Wyniki wyszukiwania - Biblioteka Nauki

1

Differential equations in banach space and henstock-kurzweil integrals

100%

Kubiaczyk I. , Sikorska A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1999

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tom 19

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nr 1-2

35-43

EN

In this paper, using the properties of the Henstock-Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x' = f(t,x) and inclusion x' ∈ F(t,x) in a Banach space, where f is Henstock-Kurzweil integrable and satisfies some conditions.

2

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

100%

Cichoń M. , Kubiaczyk I.

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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.

3

Existence theorem for the Hammerstein integral equation

100%

Cichoń M. , Kubiaczyk I.

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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

Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

100%

Kubiaczyk I. , Sikorska-Nowak A.

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nr 1

113-126

EN

In the paper, we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem $x^Δ(t) = f(t,x(t))$, t ∈ T, x(0) = x₀, where T denotes an unbounded time scale (a nonempty closed subset of R and such that there exists a sequence (xₙ) in T and xₙ → ∞) and f is continuous or satisfies Carathéodory's conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. The results presented in the paper are new not only for Banach valued functions, but also for real-valued functions.

5

Oscillation of nonlinear neutral delay differential equations of second order

100%

Kubiaczyk I. , Saker S.

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nr 2

185-212

EN

Oscillation criteria, extended Kamenev and Philos-type oscillation theorems for the nonlinear second order neutral delay differential equation with and without the forced term are given. These results extend and improve the well known results of Grammatikopoulos et. al., Graef et. al., Tanaka for the nonlinear neutral case and the recent results of Dzurina and Mihalikova for the neutral linear case. Some examples are considered to illustrate our main results.

6

Structure of the sets of weak solutions of an ordinary differential equation in a Banach space

100%

Kubiaczyk I.

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tom 44

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nr 1

67-72

7

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

80%

Yantir A. , Kubiaczyk I. , Sikorska-Nowak A.

Open Mathematics

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2015

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tom 13

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nr 1

EN

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s fixed point theorem is used to prove the main result. Finally, we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness.