Abstract
In this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian Δ(Δu(k-1)p-2Δu(k-1))+a(k)u(k)p-2u(k)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Delta (\\left| \\Delta u(k-1)\\right| ^{p-2}\\Delta u(k-1))+a(k)\\left| u(k)\\right| ^{p-2}u(k)=0 \\end{aligned}$$\\end{document}with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.
Highlights
According to the famous Lyapunov inequality [22], for a continuous function a = a(x) on the interval [b, c] ⊂ R, the validity of c |a(x)| d x > (1)b c−b is a necessary condition for the existence of a non-trivial solution u to the boundary value problem u + a(x)u = 0, in (b, c), u(b) = 0 = u(c).Many generalizations of (1) have been established in the literature
B c−b is a necessary condition for the existence of a non-trivial solution u to the boundary value problem u + a(x)u = 0, in (b, c), u(b) = 0 = u(c)
Since the second-order difference equation (PD) can be expressed as an equivalent Hamiltonian system, we can deduce new Lyapunov-type inequalities from Lyapunov-type inequalities obtained for discrete systems
Summary
From Zhang [28] we can read the following optimal Lyapunov-type inequalities for problem (2): 1. The aim of this paper is to give sharp Lyapunov-type inequalities for the second-order difference problem with p-laplacian (φp( u(k − 1))) + a(k)φp(u(k)) = 0, k ∈ [1, T ] Bl (u) = 0,. Since the second-order difference equation (PD) can be expressed as an equivalent Hamiltonian system, we can deduce new Lyapunov-type inequalities from Lyapunov-type inequalities obtained for discrete systems. Such results can be found in [15,16,26,27,29,30,32]. The proof is based on finding a minimum of some especial minimization problems, see [8] for the use of such methods
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