The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form lα[x]≡∑j=0n(−1)j(pjφ(x(j)))(j) and Lα[y]≡∑j=0n(−1)j(Pjφ(y(j)))(j) where pjpj and PjPj, j=0,…,nj=0,…,n, are continuous functions defined on [a,b][a,b] and φ(u)≔|u|αsgnu,α>0, is derived and then the Sturmian comparison theory for the corresponding 2n2nth-order equations lα[x]=0lα[x]=0 and Lα[y]=0Lα[y]=0 based on this identity is developed.