Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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.


Introduction
According to the famous Lyapunov inequality [22], for a continuous function a = a(x) on the interval [b, c] ⊂ R, the validity of is a necessary condition for the existence of a non-trivial solution u to the boundary value problem Many generalizations of (1) have been established in the literature. For our purposes, the generalization to second-order half-linear equation B Robert Stegliński robert.steglinski@p.lodz.pl 1 Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland is important. Here p > 1 and φ p (u) = |u| p−2 u. From Zhang [28] we can read the following optimal Lyapunov-type inequalities for problem (2): 1. if (2) with Dirichlet, Neumann or anti-periodic boundary conditions has a non-trivial solution, then 2. if (2) with periodic boundary conditions has a non-trivial solution, then Here, and in what follows, a + (x) = max{a(x), 0}. The Lyapunov-type inequalities have proved to be useful tools in the oscillation theory, eigenvalue problems, disconjugacy, and numerous other applications for theories of differential and difference equations (we refer the reader to some survey articles and books [8,10,23,25] and the references therein). Compared to the large number of references to continuous Lyapunov-type inequalities, little has been done for discrete Lyapunov-type inequalities. Let us list the works in which the Lyapunov-type inequality appears in a discrete context: for difference equations see [9][10][11][18][19][20][21]31], for discrete systems see [15,16,26,27,29,30,32], for fractional discrete problems see [12][13][14] and for problems on time scales (as these kind of problems include difference equations) see [1][2][3][4][5][6]17,24].
The aim of this paper is to give sharp Lyapunov-type inequalities for the second-order difference problem with p-laplacian where B D (u) = 0 is an abbreviation for u(0) = 0 = u(T + 1), i.e. the problem (P l ) is considered with Dirichlet (D), Neumann (N), mixed (DN and ND), periodic (P) and anti-periodic (AP) boundary conditions. Here 1 < p < ∞,, T ∈ N, T ≥ 2, Partial results can be found in the literature. In 1983, Cheng [9] first obtained the following Lyapunov-type inequality, which is an optimal discrete analogue of (1): if T is odd, for problem (P D ) with p = 2 and nonnegative a ∈ R T . Since the second-order difference equation (P D ) can be expressed as an equivalent Hamiltonian system, we can deduce new Lyapunov-type inequalities from Lyapunov-type inequalities obtained for discrete systems.
Problems on time scales include difference equations. In [3, Corrolary 4.1.7], we find the following Lyapunov-type inequality: Lyapunov-type inequalities for higher order difference equations can be found in [19][20][21]31]. In [21], the second order case reads as follows which differs from (P AP ) even for p = 2. By [21,Corollary 3], the existence of non-trivial solution for problem (5) implies where p is a conjugate exponent, i.e. 1/ p + 1/ p = 1.
The structure of the paper is the following: in Sect. 2, we formulate our main result in which we give the sharp Lyapunov-type inequalities for problems (P l ). In Sect. 3, we prove our main theorem. The proof is based on finding a minimum of some especial minimization problems, see [8] for the use of such methods.

The main result
In this section, we state our main result. By solution space of problem (P l ) we mean Tdimensional vector space for l = D, N , DN , N D, P and AP, respectively. Theorem 1 Let 1 < p < ∞ and a ∈ R T . The following statements hold.

If (P D ) has a non-trivial solution in X D , then
3. If (P DN ) or (P N D ) has a non-trivial solution in X DN and X N D , respectively, then Moreover, the inequalities are sharp in the sense that there are a ∈ R T satisfying above conditions with equality and the corresponding problems have a non-trivial solution.
Let us make a point of the important difference between the discrete and the continuous case: in the optimal Lyapunov-type inequalities (3) and (4), the inequality is strict (see [28] and [8,Remark 2.4]).

Proof of the main result
Using the summation by parts formula: Then, it is easy to check that u ∈ X l is a solution to problem (P l ) if and only if u ∈ X l satisfies On X l , l = D, N , DN , N D, P, AP, we define norm [1,T ] |u(k)| .
We will also use the following notation To prove our main result, we define some numbers: if l = N , P, where C denotes the set of constant functions on [0, T + 1]. First, we find the values of numbers λ l .
The following statements hold. (i) and λ N D is attained by (iv) Proof (i) For the proof see [7,Lemma 4].
(ii) First, we observe that the supremum in (7) with respect to r is attained whenever since, if max(u+r ) = − min(u+r ) we can change r a little and decrease the norm u + r ∞ . Moreover, since for any u ∈ X N and r ∈ R we have (u + r ) = u and functions in the numerator and denominator in (7) are homogeneous of degree p, we have [1,T ] u(k) = 1}.
Next, since Z N is compact, there exists u N ∈ Z N which minimizes · p p over Z N and there are 1 ≤ k 1 There is no loss of generality in assuming u N (k 1 ) = −1 and u N (k 2 ) = 1. Now, the minimality property of u N allows us to infer about the geometry of such a function.
First, we deduce that Suppose, to derive a contradiction, that there is i ∈ [k 1 + 1, Then we can find some j ∈ [i, k 2 − 1] fulfilling Thus, setting , which is impossible. Now, we prove that arguing by contradiction. So, assume that k 1 > 1. Since we have u N (k 1 + 1) = −1 + α for some α > 0, we define w ∈ Z N by putting w(k 1 ) = −1 + α/2 and w(k) = u N (k) for k = k 1 and we get Similar arguments lead to Relations (8), (9) and (10)  By the Hölder inequality, we have . Now, the formula on u DN follows easily. Similar arguments we can apply to λ N D .
(iv) As in the proof of (ii), we show that λ P = inf u∈Z P u p p , where Z P = {u ∈ X P : max k∈ [1,T ] u(k) = − min k∈ [1,T ] u(k) = 1}. There exists u P ∈ Z P which minimizes · p p over Z P and there are 1 ≤ k 1 < k 2 ≤ T such that |u P (k 1 )| = |u P (k 2 )| = 1, Without restriction of generality, we can assume that u P (k 1 ) = −1 and u P (k 2 ) = 1. The analysis similar to that in the proof of (ii) shows first that u P is nondecreasing on [k 1 , k 2 ], next that u P is nonincreasing on where the equality holds if Hence, by the minimality property of u P , we must have  attains its minimum at n 0 = T +1 2 if T is odd and at n 0 = T 2 or, equivalently, n 0 = T +2 2 if T is even. This gives λ P in (iv). Moreover, to give an explicit formula for u P we can arbitrarily choose k 1 from [1, n 0 − 1], so if k 1 = 1, u P is given by the formula in (iv).