scholarly journals Spectrum and fine spectrum of generalized second order forward difference operator Δuvw2 $\Delta _{uvw}^2 $ on sequence space l1

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
B. L. Panigrahi ◽  
P. D. Srivastava
Keyword(s):  
Sequence Space ◽  
Difference Operator ◽  
Second Order ◽  
Fine Spectrum ◽  
Forward Difference

AbstractThe purpose of this paper is to determine spectrum and fine spectrum of newly introduced operator

scholarly journals On the spectrum of 2-nd order generalized difference operator $\delta^2$ over the sequence space $c_0$

2013 ◽  
Vol 31 (2) ◽  
pp. 235 ◽  
Author(s):  
S. Dutta ◽  
Pinakadhar Baliarsingh
Keyword(s):  
Sequence Space ◽  
Difference Operator ◽  
Second Order ◽  
Order Difference ◽  
Fine Spectrum

The main purpose of  this article is to  determine the spectrum and the fine spectrum  of second order  difference operator $\Delta^2$  over the sequence space $c_0$. For any sequence $(x_k)_0^\infty$ in $c_0$, the generalized second order  difference operator $\Delta^2$  over  $c_0$ is defined by $\Delta^2(x_k)= \sum_{i=0}^2(-1)^i\binom{2}{i}x_{k-i}=x_k-2x_{k-1}+x_{k-2}$, with $ x_{n}  = 0$ for $n<0$.Throughout we use the convention that a term with a negative subscript is equal to zero.

scholarly journals On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence space $\ell_1$ and $bv$

2006 ◽  
Vol 35 (4) ◽  
pp. 893-904 ◽  
Author(s):  
Hasan FURKAN ◽  
H\"usey\.in B\.ILG\.I\C C ◽  
Kuddus\.i KAYADUMAN
Keyword(s):  
Sequence Space ◽  
Difference Operator ◽  
Fine Spectrum

scholarly journals On the Spectral Properties of the Weighted Mean Difference Operator Gu,v;Δ over the Sequence Space ℓ1

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
P. Baliarsingh ◽  
S. Dutta
Keyword(s):  
Sequence Space ◽  
Difference Operator ◽  
Point Spectrum ◽  
Weighted Mean Difference ◽  
Mean Difference ◽  
Weighted Mean ◽  
Residual Spectrum ◽  
Fine Spectrum ◽  
Special Cases ◽  
The Difference

In the present work the generalized weighted mean difference operator Gu,v;Δ has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences u=(uk) and v=(vk). For any two sequences u and v of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator Gu,v;Δ is defined by (G(u,v;Δ)x)k=∑i=0k‍ukvi(xi-xi-1) with xk=0 for all k<0. Furthermore, we compute the spectrum and the fine spectrum of the operator Gu,v;Δ over the sequence space l1. In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space l1.

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.

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.

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