# Weak solutions of differential equations in Banach spaces

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

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

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topMieczysław 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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- [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.
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