## Abstract and Applied Analysis

### Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals

Aneta Sikorska-Nowak

#### Abstract

We prove existence theorems for integro-differential equations ${x}^{\Delta}(t)=f(t,x(t),\int_{0}^{t}k(t,s,x(s))\Delta s)$, $x(0)={x}_{0}$, $t\in {I}_{a}=[0,a]\cap T$, $a\in {R}_{+}$, where $T$ denotes a time scale (nonempty closed subset of real numbers $R$), and ${I}_{a}$ is a time scale interval. The functions $f,\,\,k$ are weakly-weakly sequentially continuous with values in a Banach space $E$, and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions $f$ and $k$ satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 836347, 17 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620745

Digital Object Identifier
doi:10.1155/2010/836347

Mathematical Reviews number (MathSciNet)
MR2669088

Zentralblatt MATH identifier
1205.34135

#### Citation

Sikorska-Nowak, Aneta. Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals. Abstr. Appl. Anal. 2010 (2010), Article ID 836347, 17 pages. doi:10.1155/2010/836347. https://projecteuclid.org/euclid.aaa/1288620745