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

Sikorska-Nowak A.

Demonstratio Mathematica

2007

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

Sikorska-Nowak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

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

Sikorska-Nowak A.

Demonstratio Mathematica

2002

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.