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

• Main Authors: Domoshnitsky Alexander, Maghakyan Abraham, Shklyar Roman
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2009/141959

<p/> <p>We obtain the maximum principles for the first-order neutral functional differential equation <inline-formula> <graphic file="1029-242X-2009-141959-i1.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-141959-i2.gif"/></inline-formula> where <inline-formula> <graphic file="1029-242X-2009-141959-i3.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-141959-i4.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-141959-i5.gif"/></inline-formula> are linear continuous operators, <inline-formula> <graphic file="1029-242X-2009-141959-i6.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-141959-i7.gif"/></inline-formula> are positive operators, <inline-formula> <graphic file="1029-242X-2009-141959-i8.gif"/></inline-formula> is the space of continuous functions, and <inline-formula> <graphic file="1029-242X-2009-141959-i9.gif"/></inline-formula> is the space of essentially bounded functions defined on <inline-formula> <graphic file="1029-242X-2009-141959-i10.gif"/></inline-formula>. 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.</p>