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Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

100%

Cichoń M., Kubiaczyk I.

Annales Polonici Mathematici

1995

tom 62

nr 1

13-21

We investigate the structure of the set of solutions of the Cauchy problem x' = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.

Weak solutions of differential equations in Banach spaces

100%

Cichoń M.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1995

tom 15

nr 1

5-14

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2 Cichoń M.

1 Kubiaczyk I.

2 1995

Wyniki wyszukiwania

Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

Weak solutions of differential equations in Banach spaces