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Lyapunov’s inequality on timescales
Authors:Fu-Hsiang Wong  Shiueh-Ling Yu  Cheh-Chih Yeh  Wei-Cheng Lian  
Institution:

aDepartment of Mathematics, National Taipei Teacher’s College, Taipei 10659, Taiwan, ROC

bHolistic Education Center, St. John’s and St. Mary’s Institute of Technology, Tamsui, Taipei, Taiwan, ROC

cDepartment of Information Management, Lunghwa University of Science and Technology, Kueishan Taoyuan, 333 Taiwan, ROC

dDepartment of Information Management, National Kaohsiung Institute of Marine Technology, No.142, Hai Chuan Road, Nan-Tzu Dist, Kaohsiung, Taiwan, ROC

Abstract:The purpose of this work is to establish the timescale version of Lyapunov’s inequality as follows: Let x(t) be a nontrivial solution of (r(t)xΔ(t))Δ+p(t)xσ(t)=0on a,b] satisfying x(a)=x(b)=0. Then, under suitable conditions on p, r, a and b, we have abp+(t)Δt{r(a)r(b)baf(d),if r is increasing,r(b)r(a)baf(d),if r is decreasing, where p+(t)=max{p(t),0},f(t)=(ta)(bt) and dT satisfies |a+b2d|=min{|a+b2s|sa,b]T} if a+b2T. Here T is a timescale (see below).
Keywords:Timescales  Lyapunov’s inequality
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