Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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OSCILLATORY BEHAVIOR AND COMPARISON FOR HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS ON TIME SCALES

  • Sun, Taixiang (College of Mathematics and Information Science, Guangxi University) ;
  • Yu, Weiyong (College of Mathematics and Information Science, Guangxi University) ;
  • Xi, Hongjian (Department of Mathematics, Guangxi College of Finance and Economics)
  • Received : 2011.03.14
  • Accepted : 2011.07.06
  • Published : 2012.01.30
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Abstract

In this paper, we study asymptotic behaviour of solutions of the following higher order nonlinear dynamic equations $$S_n^{\Delta}(t,x)+{\delta}p(t)f(x(g(t)))=0$$ and $$S_n^{\Delta}(t,x)+{\delta}p(t)f(x(h(t)))=0$$ on an arbitrary time scale $\mathbb{T}$ with sup $\mathbb{T}={\infty}$, where n is a positive integer, ${\delta}=1$ or -1 and $$S_k(t,x)=\{\array x(t),\;if\;k=0,\\a_k(t)S_{{\kappa}-1}^{\Delta}(t),\;if\;1{\leq}k{\leq}n-1,\\a_n(t)[S_{{\kappa}-1}^{\Delta}(t)]^{\alpha},\;if\;k=n,$$ with ${\alpha}$ being a quotient of two odd positive integers and every $a_k$ ($1{\leq}k{\leq}n$) being positive rd-continuous function. We obtain some sufficient conditions for the equivalence of the oscillation of the above equations.

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References

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