Special cases of critical linear difference equations

Abstract

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

Higher order difference equations with homogeneous governing functions nonincreasing in each variable with unbounded solutions

“…Quite recently in [28] we have shown that the claim in Theorem 1 is not correct by presenting a counterexample in the class of difference equations with interlacing indices. The class of equations seems quite suitable for providing some counterexamples in the theory of difference equations (see, e.g., recent paper [4]).…”

“…Box 11099, Taif 21944, Saudi Arabia. 4 Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia. 5 Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, 36000 Kraljevo, Serbia.…”

Criticality of general two‐term even‐order linear difference equation via a chain of recessive solutions

“…The equations are then classified on the basis of their criticality. The high topicality of the investigated issue is evidenced by a number of recent articles on this topic [8,15,17,27], but the issue under investigation is also related to the rapidly developing theory of the critical constant [18][19][20][21][22] (in other time domains, see [23,24,50,52] and the references cited therein).…”

“…Among other important features, difference equations are related to the discretization of differential equations [6,27,29] (see also [32]). Nevertheless, the discrete calculus allows us to solve various problems directly [2, 29, chapter 1].…”

Closed‐form solutions of second‐order linear difference equations close to the self‐adjoint Euler type

“…See, for example, the concept of conditional oscillation and the oscillation constant in Hasil and Veselý. 8 We also refer the reader to see related articles, [9][10][11][12][13][14][15] and also in the continuous and time-scale domain, we recommend related references. [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] Furthermore, oscillation and other properties of perturbed equations are investigated in previous studies [31][32][33][34][35][36] as well.…”