Home Properties of critical and subcritical second order self-adjoint linear equations
Article
Licensed
Unlicensed Requires Authentication

Properties of critical and subcritical second order self-adjoint linear equations

  • Jan Jekl
Published/Copyright: October 4, 2021
Become an author with De Gruyter Brill

Abstract

We discuss critical and subcritical linear second-order difference equations, and we observe several identities and inequalities which such equations satisfy depending on their coefficients. Later, we investigate the limit behaviour depending on the coefficients of solutions and of the sequences which appear when finding said solutions. We will see that certain identity is preserved in limit under weaker assumptions. Finally, we investigate a class of fourth-order linear difference equations and show that they are always 1-critical.


This research is supported by Czech Science Foundation under Grant GA20-11846S and by Masaryk University under Grant MUNI/A/0885/2019.


  1. (Communicated by Michal Fečkan)

References

[1] Chatzarakis, G. E.—Dmitrović, L. H.—Pašić, M.: Oscillation tests for difference equations with several non-monotone deviating arguments, Math. Slovaca 68(5) (2018), 1083–1096.10.1515/ms-2017-0170Search in Google Scholar

[2] Došlý, O.—Řehák, P.: Nonoscillation criteria for half-linear second-order difference equations, Comput. Math. Appl. 42(3) (2001), 453–464.10.1016/S0898-1221(01)00169-9Search in Google Scholar

[3] Došlý, O.—Řehák, P.: Recessive solution of half-linear second order difference equations, J. Difference Equ. Appl. 9(1) (2003), 49–61.10.1080/10236100309487534Search in Google Scholar

[4] Došlý, O.—Hasil, P.: Critical higher order Sturm-Liouville difference operators, J. Difference Equ. Appl. 17(9) (2011), 1351–1363.10.1080/10236190903527251Search in Google Scholar

[5] Došlý, O.—Komenda, J.: Conjugacy criteria and principal solutions of self-adjoint differential equations, Arch. Math. (Brno) 31(3) (1995), 217–238.Search in Google Scholar

[6] Došlý, O.—Růžička, V.: Nonoscillation of higher order half-linear differential equations, Electron. J. Qual. Theory Differ. Equ. 19 (2015), 1–15.10.14232/ejqtde.2015.1.19Search in Google Scholar

[7] Došlý, O.—Růžička, V.: Nonoscillation criteria and energy functional for even-order half-linear two-term differential equations, Electron. J. Qual. Theory Differ. Equ. 95 (2016), 1–17.10.14232/ejqtde.2015.1.19Search in Google Scholar

[8] Gesztesy, F.—Zhao, Z.: On critical and subcritical Sturm-Liouville operators, J. Funct. Anal. 98(2) (1991), 311–345.10.1016/0022-1236(91)90081-FSearch in Google Scholar

[9] Gesztesy, F.—Zhao, Z.: Critical and subcritical jacobi operators defined as friedrichs extensions, J. Differential Equations 103(1) (1993), 68–93.10.1006/jdeq.1993.1042Search in Google Scholar

[10] Gesztesy, F.—Zhao, Z.: On positive solutions of critical Schrödinger operators in two dimensions, J. Funct. Anal. 127(1) (1995), 235–256.10.1006/jfan.1995.1010Search in Google Scholar

[11] Grigorian, G. A.: Oscillatory criteria for the second order linear ordinary differential equations, Math. Slovaca 69(4) (2019), 857–870.10.1515/ms-2017-0274Search in Google Scholar

[12] Hooker, J. W.—Patula, W. T.: Riccati type transformations for second-order linear difference equations, J. Math. Anal. Appl. 82(2) (1981), 451–462.10.1016/0022-247X(81)90208-0Search in Google Scholar

[13] Hasil, P.: On positivity of the three term 2n-order difference operators, Studies of the University of Žilina, Mathematical Series 23(1) (2009) 51–58.Search in Google Scholar

[14] Hasil, P.: Criterion of p-criticality for one term 2n-order difference operators, Arch. Math. (Brno) 47(2) (2011), 99–109.Search in Google Scholar

[15] Hasil, P.—Zemánek, P.: Critical second order operators on time scales, Discrete Contin. Dyn. Syst. Series A (2011), 653–659.Search in Google Scholar

[16] Hasil, P.: Conjugacy of self-adjoint difference equations of even order, Abstr. Appl. Anal. 2011 (2011), Art. ID 814962.10.1155/2011/814962Search in Google Scholar

[17] Hasil, P.—Veselý, M.: Critical oscillation constant for difference equations with almost periodic coefficients, Abstr. Appl. Anal. 2012 (2012), Art. ID 471435.10.1155/2012/471435Search in Google Scholar

[18] Hasil, P.—Veselý, M.: Oscillation and non-oscillation criteria for linear and half-linear difference equations, J. Math. Anal. Appl. 452(1) (2017), 401–428.10.1016/j.jmaa.2017.03.012Search in Google Scholar

[19] Hasil, P.—Juránek, J.—Veselý, M.: Non-oscillation of half-linear difference equations with asymptotically periodic coefficients, Acta Math. Hungar. 159(1) (2019), 323–348.10.1007/s10474-019-00940-7Search in Google Scholar

[20] Ichihara, N.: Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control, J. Math. Pures Appl. 100(3) (2013), 368–390.10.1016/j.matpur.2013.01.005Search in Google Scholar

[21] Jekl, J.: Linear even order homogenous difference equation with delay in coefficient, Electron. J. Qual. Theory Differ. Equ. 45 (2020), 1–19.10.14232/ejqtde.2020.1.45Search in Google Scholar

[22] Kelley, W. G.—Peterson, A. C.: Difference Equations: an Introduction with Applications, 2nd ed., United States: Academic Press, 2001.Search in Google Scholar

[23] Kratz, W.: Banded matrices and difference equations, Linear Algebra Appl. 337(1) (2001), 1–20.10.1016/S0024-3795(01)00328-7Search in Google Scholar

[24] Lucia, M.—Prashanth, S.: Criticality theory for Schrödinger operators with singular potential, J. Differential Equations 265(8) (2018), 3400–3440.10.1016/j.jde.2018.05.006Search in Google Scholar

[25] Murata, M.: Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal. 56(3) (1984), 300–310.10.1016/0022-1236(84)90079-XSearch in Google Scholar

[26] Migda, J.—Migda, M.—Zdanowicz, M.: Asymptotic properties of solutions to fourth order difference equations, J. Comput. Appl. Math. 362 (2019), 68–82.10.1016/j.cam.2019.05.018Search in Google Scholar

[27] Migda, M.—Dutkiewicz, A.: Asymptotic behavior of solutions of second-order difference equations of Volterra type, Turkish J. Math. 43(5) (2019), 2203–2217.10.3906/mat-1904-30Search in Google Scholar

[28] Migda, J.: Asymptotic properties of solutions to second-order difference equations, Turkish J. Math. 44(1) (2020), 19–30.10.3906/mat-1908-69Search in Google Scholar

[29] Patula, W. T.: Growth and oscillation properties of second order linear difference equations, SIAM J. Math. Anal. 10(1) (1979), 55–61.10.1137/0510006Search in Google Scholar

[30] Pinchover, Y.: Criticality and ground states for second-order elliptic equations, J. Differential Equations 80(2) (1989), 237–250.10.1016/0022-0396(89)90083-1Search in Google Scholar

[31] Pinchover, Y.: On criticality and ground states of second order elliptic equations, II, J. Differential Equations 87(2) (1990), 353–364.10.1016/0022-0396(90)90007-CSearch in Google Scholar

[32] Řehák, P.: Asymptotic formulae for solutions of linear second-order difference equations, J. Difference Equ. Appl. 22(1) (2016), 107–139.10.1080/10236198.2015.1077815Search in Google Scholar

[33] Řehák, P.: An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation, J. Math. Anal. Appl. 454(2) (2017), 829–882.10.1016/j.jmaa.2017.05.034Search in Google Scholar

[34] Stević, S.: Growth theorems for homogeneous second-order difference equations, ANZIAM J. 43 (2002), 559–566.10.1017/S1446181100012141Search in Google Scholar

[35] Stević, S.: Asymptotic behaviour of second-order difference equations, ANZIAM J. 46 (2004), 157–170.10.1017/S1446181100013742Search in Google Scholar

[36] Stević, S.: Growth estimates for solutions of nonlinear second-order difference equations, ANZIAM J. 46(3) (2005), 439–448.10.1017/S1446181100008361Search in Google Scholar

[37] Veselý, M.—Hasil, P.: Criticality of one term 2n-order self-adjoint differential equations, Electron. J. Qual. Theory Differ. Equ. 18 (2012), 1–12.10.14232/ejqtde.2012.3.18Search in Google Scholar

Received: 2020-07-08
Accepted: 2020-11-13
Published Online: 2021-10-04
Published in Print: 2021-10-26

© 2021 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
  3. Polynomial functions on rings of dual numbers over residue class rings of the integers
  4. Sufficient conditions for p-valent functions
  5. Upper bounds for analytic summand functions and related inequalities
  6. Global structure for a fourth-order boundary value problem with sign-changing weight
  7. On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
  8. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
  9. Properties of critical and subcritical second order self-adjoint linear equations
  10. Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
  11. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
  12. Poisson like matrix operator and its application in p-summable space
  13. On the homological and algebraical properties of some Feichtinger algebras
  14. Disjoint topological transitivity for weighted translations generated by group actions
  15. On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
  16. Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
  17. The shifted Gompertz-G family of distributions: Properties and applications
  18. On the testing hypothesis in uniform family of distributions with nuisance parameter
  19. Clarkson inequalities related to convex and concave functions
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2021-0045/html
Scroll to top button