Nonoscillation of first order impulse differential equations with delay

[摘要] Oscillation properties of impulse functional-differential equations are studied for equations of the type [GRAPHICS] x(xi) = 0, xi is not an element of [a, b], x(t(j)) = beta(j)x(t(j) - 0), j = 1, ..., k, a < t(1) < t(2) < ... < t(k) < b. The proven test for oscillation generalizes the known ones and allows consideration of the solvability of boundary value problems for the corresponding nonhomogeneous impulse equations. In particular, for the scalar impulse equation xover dot(t) + p(t)x(t - tau)) = 0, t is an element of [0, infinity), x(xi) = 0 for xi < 0, x(t(j)) = beta(j)x(t(j) - 0), beta(j) > 0, j = 1, 2, ..., denote B(t) = Pi(j is an element of Dt) beta(j), where D-t = {i: t(i) is an element of [t - tau(t), t]}, p(+)(t) = max{p(t), 0}. PROPOSITION. Let 1 + ln B(t)/e greater than or equal to integral(r(t))(t)p(+) (s) ds where r(t) = max{t - tau(t), 0}, t > 0. Then the nontrivial solution of this equation has no zeros on [0, infinity). (C) 1997 Academic Press.

[发布日期] 1997-02-01