
language = {eng},
number = {1},
pages = {71-90},
title = {Integro-differential equations on time scales with Henstock-Kurzweil delta integrals},
url = {http://eudml.org/doc/271148},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Aneta Sikorska-Nowak
TI - Integro-differential equations on time scales with Henstock-Kurzweil delta integrals
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2011
VL - 31
IS - 1
SP - 71
EP - 90
AB - In this paper we prove existence theorems for integro - differential equations $x^Δ (t) = f(t,x(t),∫₀^t k(t,s,x(s))Δs)$, 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.
LA - eng
KW - 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; time scale; Henstock-Lebesgue delta integral
UR - http://eudml.org/doc/271148
ER -

References

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  1. [1] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Result Math. 35 (1999), 3-22. Zbl0927.39003
  2. [2] R.P. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales, a survey, Math. Inequal. Appl. 4 (4) (2001), 535-557. Zbl1021.34005
  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. Zbl1086.34014
  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. Zbl0978.39001
  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 Zbl1025.34001
  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 Zbl1092.39017
  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. Zbl1205.26050
  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 Zbl1198.34192
  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. Zbl0938.34027
  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 Zbl1042.45504
  17. [17] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996. Zbl0866.45004
  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 Zbl1039.26007
  21. [21] R. Henstock, The General Theory of Integration, Oxford Math. Monographs, Clarendon Press, Oxford, 1991. Zbl0745.26006
  22. [22] S. Hilger, Ein Maβkettenkalkül mit Anvendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität at Würzburg, Germany, 1988. Zbl0695.34001
  23. [23] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56. Zbl0722.39001
  24. [24] B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Akademic Publishers, Dordrecht, 1996. Zbl0869.34039
  25. [25] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2001. Zbl0986.05001
  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 Zbl1031.39005
  27. [27] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614. Zbl0607.34055
  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 Zbl1198.34197
  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 Zbl1014.45007
  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. Zbl0858.34068
  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. Zbl0943.45004
  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 Zbl1110.26011
  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. Zbl0781.34003
  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. Zbl0218.46016
  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 Zbl1151.34005
  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. Zbl1065.45005
  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 Zbl1130.45009
  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 Zbl0888.45007
  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 Zbl0871.39003

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