Systems of Functional Differential Equations on Finite Intervals

Ravi P. Agarwal, Leonid Berezansky, Elena Braverman and Alexander Domoshnitsky
Additional contact information
Ravi P. Agarwal: Texas A&M University—Kingsville, Department of Mathematics
Leonid Berezansky: Ben-Gurion University of the Negev, Department of Mathematics
Elena Braverman: University of Calgary, Department of Mathematics
Alexander Domoshnitsky: Ariel University Center of Samaria, Department of Computer Sciences and Mathematics

Chapter Chapter 16 in Nonoscillation Theory of Functional Differential Equations with Applications, 2012, pp 399-428 from Springer

Abstract: Abstract In Chap. 16, boundary value problems for system of linear functional differential equations are studied, and results on positivity of Green’s matrices are obtained. A necessary and sufficient condition for nonnegativity of all the entries of the Cauchy matrix for linear systems of ordinary differential equations of the first order (obtained by Wazewski) is nonpositivity of off-diagonal coefficients. In the first part of this chapter, extension of this result to various boundary value problems for functional differential equations is obtained, and all the results include an analogue of Wazewski’s sign condition. The main results of the chapter are based on the idea to construct a first-order functional differential equation for the r-th component of the solution vector. Even in the case of systems of ordinary differential equations, this equation is functional differential, with a sufficiently complicated operator. Next, the assertions on differential inequalities for first-order functional differential equations from Chap. 15 are used. This approach explains the importance of considering first-order equations in the operator form, which describes only properties of the operators and does not assume their concrete form. Assertions on differential inequalities for the r-th component of the solution vector or, in the other terminology, about positivity of the entries of the r-th row of Green’s matrices are obtained in a form of theorems on several equivalences, one of the equivalent properties is nonoscillation. It is demonstrated that many tests obtained from these general theorems for one of the solution’s components are unimprovable in a certain sense.

Keywords: Scalar Functional Differential Equations; Cauchy Matrix; Wazewski; Solution Vector; Nonpositivity (search for similar items in EconPapers)
Date: 2012
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-3455-9_16

Ordering information: This item can be ordered from
http://www.springer.com/9781461434559

DOI: 10.1007/978-1-4614-3455-9_16

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().