Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 141959, 26 pages
Research Article

Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations

Department of Mathematics and Computer Science, Ariel University Center of Samaria, Ariel 44837, Israel

Received 11 April 2009; Revised 25 August 2009; Accepted 3 September 2009

Academic Editor: Marta A D García-Huidobro

Copyright © 2009 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.


We obtain the maximum principles for the first-order neutral functional differential equation (Mx)(t)x(t)(Sx)(t)(Ax)(t)+(Bx)(t)=f(t),  t[0,ω], where A:C[0,ω]L[0,ω],B:C[0,ω]L[0,ω], and S:L[0,ω]L[0,ω] are linear continuous operators, A and B are positive operators, C[0,ω] is the space of continuous functions, and L[0,ω] is the space of essentially bounded functions defined on [0,ω]. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.