On the half-linear second order differential equations

Basis properties of eigenfunctions of the 𝑝-Laplacian

“…Generalized sine functions S p , 1 < p < ∞, were studied by Elbert [6] as Dirichlet eigenfunctions for the one-dimensional p-Laplacian equation ( is the first zero of S p , which is the eigenfunction (normalized by S p (0) = 1) corresponding to the minimal Dirichlet eigenvalue λ = 1 of (1). Elbert also deduced the relation…”

Lyapunov inequality for a class of fractional differential equations with Dirichlet boundary conditions

“…We refer the reader to [1,17] and the references therein. For the nonlinear case, Elbert [5] firstly extended the inequality (1.2) to the following p-Laplacian problem (|u | (1.3) For p = 2, then the linear problem (1.1) is recovered. Nápoli and Pinasco [4] extended the inequality (1.3) to the following more generalized nonlinear problems…”

“…Explicitly, the author showed that if the above problem (1.4) has a nontrivial solution, then 5) which yields the standard Lyapunov inequality (1.2) if we take α = 2 in (1.5), where Γ is the gamma function. From then on, some Lyapunov-type inequalities for other fractional boundary value problems were established, see, for example, [7,10,11,14,16] and the references listed therein.…”

Positive solutions of a weakly singular periodic eco-economic system with changing-sign perturbation

“…In [9], Elbert extended inequality (1.1) to the one-dimensional p-Laplacian equation. More precisely, he proved that, if u is a nontrivial solution of the problem Observe that for p = 2, (1.3) reduces to (1.1).…”