An integral oscillation criterion for even order half-linear differential equations (Q2176491)

The author deals with the even order half-linear differential equation \[L_n x+q(t)|x|^\beta \mathop{\mathrm{sgn}}{x}=0,\quad t\geq a>0,\tag{A}\] where \(n\geq 4\) is an even integer, the quasi-derivatives \(L_n\) are defined by \[L_1 x(t)=x'(t),\quad L_{i+1}x(t)=[p_i(t)|L_i x(t)|^{\alpha_i}\mathop{\mathrm{sgn}}{L_i x(t)}] \quad (i=1,\ldots,n-1),\] \(\alpha_i\) (\(i=1,\ldots,n-1\)) and \(\beta\) are positive constants, and \(p_i,q\colon [a,\infty)\to (0,\infty)\) (\(i=1,\ldots,n-1\)) are continuous functions with \[\int_\alpha^\infty \! \frac{1}{(p_i(t))^{1/{\alpha_i}}} \, dt = \infty.\] In the literature, oscillation criteria for equation (A) are known in the case of \(\beta\neq a_1\cdots a_{n-1}\). On the other hand, in this paper, the author focuses on the case of \(\beta=\alpha_1\cdots \alpha_{n-1}\). The author establishes sufficient conditions under which equation (A) has no nonoscillatory solution of Kigradze's degree \(l\), i.e., \[x(t)L_j x(t)>0 \;\;\text{for}\;\; j=0,1,\ldots,l\] and \[(-1)^{n+j}x(t)L_j x(t)<0 \;\;\text{for}\;\; j=l+1,l+2,\ldots,n.\] The author also includes some interesting corollaries, which show that all solutions of equation (A) are oscillatory under certain conditions.