Lyapunov’s inequality on timescales

The purpose of this work is to establish the timescale version of Lyapunov’s inequality as follows: Let x(t)x(t) be a nontrivial solution of (r(t)xΔ(t))Δ+p(t)xσ(t)=0on [a,b] satisfying x(a)=x(b)=0x(a)=x(b)=0. Then, under suitable conditions on pp, rr, aa and bb, we have ∫abp+(t)Δt≥{r(a)r(b)b−af(d),if r is increasing,r(b)r(a)b−af(d),if r is decreasing, where p+(t)=max{p(t),0},f(t)=(t−a)(b−t)p+(t)=max{p(t),0},f(t)=(t−a)(b−t) and d∈Td∈T satisfies |a+b2−d|=min{|a+b2−s|∣s∈[a,b]∩T} if a+b2∈T. Here TT is a timescale (see below).