A Picone identity for variable exponent operators and applications

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇ u|p(x)−2 ∇ u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)??(y'))' + Xn,i=1 qi(t)??i(y)=0(1) and the second is the half-linear differential equations (k(t)??(x'))'+ p(t)??(x) = 0 (2) where ?*(s) = |s|*-1s and ?1 >...> ?m > ? > ?m+1 > ... > ?n > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone?s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

Sturm-Picone comparison theorem for matrix systems on time scales

A well-known Picone identity is extended and generalized to second-order dynamic matrix equations on arbitrary time scales. A comparison theorem is obtained in the spirit of the classical Sturm-Picone comparison theorem that extends known scalar results to matrix equations that include the linear homogeneous and inhomogeneous cases, and nonlinear unperturbed and perturbed cases.