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The existence theory for the differential equation xm(t)=f(t,x) in Banach spaces and Henstock-Kurzweil integral

100%

Sikorska-Nowak A.

Demonstratio Mathematica

2007

tom Vol. 40, nr 1

115-124

Using the properties of the Henstock–Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x(m)(t) = f(t, x) in a Banach space, where f is HL integrable and satis.es certain conditions. Our fundamental tool is the measure of noncompactness developed by Kuratowski and Hausdorff.

Existance of solutions of nonlinear integral equations and Henstock-Kurzweil integrals

100%

Sikorska-Nowak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

tom Vol. 47, [Z] 2

227-238

We prove an existence theorems for the nonlinear integral equation... [formuła matematyczna]... where f, g, x are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals

100%

Sikorska-Nowak A.

Demonstratio Mathematica

2002

tom Vol. 35, nr 1

49-60

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

Dynamic equations (…) on time scales

100%

Sikorska-Nowak A.

Demonstratio Mathematica

2011

tom Vol. 44, nr 2

317-333

In this paper we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem (…), where (…) denotes a mth order (…) - derivative,T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence (…) E -a Banach space and f is a continuous function or satis .es 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. As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

Existence of solutions of the dynamic Cauchy problem in Banach spaces

51%

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

Demonstratio Mathematica

2012

tom Vol. 45, nr 3

561-573

In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.