In this paper, we establish some new reverse dynamic inequalities and use them to prove some higher integrability theorems for decreasing functions on time scales. In order to derive our main results, we first prove a new dynamic inequality for convex functions related to the inequality of Hardy, Littlewood and Pólya, known from the literature. Then, we prove a refinement of the famous Hardy inequality on time scales for a class of decreasing functions. As an application, our results are utilized to formulate the corresponding reverse integral and discrete inequalities, which are essentially new.

We study the Dirichlet boundary value problem for the p(x)-Laplacian of the formWe introduce a new variational technic that allows us to investigate problem (P) without need of the Ambrosetti and Rabinowitz condition on the nonlinearity f .