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Abstract and Applied Analysis
Volume 2010, Article ID 836347, 17 pages
Research Article

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

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Received 11 February 2010; Accepted 17 June 2010

Academic Editor: H. Bevan Thompson

Copyright © 2010 Aneta Sikorska-Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We prove existence theorems for integro-differential equations π‘₯ Ξ” ∫ ( 𝑑 ) = 𝑓 ( 𝑑 , π‘₯ ( 𝑑 ) , 𝑑 0 π‘˜ ( 𝑑 , 𝑠 , π‘₯ ( 𝑠 ) ) Ξ” 𝑠 ) , π‘₯ ( 0 ) = π‘₯ 0 , 𝑑 ∈ 𝐼 π‘Ž = [ 0 , π‘Ž ] ∩ 𝑇 , π‘Ž ∈ 𝑅 + , where 𝑇 denotes a time scale (nonempty closed subset of real numbers 𝑅 ), and 𝐼 π‘Ž is a time scale interval. The functions 𝑓 , π‘˜ 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 π‘˜ satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.