Abstract and Applied Analysis
Volume 2011 (2011), Article ID 242965, 26 pages
http://dx.doi.org/10.1155/2011/242965
Research Article

Linear Hyperbolic Functional-Differential Equations with Essentially Bounded Right-Hand Side

Alexander Domoshnitsky,1 Alexander Lomtatidze,2 Abraham Maghakyan,1 and Jiří Šremr2

1Department of Mathematics and Computer Science, Ariel University Center of Samaria, 40700 Ariel, Israel
2Institute of Mathematics, Academy of Sciences of the Czech Republic, Branch in Brno, Žižkova 22, 616 62 Brno, Czech Republic

Received 14 March 2011; Revised 18 May 2011; Accepted 19 May 2011

Academic Editor: Stefan Siegmund

Copyright © 2011 Alexander Domoshnitsky et al. 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

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem 𝑢 ( 1 , 1 ) ( 𝑡 , 𝑥 ) = ℓ 0 ( 𝑢 ) ( 𝑡 , 𝑥 ) + ℓ 1 ( 𝑢 ( 1 , 0 ) ) ( 𝑡 , 𝑥 ) + ℓ 2 ( 𝑢 ( 0 , 1 ) ) ( 𝑡 , 𝑥 ) + 𝑞 ( 𝑡 , 𝑥 ) , 𝑢 ( 𝑡 , 𝑐 ) = 𝛼 ( 𝑡 ) for 𝑡 ∈ [ 𝑎 , 𝑏 ] , 𝑢 ( 𝑎 , 𝑥 ) = 𝛽 ( 𝑥 ) f o r 𝑥 ∈ [ 𝑐 , 𝑑 ] given on the rectangle [ 𝑎 , 𝑏 ] × [ 𝑐 , 𝑑 ] are established, where the linear operators ℓ 0 , ℓ 1 , ℓ 2 map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations.