吴英柱. 高阶变系数函数方程解的振动准则[J]. 华南师范大学学报(自然科学版), 2016, 48(2): 107-110. doi: 10.6054/j.jscnun.2015.08.024
引用本文: 吴英柱. 高阶变系数函数方程解的振动准则[J]. 华南师范大学学报(自然科学版), 2016, 48(2): 107-110. doi: 10.6054/j.jscnun.2015.08.024
WU Yingzhu*. Oscillation Criteria of Solutions to Higher Order Variable Coefficient Functional Equations[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(2): 107-110. doi: 10.6054/j.jscnun.2015.08.024
Citation: WU Yingzhu*. Oscillation Criteria of Solutions to Higher Order Variable Coefficient Functional Equations[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(2): 107-110. doi: 10.6054/j.jscnun.2015.08.024

高阶变系数函数方程解的振动准则

Oscillation Criteria of Solutions to Higher Order Variable Coefficient Functional Equations

  • 摘要: 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.

     

    Abstract: 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.

     

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