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Oscillation Criteria for First‐Order Delay Equations
Author(s) -
Sficas Y. G.,
Stavroulakis I. P.
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001662
Subject(s) - mathematics , oscillation (cell signaling) , order (exchange) , differential equation , mathematics subject classification , delay differential equation , combinatorics , functional differential equation , mathematical analysis , mathematical physics , genetics , finance , economics , biology
This paper is concerned with the oscillatory behaviour of first‐order delay differential equations of the formx ′ ( t ) + p ( t ) x ( τ ( t ) ) = 0 ,t ⩾ t 0 ,(1) where p , τ ∈ C ( [ t 0 , ∞ ) , τ ( t ) )is non‐decreasing, τ(t) < t for t ⩾ t 0 andlim t → ∞ τ ( t ) = ∞ . Let the numbers k and L be defined byk = lim inf ∫ τ ( t ) t p ( s ) d sx → ∞a n dL = lim x → ∞ sup∫ τ ( t ) t p ( s ) d sIt is proved here that when L < 1 and 0 < k ⩽ 1/ e all solutions of equation (1) oscillate in several cases in which the condition L > ln λ 1 − 1 + 5 − 2 λ 1 + 2 k λ 1λ 1holds, where λ 1 is the smaller root of the equation λ = e kλ . 2000 Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).