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Existence theorem for the Hammerstein integral equation

100%

Cichoń M., Kubiaczyk I.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1996

tom 16

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.

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2 Cichoń M.

2 Kubiaczyk I.

1 1996

1 1995

Wyniki wyszukiwania

Existence theorem for the Hammerstein integral equation

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