Bounded and zero convergent solutions of second-order difference equations. (English) Zbl 0698.39002

Cheng, Sui-Sun; Li, Horng-Jaan

J. Math. Anal. Appl. 141, No. 2, 463-483 (1989).

This paper is devoted to the study of the linear difference equation \((1)\quad a(k+1)x(k+2)-(a(k)+a(k+1)+b(k+1))x(k+1)+a(k)x(k)=0\) \((k=0,1,2,...)\) over the reals. Here \(a(k)>0\) for \(k\geq 0\) and b(k)\(\geq 0\) for \(k\geq 1\). Two linearly independent, eventually positive solutions y and z of (1) are called recessive and dominant, respectively, if \(\lim_{k\to \infty}(y(k)/z(k))=0\). For equation (1) results concerning the existence of recessive and dominant solutions, the monotonicity of solutions, the boundedness of solutions and the convergence of solutions to zero as well as comparison results are derived.

Reviewer: H.L nger

MSC:

39A10

Additive difference equations

Keywords:

second-order difference equations; recessive solution; linear difference equation; positive solutions; dominant solutions; monotonicity of solutions; boundedness of solutions; convergence; comparison results

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