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1

On a fixed point theorem for weakly sequentially continuous mapping

100%
Kubiaczyk I.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1995
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tom 15
|
nr 1
15-20
EN
Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅({x} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous results of this type, in this theorem the continuity hypothesis on f is essentially weakened. Our results generalize those of [1,7,15,17].
2
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Oscillation of nonlinear neutral delay differential equations of second order

64%
Kubiaczyk I., Saker S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2002
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tom 22
|
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.
3

Existence theorem for the Hammerstein integral equation

64%
Cichoń M., Kubiaczyk I.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1996
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tom 16
|
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

Differential equations in banach space and henstock-kurzweil integrals

64%
Kubiaczyk I., Sikorska A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1999
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tom 19
|
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.
5
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Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

64%
Kubiaczyk I., Sikorska-Nowak A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2009
|
tom 29
|
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.
6
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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
|
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.
7
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Structure of the sets of weak solutions of an ordinary differential equation in a Banach space

63%
Kubiaczyk I.
Annales Polonici Mathematici
|
1984
|
tom 44
|
nr 1
67-72
8
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Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

51%
Yantir A., Kubiaczyk I., Sikorska-Nowak A.
Open Mathematics
|
2015
|
tom 13
|
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.
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