Oscillation and nonoscillation criteria for second-order nonlinear difference equations of Euler type

AbstractIn this study, we are concerned with a boundary value problem (BVP) for nonlinear difference equations on the set of all integers Z. under the assumption that the left-hand side is a second-order linear difference expression which belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l2 and includes boundary conditions at infinity. Existence and uniqueness results for solution of the considered BVP are established.

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MODIFIED OSCILLATION CRITERIA OF SECOND-ORDER NEUTRAL DIFFERENCE EQUATIONS

Functional Differential Equations ◽

10.26351/fde/28/1-2/2 ◽

2021 ◽

Vol 28 (1-2) ◽

pp. 19-30

Author(s):

G. CHATZARAKIS G. CHATZARAKIS ◽

R. KANAGASABAPATHI R. KANAGASABAPATHI ◽

S. SELVARANGAM S. SELVARANGAM ◽

E. THANDAPANI E. THANDAPANI

Keyword(s):

Difference Equations ◽

Second Order ◽

Riccati Transformation ◽

Nonlinear Difference Equations ◽

Transformation Technique ◽

Neutral Difference Equations ◽

Oscillation Criteria ◽

Neutral Term

In this paper we shall consider a class of second-order nonlinear diﬀerence equations with a negative neutral term. Some new oscillation criteria are obtained via Riccati transformation technique. These criteria improve and modify the existing results mentioned in the literature. Some examples are given to show the applicability and signiﬁcance of the main results.

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Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

Georgian Mathematical Journal ◽

10.1515/gmj.1999.401 ◽

1999 ◽

Vol 6 (5) ◽

pp. 401-414

Author(s):

T. Chantladze ◽

N. Kandelaki ◽

A. Lomtatidze

Keyword(s):

Linear Equation ◽

Integrable Function ◽

Second Order ◽

Order Linear ◽

Locally Integrable Function ◽

Oscillation And Nonoscillation Criteria ◽

Oscillation And Nonoscillation

Abstract New oscillation and nonoscillation criteria are established for the equation 𝑢″ + 𝑝(𝑡)𝑢 = 0, where 𝑝 : ]1, + ∞[ → 𝑅 is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

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Weighted Averaging Techniques in Oscillation Theory for Second Order Difference Equations

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-009-9 ◽

1992 ◽

Vol 35 (1) ◽

pp. 61-69 ◽

Cited By ~ 13

Author(s):

Lynn H. Erbe ◽

Pengxiang Yan

Keyword(s):

Oscillation Theory ◽

Second Order ◽

Weighted Averaging ◽

Real Numbers ◽

Order Difference ◽

The Matrix ◽

System 2 ◽

Oscillation And Nonoscillation Criteria ◽

Oscillation And Nonoscillation ◽

Averaging Techniques

AbstractWe consider the self-adjoint second-order scalar difference equation (1) Δ(rnΔxn) +pnXn+1 = 0 and the matrix system (2) Δ(RnΔXn) + PnXn+1 = 0, where are seQuences of real numbers (d x d Hermitian matrices) with rn > 0(Rn > 0). The oscillation and nonoscillation criteria for solutions of (1) and (2), obtained in [3, 4, 10], are extended to a much wider class of equations by Riccati and averaging techniques.

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