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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.

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.

Existence theorem for the Hammerstein integral equation

100%

Cichoń M. , Kubiaczyk I.

nr 2

171-177

In this paper we prove an existence theorem for the Hammerstein integral equation $x(t) = p(t) + λ ∫_I K(t,s)f(s,x(s))ds$, where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

On some solutions of nonlocal Cauchy problems

88%

Cichoń M. , Majcher P.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2003

tom [Z] 43(2)

185-199

The aim of the paper is to give a theorem about the existence of solutions of a semilinear nonlocal Cauchy problem for ordinary differential equations in Banach spaces. Since all assumptions are expressed in terms of the weak topology, we obtain the existence of pseudo-integral-solutions instead of integral, Caratheodory or classical ones considered in previous papers.