×

On existence of positive solutions and bounded oscillations for neutral difference equations. (English) Zbl 0763.39002

The authors consider the linear homogeneous difference equation \(\Delta^ \alpha(x_ n-cx_{n-m})+p_ n x_{n-k}=0\), where \(c,p_ n\geq 0\). Here \(\Delta\) is the forward difference operator, \(\Delta x_ n=x_{n+1}-x_ n\), and \(\alpha=1,2\). It is shown that positive solutions tending to zero or \(\infty\) or remaining bounded, or oscillatory solutions, respectively, exist under appropriate conditions on \(c\) and \(p_ n\).
Reviewer’s remark: The right-hand sides of equations (2.17) and (2.20) should read \((n+3)/(n^ 2(n+1))\) and \((n+3)/(n(n+1))\), respectively, for these equations to have the indicated solutions.

MSC:

39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brayron, R. K.; Willoughby, R. A., On the numerical integration of a symmetric system of difference differential equations of neutral type, J. Math. Anal. Appl., 18, 182-189 (1967) · Zbl 0155.47302
[2] Erbe, L. H.; Zhang, B. G., Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2, No. 3, 300-309 (1989) · Zbl 0723.39004
[3] Gedrgion, D. A.; Grove, E. A.; Ladas, G., Oscillations of neutral difference equations, Appl. Anal., 33, Nos. 3-4, 243-253 (1989) · Zbl 0685.39003
[4] Gyori, I.; Ladas, G., Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations, 2, 123-131 (1989) · Zbl 0723.34058
[5] Ladas, G., Explicit conditions for the oscillation of difference equations, J. Math. Anal. Appl., 153, 276-287 (1990) · Zbl 0718.39002
[6] Ladas, G., Recent developments in the oscillations of delay difference equations, (Differential Equations: Stability and Control (1990), Dekker: Dekker New York) · Zbl 0731.39002
[7] Ladas, G.; Philos, C. G.; Sficas, Y. G., Necessary and sufficient conditions for the oscillation of difference equations, Libertas Math., 9, 121-125 (1989) · Zbl 0689.39002
[8] G. Ladas, C. G. Philos, and Y. G. SficasJ. Appl. Math. Simulation; G. Ladas, C. G. Philos, and Y. G. SficasJ. Appl. Math. Simulation · Zbl 0685.39004
[9] Lalli, B. S.; Zhang, B. G.; Zhao, Li Jan, On oscillations and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl., 158, 213-233 (1991) · Zbl 0732.39002
[10] Jurang Yan and Chuanxi QianJ. Math. Anal. Appl.; Jurang Yan and Chuanxi QianJ. Math. Anal. Appl.
[11] Moore, R. E., Computational Functional Analysis (1985), Ellis Harwood · Zbl 0574.46001
[12] Agarwal, R. P., Difference calculus with applications to difference equations, (International Series of Numerical Mathematics, Vol. 71 (1984), Birkhauser-Verlag: Birkhauser-Verlag Basel), 95-110
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.