On the spectral analysis of self adjoint operators generated by second order difference equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 139-151 ◽  
Author(s):  
Dale T. Smith
Keyword(s):  
Essential Spectrum ◽  
Difference Equations ◽  
Difference Operator ◽  
Simple Criterion ◽  
Order Difference ◽  
Forward Difference ◽  
Other Information ◽  
Image Position ◽  
Adjoint Operators ◽  
Oscillation Constant

SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

scholarly journals Oscillation criteria for certain second order nonlinear difference equations

1999 ◽  
Vol 60 (1) ◽  
pp. 95-108 ◽  
Author(s):  
S.R. Grace ◽  
H.A. El-Morshedy
Keyword(s):  
Difference Equations ◽  
Difference Operator ◽  
Nonnegative Function ◽  
Second Order ◽  
Real Sequence ◽  
Nonlinear Difference Equations ◽  
Oscillation Criteria ◽  
Forward Difference ◽  
Image Position

This paper is concerned with nonlinear difference equations of the formwhere δ is the forward difference operator defined by δun−1 = un − un −1 δ2un −1= δ(δun-1) and {an} is a real sequence which is not assumed to be nonnegative. The function f is such that uf(u) < 0 for all u ≠ 0 and f(u) − f(v) = g(u, v)(u − v), for all u, v ≠ 0, and for some nonnegative function g. Our results are not only new but also improve and generalise some recent oscillation criteria. Examples illustrating the importance of our main results are also given.

scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

On the essential spectrum of self-adjoint operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 261-274
Author(s):  
B. J. Harris
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Adjoint Operator ◽  
Matrix Differential ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe provide estimates of the formfor the length of gap centre μ in the essential spectrum of a self-adjoint operator generated by a matrix differential expression.

scholarly journals A stability condition for nth order difference equations

1971 ◽  
Vol 12 (1) ◽  
pp. 24-30
Author(s):  
Russell A. Smith
Keyword(s):  
Difference Equations ◽  
Hermitian Form ◽  
Real Variable ◽  
Necessary Condition ◽  
Quadratic Lyapunov Function ◽  
Globally Asymptotically Stable ◽  
Order Difference ◽  
Constant Coefficients ◽  
Image Position ◽  
Special Case

Consider the system of difference equationsin which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

Asymptotic behavior in neutral difference equations with negative coefficients

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
G. Chatzarakis ◽  
G. Miliaras
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Retarded Argument ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Deviated Argument

AbstractIn this paper, we study the asymptotic behavior of the solutions of a neutral difference equation of the form $\Delta [x(n) + cx(\tau (n))] - p(n)x)(\sigma (n)) = 0,$, where τ(n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ ℝ, (−p(n))n≥0 is a sequence of negative real numbers such that p(n) ≥ p, p ∈ ℝ+, and Δ denotes the forward difference operator Δx(n) = x(n+1)−x(n).

Non-self-adjoint difference operators and their spectrum

2005 ◽  
Vol 461 (2057) ◽  
pp. 1505-1531 ◽  
Author(s):  
Robert Howard Wilson
Keyword(s):  
Difference Operator ◽  
Discrete Analogue ◽  
Second Order ◽  
Difference Operators ◽  
Essential Difference ◽  
Order Difference ◽  
Second Order Differential Equations ◽  
Forward Difference ◽  
Difference Expression ◽  
Complex Coefficients

Initially, this paper is a discrete analogue of the work of Brown et al. (1999 Proc. R. Soc. A 455 , 1235–1257) on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expression where the coefficients p n and q n are complex and Δ is the forward difference operator, i.e. Δ x n = x n +1 − x n . Properties of the so-called Hellinger–Nevanlinna m -function for the recurrence relation Mx n = λ w n x n , where the w n are real and positive, are examined, and relationships between the properties of the m -function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

scholarly journals Oscillation for Super Linear/Linear Second Order Neutral Difference Equations with Variable Several Delays

2020 ◽  
Vol 5 (4) ◽  
pp. 663-681
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Several Delays

This article, is concerned with finding sufficient conditions for the oscillation and non oscillation of the solutions of a second order neutral difference equation with multiple delays under the forward difference operator, which generalize and extend some existing results.This could be possible by extending an important lemma from the literature.

Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

2021 ◽  
Vol 71 (4) ◽  
pp. 941-960
Author(s):  
Ajit Kumar Bhuyan ◽  
Laxmi Narayan Padhy ◽  
Radhanath Rath
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Unbounded Solution ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Different Types

Abstract In this article, sufficient conditions are obtained so that every solution of the neutral difference equation Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) = 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$ or every unbounded solution of Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) − u n H ( y α ( n ) ) = 0 , n ≥ n 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$ oscillates, where m=2 is any integer, Δ is the forward difference operator given by Δy n = y n+1 − y n ; Δ m y n = Δ(Δ m−1 y n ) and other parameters have their usual meaning. The non linear function L ∈ C (ℝ, ℝ) inside the operator Δ m includes the case L(x) = x. Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. Further, all the three possible cases, p n ≥ 0, p n ≤ 0 and p n changing sign, are considered. The results of this paper generalize and extend some known results.

scholarly journals Oscillation of solutions of impulsive neutral difference equations with continuous variable

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Gengping Wei ◽  
Jianhua Shen
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Continuous Variable ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
All Solutions ◽  
Positive Integers ◽  
Oscillation Of Solutions

We obtain sufficient conditions for oscillation of all solutions of the neutral impulsive difference equation with continuous variableΔτ(y(t)+p(t)y(t−mτ))+Q(t)y(t−lτ)=0,t≥t0−τ,t≠tk,y(tk+τ)−y(tk)=bky(tk),k∈ℕ(1), whereΔτdenotes the forward difference operator, that is,Δτz(t)=z(t+τ)−z(t),p(t)∈C([t0−τ,∞),ℝ),Q(t)∈C([t0−τ,∞),(0,∞)),m,lare positive integers,τ>0andbkare constants,0≤t0<t1<t2<⋯<tk<⋯withlimk→∞tk=∞.

scholarly journals Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
R. N. Rath ◽  
J. G. Dix ◽  
B. L. S. Barik ◽  
B. Dihudi
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Necessary Conditions ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
Neutral Delay ◽  
Forward Difference ◽  
Delay Difference Equations ◽  
Delay Difference

We find necessary conditions for every solution of the neutral delay difference equationΔ(rnΔ(yn−pnyn−m))+qnG(yn−k)=fnto oscillate or to tend to zero asn→∞, whereΔis the forward difference operatorΔxn=xn+1−xn, andpn, qn, rnare sequences of real numbers withqn≥0, rn>0. Different ranges of{pn}, includingpn=±1, are considered in this paper. We do not assume thatGis Lipschitzian nor nondecreasing withxG(x)>0forx≠0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.

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