Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals

A. Sikorska-Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

  • Volume: 27, Issue: 2, page 315-327
  • ISSN: 1509-9407

Abstract

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We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.

How to cite

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A. Sikorska-Nowak. "Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.2 (2007): 315-327. <http://eudml.org/doc/271205>.

@article{A2007,
abstract = {We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.},
author = {A. Sikorska-Nowak},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {pseudo-solution; Pettis integral; Henstock-Kurzweil integral; Henstock-Kurzweil-Pettis integral; Cauchy problem},
language = {eng},
number = {2},
pages = {315-327},
title = {Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals},
url = {http://eudml.org/doc/271205},
volume = {27},
year = {2007},
}

TY - JOUR
AU - A. Sikorska-Nowak
TI - Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2007
VL - 27
IS - 2
SP - 315
EP - 327
AB - We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.
LA - eng
KW - pseudo-solution; Pettis integral; Henstock-Kurzweil integral; Henstock-Kurzweil-Pettis integral; Cauchy problem
UR - http://eudml.org/doc/271205
ER -

References

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  1. [1] Z. Artstein, Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations 23 (1977), 224-243. Zbl0353.34044
  2. [2] J.M. Ball, Weak continuity properties of mappings and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280. 
  3. [3] J. Banaś, Demicontinuity and weak sequential continuity of operators in the Lebesgue space, Proceedings of the 1th Polish Symposium on Nonlinear Analysis, Łódź (1997), 124-129. 
  4. [4] 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. Zbl0441.47056
  5. [5] F.S. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259-262. 
  6. [6] S.S. Cao, The Henstock integral for Banach valued functions, SEA Bull. Math. 16 (1992), 36-40. 
  7. [7] T.S. Chew, On Kurzweil generalized ordinary differential equations, J. Differential Equations 76 (1988), 286-293. Zbl0666.34041
  8. [8] T.S Chew and F. Flordeliza, On x' = f(t,x) and Henstock-Kurzweil integrals, Differential and Integral Equations 4 (1991), 861-868. Zbl0733.34004
  9. [9] T.S. Chew, W. van Brunt and G.C. Wake, On retarded functional differential equations and Henstock-Kurzweil integrals, Differential and Integral Equations 9 (1996), 569-580. Zbl0873.34054
  10. [10] T.S. Chew and T.L. Toh, On functional differential equation with unbounded delay and Henstock-Kurzweil integrals, New Zeland Journal of Mathematics 28 (1999), 111-123. Zbl0961.34053
  11. [11] M. Cichoń, Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungarica 92 (2001), 75-82. Zbl1001.26003
  12. [12] M. Cichoń, Weak solutions of differential equations in Banach spaces, Disc. Math. Differ. Incl. 15 (1995), 5-14. 
  13. [13] M. Cichoń, I. Kubiaczyk and A. Sikorska, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czech. Math. J. 54 (129) (2004), 279-289. Zbl1080.34550
  14. [14] M.C. Deflour and S.K. Mitter, Hereditary differential systems with constant delays, I General case, J. Differential Equations 9 (1972), 213-235. Zbl0242.34055
  15. [15] R.F.Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1991), 81-86. Zbl0506.28007
  16. [16] R.A. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math. 21 (1991), 923-949. Zbl0764.28008
  17. [17] R.A. Gordon, The Denjoy extension of the Bochner, Pettis and Dunford integrals, Studia Math. 92 (1989), 73-91. Zbl0681.28006
  18. [18] R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math. Soc., Providence, R.I. 1994. Zbl0807.26004
  19. [19] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. Math. 34 (1990), 557-567. Zbl0685.28003
  20. [20] J. Hale, Functional Differential Equations, Springer-Verlag, 1971. Zbl0222.34003
  21. [21] R. Henstock, The General Theory of Integration, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1991. Zbl0745.26006
  22. [22] W.J. Knight, Solutions of differential equations in Banach spaces, Duke Math. J. 41 (1974), 437-442 Zbl0288.34063
  23. [23] I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mappings, Disc. Math. Differ. Incl. 15 (1995), 15-20. Zbl0832.47046
  24. [24] I. Kubiaczyk, A. Sikorska, Differential equations in Banach spaces and Henstock-Kurzweil integrals, Disc. Math. Differ. Incl. 19 (1999), 35-43. Zbl0962.34043
  25. [25] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957), 642-659. Zbl0642.26004
  26. [26] A.R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear Equations in Abstract Spaces, (V. Lakshmikantham, ed.), 1978, 378-404. 
  27. [27] P.Y. Lee, Lanzhou Lectures on Henstock Integration, Ser. Real Anal. 2, World Sci., Singapore, 1989. 
  28. [28] B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. Zbl0019.41603
  29. [29] A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals, Demonstratio Math. 35 (2002), 49-60. Zbl1011.34066

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