Recently there has been an increasing interest in studying p(t)-Laplacian equations, an example of which is given in the following form (|u ′ (t)| p(t)−2 u ′ (t))′ + c(t)|u(t)| q(t)−2 u(t) = 0, t > 0. In particular, the first study of sufficient conditions for oscillatory solution of p(t)-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with p(t)- Laplacian via Riccati method. The results obtained are new and rare, except for a work of Zhang (2007). We present more detailed results than Zhang (2007).
In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation $(r(t)u^{\prime }(t))^{\prime }+p(t)u(t)=0$ are derived. One of them deals with the case $\int ^{\infty }\frac{{\mathrm d}t}{r(t)}=\infty $, and the other with the case $\int ^{\infty }\frac{{\mathrm d}t}{r(t)}<\infty $.