Search results for: Aneta Sikorska-Nowak

Items from 1 to 9 out of 9 results

article

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

Ahmet Yantir, Ireneusz Kubiaczyk, Aneta Sikorska-Nowak

Open Mathematics > 2015 > 13 > 1

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...

article

Impulsive nonlocal differential equations through differential equations on time scales

Mieczysław Cichoń, Bianca Satco, Aneta Sikorska-Nowak

Applied Mathematics and Computation > 2011 > 218 > 6 > 2449-2458

We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.

article

Integro-differential equations on time scales with Henstock-Kurzweil delta integrals

Aneta Sikorska-Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2011 > 31 > 1 > 71-90

In this paper we prove existence theorems for integro - differential equations$x^Δ (t) = f(t,x(t),∫₀^t k(t,s,x(s))Δs)$,t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊,x(0) = x₀where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral,...

article

Theorems for generalized Favard-Kantorovich and Favard-Durrmeyer operators in exponential function spaces

Grzegorz Nowak, Aneta Sikorska-Nowak

Mathematica Slovaca > 2010 > 60 > 2 > 265-278

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove some direct approximation theorems for functions f such that w _σ f ∈ L _p(R), where 1 ≤ p ≤ ∞ and w _σ(x) = exp(−σx ²), σ > 0.

article

Weak solutions for the dynamic Cauchy problem in Banach spaces

Mieczysław Cichoń, Ireneusz Kubiaczyk, Aneta Sikorska-Nowak, Ahmet Yantir

Nonlinear Analysis > 2009 > 71 > 7-8 > 2936-2943

This paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete Cauchy problem in Banach spaces. We offer the existence of the weak solution of dynamic Cauchy problem on an infinite time scale. The measure of weak noncompactness and the fixed point theorem of Kubiaczyk are used to prove the main result.

article

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

Ireneusz Kubiaczyk, Aneta Sikorska-Nowak

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2009 > 29 > 1 > 113-126

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...

article

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

Aneta Sikorska-Nowak

Commentationes Mathematicae > 2007 > 47 > 2

We prove an existence theorems for the nonlinear integral equation \[ x(t) = f (t) + \int_{0}^a k_1 (t, s)x(s)ds + \int_{0}^a k_2(t, s)g(s, x(s))ds,\quad t \in I_a = [0, a], a \in \mathbb{R} _+, \] 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.

article

On the Generalized Favard-Kantorovich and Favard-Durrmeyer Operators in Exponential Function Spaces

Grzegorz Nowak, Aneta Sikorska-Nowak

Journal of Inequalities and Applications > 2006 > 2007 > 1 > 1-12

We consider the Kantorovich- and the Durrmeyer-type modifications of the generalized Favard operators and we prove an inverse approximation theorem for functions such that , where and , .

article

On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals

Aneta Sikorska-Nowak

Annales Polonici Mathematici > 2004 > 83 > 3 > 257-267

We prove some existence theorems for nonlinear integral equations of the Urysohn type $x(t) = φ(t) + λ∫_0^a f(t,s,x(s))ds$ and Volterra type $x(t) = φ(t) + ∫_0^tf(t,s,x(s))ds$, $t ∈ I_a = [0,a]$, where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

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