高阶变系数函数方程解的振动准则

利用反复迭代的思想方法，讨论了一类高阶变系数函数方程x(g(t))=p(t)x(t)+〖DD(〗m〖〗i=1〖DD)〗Q_i(t)〖DD(〗s〖〗j=1〖DD)〗〖JB(|〗x(gk_j+i(t))〖JB)|〗a_jsgnx(gk_j+i(t))解的振动性，给出了这类函数方程一切解振动的几个充分条件:如果存在整数n0，使得lim〖DD(X〗t〖DD)〗sup〖DD(〗m〖〗i=1〖DD)〗Qi(t)〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i-1〖〗k=1〖DD)〗p(gk(t))〖JB2*］〗aj1〖KG1.5mm〗(t〖XC152HSW1.TIF；%85%85,JZ〗I),则上述方程的一切解振动；如果存在一个整数n0，使得lim〖DD(X〗t〖DD)〗sup〖JB2*［〗p(g(t))〖DD(〗m〖〗i=1〖DD)〗Qi(t)〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i-2〖〗k=1〖DD)〗pn(gk(t))〖JB2*］〗j+〖DD(〗m〖〗i=1〖DD)〗Qi(g(t))〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i〖〗k=2〖DD)〗pn(gk(t))〖JB2*］〗j〖JB2*］〗1〖KG1.5mm〗(t〖XC152HSW1.TIF；%85%85,JZ〗I),则上述方程的一切解也振动. 并且给出了该方程在差分方程中的若干应用.

Oscillation Criteria of Solutions to Higher Order Variable Coefficient Functional Equations

By utilizing iterative method, oscillation of solutions to high-order variable coefficient functional differential equations of the form x(g(t))〖KG-*4〗=p(t)x(t)+〖DD(〗m〖〗i=1〖DD)〗Qi(t)〖DD(〗s〖〗j=1〖DD)〗〖JB(|〗x(gk_j+i(t))〖JB)|〗ajsgn x(gkj+i(t)) is discussed. When n0, n is an integer, and lim〖DD(X〗t〖DD)〗sup〖DD(〗m〖〗i=1〖DD)〗Qi(t)〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i-1〖〗k=1〖DD)〗p(gk(t))〖JB2*］〗aj1〖KG0.8mm〗(t〖XC152HSW1.TIF；%85%85,JZ〗I), all the solutions of the above equations are oscillation. When n0, n is an integer, and lim〖DD(X〗t〖DD)〗sup〖JB2*［〗p(g(t))〖DD(〗m〖〗i=1〖DD)〗Qi(t)〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i-2〖〗k=1〖DD)〗pn(gk(t))〖JB2*］〗j+〖DD(〗m〖〗i=1〖DD)〗Qi(g(t))〖DD(〗s〖〗j=1〖DD)〗〖JB2*［〗〖DD(〗kj+i〖〗k=2〖DD)〗pn(gk(t))〖JB2*］〗j〖JB2*］〗1〖KG1.5mm〗(t〖XC152HSW1.TIF；%85%85,JZ〗I), all the solutions of the above equations are oscillation. Some sufficient conditions for these equations are established. Some applications in difference equations are given.