This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real value...This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real valued functions such that g(t) not equivalent to t, lim(t-->infinity) g(t) = infinity. It is proved here that when 0 less than or equal to m := lim inf(t-->infinity) Q(t)P(g(t)) less than or equal to 1/4 all solutions of this equation oscillate if the condition lim(t-->infinity) sup Q(t)P(g(t)) > (1 + root1 -4m/2)(2) (*) is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.展开更多
In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existe...In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.展开更多
The authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k an...The authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k and l are integers with k ≥ 1 and l 〉 0. And the asymptotic behavior of nonoscillatory solutions of （＊）. An example is given to show the difference between the equations with and without ＂maxima＂ is studied.展开更多
文摘This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real valued functions such that g(t) not equivalent to t, lim(t-->infinity) g(t) = infinity. It is proved here that when 0 less than or equal to m := lim inf(t-->infinity) Q(t)P(g(t)) less than or equal to 1/4 all solutions of this equation oscillate if the condition lim(t-->infinity) sup Q(t)P(g(t)) > (1 + root1 -4m/2)(2) (*) is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.
基金supported by the National Natural Science Foundation of China(12071491,12001113)。
文摘In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.
基金the Natural Science Foundation of Hebei Province (103141)Key Science Foundation of Hebei Normal University (1301808)
文摘The authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k and l are integers with k ≥ 1 and l 〉 0. And the asymptotic behavior of nonoscillatory solutions of （＊）. An example is given to show the difference between the equations with and without ＂maxima＂ is studied.
