Wyniki wyszukiwania

1

Weak solutions of differential equations in Banach spaces

100%

Cichoń M.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1995

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tom 15

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nr 1

5-14

EN

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.

2

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Monotonic solutions for quadratic integral equations

64%

Cichoń M., Metwali M.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2011

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tom 31

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nr 2

157-171

EN

Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line ℝ₊ for a nonlinear quadratic functional integral equation.

3

Existence theorem for the Hammerstein integral equation

64%

Cichoń M., Kubiaczyk I.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1996

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tom 16

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nr 2

171-177

EN

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.

4

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

64%

Cichoń M., Kubiaczyk I.

Annales Polonici Mathematici

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1995

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tom 62

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nr 1

13-21

EN

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.

5

Differential inclusions and multivalued integrals

51%

Cichoń K., Cichoń M., Satco B.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2013

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tom 33

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nr 2

171-191

EN

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented by a short survey about multivalued integration including Pettis and Henstock-Kurzweil-Pettis multivalued integrals.

Ograniczanie wyników

4 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Annales Polonici Mathematici

5 Cichoń M.

2 Kubiaczyk I.

1 Cichoń K.

1 Metwali M.

1 Satco B.

1 2013

1 2011

1 1996

2 1995

Weak solutions of differential equations in Banach spaces

Monotonic solutions for quadratic integral equations

Existence theorem for the Hammerstein integral equation

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

Differential inclusions and multivalued integrals