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Positive solutions of a higher order neutral differential equation
Author(s) -
Graef John R.,
Qian Chuanxi,
Yang Bo
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310067
Subject(s) - mathematics , integer (computer science) , order (exchange) , type (biology) , class (philosophy) , function (biology) , differential equation , pure mathematics , constant (computer programming) , continuous function (set theory) , functional differential equation , combinatorics , mathematical analysis , ecology , finance , artificial intelligence , evolutionary biology , computer science , economics , biology , programming language
In this paper, we consider the higher order neutral delay differential equation$$ (x(t) - x(t - r))^{n} + p(t)x(t - \sigma) = 0, \quad t \ge 0, \eqno (*) $$where p : [0, ∞) → (0, ∞) is a continuous function, r > 0 and σ > 0 are constants, and n > 0 is an odd integer. A positive solution x ( t ) of Eq. (*) is called a Class–I solution if y ( t ) > 0 and y ′( t ) < 0 eventually, where y ( t ) = x ( t ) – x ( t – r ). We divide Class–I solutions of Eq. (*) into four types. We first show that every positive solution of Eq. (*) must be of one of these four types. For three of these types, a necessary and sufficient condition is obtained for the existence of such solutions. A necessary condition for the existence of a solution of the fourth type is also obtained. The results are illustrated with examples.