On Proper Oscillatory and Vanishing at Infinity Solutions of Differential Equations with a Deviating Argument
نویسندگان
چکیده
Sufficient conditions are found for the existence of multiparametrical families of proper oscillatory and vanishing-at-infinity solutions of the differential equation u(n)(t) = g ( t, u(τ0(t)), . . . , u(τm−1(t)) ) , where n ≥ 4, m is the integer part of 2 , g : R+ × R m → R is a function satisfying the local Carathéodory conditions, and τi : R+ → R (i = 0, . . . , m− 1) are measurable functions such that τi(t) → +∞ for t → +∞ (i = 0, . . . , m− 1). Introduction In this paper we consider the differential equation u(n)(t) = g ( t, u(τ0(t)), . . . , u(τm−1(t)) ) (0.1) and its particular cases u(n)(t) = p(t) ∣ ∣u(τ(t)) ∣ ∣ λ sgnu(τ(t)), (0.2) u(n)(t) = p(t)u(τ(t)), (0.3) u(n)(t) = m−1 ∑ i=0 pi(t)u(τi(t)). (0.4) Throughout the paper it will be assumed that n ≥ 4, m is integer part of the number 2 , g : R+×R m → R is a function satisfying the local Carathéodory conditions, p : R+ → R and pi : R+ → R (i = 0, . . . , m − 1) are locally 1991 Mathematics Subject Classification. 34K15,34K10.
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تاریخ انتشار 2001