Existence of positive nondecreasing solutions of nonlinear difference equations. (English) Zbl 0805.39004

Li, Horng Jaan; Yeh, Cheh Chih

Nonlinear Anal., Theory Methods Appl. 22, No. 10, 1271-1284 (1994).

The paper is concerned with the nonlinear difference equation (1) \((\Delta y_ k)^{p-1} - (\Delta y_{k - 1})^{p - 1} + s_ ky_ k^{p - 1} = 0\), \(k = 1,2, \dots\) where \(s_ k \geq 0\), \(p>1\). The authors establish some sufficient and/or necessary conditions for equation (1) to have a positive nondecreasing solution. A nontrivial solution \(\{y_ k\}^ \infty_{k = 0}\) of (1) is said to be oscillatory if for every positive integer \(N\) there exists \(n \geq N\) such that \(y_ n y_{n-1} \leq 0\). Oscillation and nonoscillation criteria for solutions of (1) are given.

Reviewer: V.I.Tkachenko (Kiev)

MSC:

39A10

Additive difference equations

Keywords:

oscillatory solution; nonlinear difference equation; positive nondecreasing solution; nonoscillation criteria

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