Weak solutions of differential equations in Banach spaces

Mieczysław Cichoń

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1995)

  • Volume: 15, Issue: 1, page 5-14
  • ISSN: 1509-9407

Abstract

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In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.

How to cite

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Mieczysław Cichoń. "Weak solutions of differential equations in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 15.1 (1995): 5-14. <http://eudml.org/doc/275857>.

@article{MieczysławCichoń1995,
abstract = {In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.},
author = {Mieczysław Cichoń},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {pseudo-solutions; Cauchy problem; measures of noncompactness; initial value problem; Banach space; generalized solution},
language = {eng},
number = {1},
pages = {5-14},
title = {Weak solutions of differential equations in Banach spaces},
url = {http://eudml.org/doc/275857},
volume = {15},
year = {1995},
}

TY - JOUR
AU - Mieczysław Cichoń
TI - Weak solutions of differential equations in Banach spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1995
VL - 15
IS - 1
SP - 5
EP - 14
AB - In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.
LA - eng
KW - pseudo-solutions; Cauchy problem; measures of noncompactness; initial value problem; Banach space; generalized solution
UR - http://eudml.org/doc/275857
ER -

References

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  16. [16] I. Kubiaczyk, On the existence of solutions of differential equations in Banach spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 607-614. Zbl0607.34055
  17. [17] I. Kubiaczyk, S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Mat. Beograd 32 (1982), 99-103. Zbl0516.34058
  18. [18] A. R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, pp. 387-404 in: Nonlinear Equations in Abstract Spaces, ed. by V. Lakshmikantham 1978. 
  19. [19] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. Zbl0019.41603
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  21. [21] M. Talagrand, Pettis integral and measure theory, Memoires Amer. Math. Soc. 307 Vol. 51, Amer. Math. Soc., Providence, Rhode-Island 1984. 

Citations in EuDML Documents

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  1. Donal O'Regan, Fixed point theorems for weakly sequentially closed maps
  2. Mieczysław Cichoń, Ireneusz Kubiaczyk, Existence theorem for the Hammerstein integral equation
  3. Piotr Majcher, Magdalena Roszak, On the semilinear integro-differential nonlocal Cauchy problem
  4. Mieczysław Cichoń, Ireneusz Kubiaczyk, Kneser-type theorem for the Darboux problem in Banach spaces
  5. Mieczysław Cichoń, Ireneusz Kubiaczyk, Sikorska-Nowak, Aneta Sikorska-Nowak, Aneta, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem
  6. A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals
  7. Kinga Cichoń, Mieczysław Cichoń, Bianca Satco, Differential inclusions and multivalued integrals
  8. Afif Ben Amar, Some fixed point theorems and existence of weak solutions of Volterra integral equation under Henstock-Kurzweil-Pettis integrability

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