Existence criteria for positive solutions of a nonlinear difference inequality

Sui Cheng; Guang Zhang

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 3, page 197-220
  • ISSN: 0066-2216

Abstract

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This paper is concerned with a class of nonlinear difference inequalities which include many different classes of difference inequalities and equations as special cases. By means of a Riccati type transformation, necessary and sufficient conditions for the existence of eventually positive solutions and positive nonincreasing solutions are obtained. Various type of comparison theorems are also derived as applications, which extends many theorems in the literature.

How to cite

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Cheng, Sui, and Zhang, Guang. "Existence criteria for positive solutions of a nonlinear difference inequality." Annales Polonici Mathematici 73.3 (2000): 197-220. <http://eudml.org/doc/262826>.

@article{Cheng2000,
abstract = {This paper is concerned with a class of nonlinear difference inequalities which include many different classes of difference inequalities and equations as special cases. By means of a Riccati type transformation, necessary and sufficient conditions for the existence of eventually positive solutions and positive nonincreasing solutions are obtained. Various type of comparison theorems are also derived as applications, which extends many theorems in the literature.},
author = {Cheng, Sui, Zhang, Guang},
journal = {Annales Polonici Mathematici},
keywords = {Sturm type comparison theorem; limit comparison theorem; nth order linear difference equation; eventually positive solution; eventually positive nonincreasing solution; neutral difference equation; nonlinear delay difference inequality; nonlinear difference inequalities; Riccati type transformation; positive solutions; nonincreasing solutions; comparison theorems},
language = {eng},
number = {3},
pages = {197-220},
title = {Existence criteria for positive solutions of a nonlinear difference inequality},
url = {http://eudml.org/doc/262826},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Cheng, Sui
AU - Zhang, Guang
TI - Existence criteria for positive solutions of a nonlinear difference inequality
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 3
SP - 197
EP - 220
AB - This paper is concerned with a class of nonlinear difference inequalities which include many different classes of difference inequalities and equations as special cases. By means of a Riccati type transformation, necessary and sufficient conditions for the existence of eventually positive solutions and positive nonincreasing solutions are obtained. Various type of comparison theorems are also derived as applications, which extends many theorems in the literature.
LA - eng
KW - Sturm type comparison theorem; limit comparison theorem; nth order linear difference equation; eventually positive solution; eventually positive nonincreasing solution; neutral difference equation; nonlinear delay difference inequality; nonlinear difference inequalities; Riccati type transformation; positive solutions; nonincreasing solutions; comparison theorems
UR - http://eudml.org/doc/262826
ER -

References

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  1. [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992. Zbl0925.39001
  2. [2] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. Zbl0780.34048
  3. [3] I. Györi and M. Pituk, Asymptotic formulae for the solutions of a linear delay difference equation, J. Math. Anal. Appl. 195 (1995), 376-392. Zbl0846.39003
  4. [4] B. S. Lalli, Oscillation theorems for neutral difference equations, Comput. Math. Appl. 28 (1994), 191-202. Zbl0807.39004
  5. [5] B. S. Lalli, B. G. Zhang and J. Z. Li, On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl. 158 (1991), 213-233. Zbl0732.39002
  6. [6] J. Popenda, The oscillation of solutions of difference equations, Comput. Math. Appl. 28 (1994), 271-279. Zbl0807.39006
  7. [7] J. Yan, Oscillation of solutions of first order delay differential equations, Nonlinear Anal. 11 (1987), 1279-1287. Zbl0639.34067
  8. [8] J. Yan and C. Qian, Oscillation and comparison results for delay difference equations, J. Math. Anal. Appl. 165 (1992), 346-360. Zbl0818.39002
  9. [9] J. S. Yu, A note on 'Oscillation of solutions of first order delay differential equations', J. Math. Res. Exposition 10 (1990), 290-293. Zbl0772.34052
  10. [10] B. G. Zhang and S. S. Cheng, Oscillation criteria and comparison theorems for delay difference equations, Fasc. Math. 25 (1995), 13-32. Zbl0830.39005

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