A COMPARISON THEOREM AND OSCILLATION OF THE SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH RETARDED ARGUMENTS DEPENDING ON THE UNKNOWN FUNCTION

TitleA COMPARISON THEOREM AND OSCILLATION OF THE SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH RETARDED ARGUMENTS DEPENDING ON THE UNKNOWN FUNCTION
Publication TypeJournal Article
Year of Publication2005
AuthorsSimeonov, PS
Volume9
Issue3
Start Page389
Pagination8
Date Published2005
ISSN1083-2564
AMS34K15
Abstract

The ${ n − th }$ order retarded differential equations $${ r_n(t)(r_{n−1}(t)(· · ·(r_1(t)x ′ (t))′ · · ·) ′ ) ′ \ \ }$$  $${  \ \ ±F(t, x(∆_1(t, x(t))), · · · , x(∆_m(t, x(t)))) = 0  \ \ \ \ \  (±E)}$$ are considered, where ${ n ≥ 2, r_i ∈ C(J,(0, +∞)), i = 1, · · · , n; \int_{a}^{+\infty} \, \frac{dt}{r_i(t)} = +\infty, i = 1, · · · , n − 1; τ_j (t) ≤ ∆_j (t, x) ≤ t,}$ ${ t ∈ J = [a, +∞), x ∈ R, j = 1, · · · , m }$.

In the main Theorem 1 the oscillatory and asymptotic properties of equations ${ (±E)}$ are compared with those of the comparison equations $${ r_n(t)(r_n−1(t)(· · ·(r_1(t)x ′ (t))′ · · ·) ′ ) ′ }$$ $${ \ \ \ ±F(t, x(τ_1(t)), · · · , x(τ_m(t))) = 0. \ \ \ \ \ (±C) }$$

By means of Theorem 1 it becomes possible to obtain oscillation criteria for equations ${(±E)}$ using well-known oscillation results for equations ${(±C)}$. This possibility is realized in Theorems 2, 3 and 4 for some particular cases of equation ${(+E)}$.
URLhttp://www.acadsol.eu/en/articles/9/3/9.pdf
Refereed DesignationRefereed
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