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

• Volume: 31, Issue: 1, page 71-90
• ISSN: 1509-9407

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## Abstract

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In this paper we prove existence theorems for integro - differential equations ${x}^{\Delta }\left(t\right)=f\left(t,x\left(t\right),\int {₀}^{t}k\left(t,s,x\left(s\right)\right)\Delta s\right)$, t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where 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, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale.

## How to cite

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Aneta Sikorska-Nowak. "Integro-differential equations on time scales with Henstock-Kurzweil delta integrals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.1 (2011): 71-90. <http://eudml.org/doc/271148>.

@article{AnetaSikorska2011,
abstract = {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) = x₀ where 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, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale.},
author = {Aneta Sikorska-Nowak},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {integro-differential equations; nonlinear Volterra integral equation; time scales; Henstock-Kurzweil delta integral; HL delta integral; Banach space; fixed point; measure of noncompactness; Carathéodory solutions; time scale; Henstock-Lebesgue delta integral},
language = {eng},
number = {1},
pages = {71-90},
title = {Integro-differential equations on time scales with Henstock-Kurzweil delta integrals},
url = {http://eudml.org/doc/271148},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Aneta Sikorska-Nowak
TI - Integro-differential equations on time scales with Henstock-Kurzweil delta integrals
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2011
VL - 31
IS - 1
SP - 71
EP - 90
AB - 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) = x₀ where 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, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale.
LA - eng
KW - integro-differential equations; nonlinear Volterra integral equation; time scales; Henstock-Kurzweil delta integral; HL delta integral; Banach space; fixed point; measure of noncompactness; Carathéodory solutions; time scale; Henstock-Lebesgue delta integral
UR - http://eudml.org/doc/271148
ER -

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