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An existence theorem for a nonlinear difference equation. (English) Zbl 0774.39001

The authors deal with the following nonlinear difference equation \[ \Delta(\Delta y_{k-1})^{p-1}+s_ k y_ k^{p-1}=0, \qquad k=1,2,3,\dots \tag{1} \] where \(p>1\) and \(\{s_ k\}_ 1^ \infty\) is a real sequence. They establish conditions under which (1) has a positive nondecreasing solution. The main idea is to use the fixed point theorem of Schauder for the equation obtained from (1) by means of a Riccati-type transformation.

MSC:

39A10 Additive difference equations
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[2] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008
[3] Cheng, S. S.; Li, H. J.; Patula, W. T., Bounded and zero convergent solutions of second order difference equations, J. math. Analysis Applic., 141, 463-483 (1989) · Zbl 0698.39002
[4] Griffel, D. H., Applied Functional Analysis, Horwood Series Math. Applic. (1981), Chichester, U.K. · Zbl 0461.46001
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