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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 571795, 7 pages
doi:10.1155/2011/571795
Research Article

Bounds of Solutions of Integrodifferential Equations

Zdeněk Šmarda

Department of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech Republic

Received 20 January 2011; Accepted 24 February 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Zdeněk Šmarda. 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.

Abstract

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: ∫ π‘₯ β€² ( 𝑑 ) βˆ’ β„± ( 𝑑 , π‘₯ ( 𝑑 ) , 𝑑 0 π‘˜ ( 𝑑 , 𝑠 , π‘₯ ( 𝑑 ) , π‘₯ ( 𝑠 ) ) 𝑑 𝑠 ) = β„Ž ( 𝑑 ) , π‘₯ ( 0 ) = π‘₯ 0 , where β„Ž ∢ 𝑅 + β†’ 𝑅 , π‘˜ ∢ 𝑅 2 + Γ— 𝑅 2 β†’ 𝑅 , β„± ∢ 𝑅 + Γ— 𝑅 2 β†’ 𝑅 are continuous functions, 𝑅 + = [ 0 , ∞ ) .