Abstract
We show that the recessive solution of the second-order half-linear difference equation , , , where are real-valued sequences, is closely related to the divergence of the infinite series .
Highlights
We consider the second-order half-linear difference equationΔ rkΦ Δxk ckΦ xk 1 0, Φ x : |x|p−2x, p > 1, 1.1 where r, c are real-valued sequences and rk > 0, and we investigate properties of its recessive solution.Qualitative theory of 1.1 was established in the series of the papers of Rehak 1–5 and it is summarized in 6, Chapter 3
The reason why under assumptions in i or iii it is possible to formulate such a condition is that there is a substantial difference in asymptotic behavior of recessive and dominant solutions i.e., solutions which are linearly independent of the recessive solution
Denote by wk rkΦ Δhk/hk the associated solution of 2.1 and let wk be a solution of 2.1 generated by another solution linearly independent of h of 1.1. It follows from Lemma 3.1 that vk |hk|p wk − wk is a solution of 3.4, that is, vk 1 vk − H k, vk, 4.2 and suppose that this solution satisfies the condition vN < 0. This means that wN < wN and to prove that h is the recessive solution of 1.1, we need to show that there exists m ≥ N such that rm wm ≤ 0, that is, according to Lemma 3.1, vm vm∗ ≤ 0
Summary
Δ rkΦ Δxk ckΦ xk 1 0, Φ x : |x|p−2x, p > 1, 1.1 where r, c are real-valued sequences and rk > 0, and we investigate properties of its recessive solution. We will recall basic facts of the oscillation theory of 1.1 . There are several attempts in literature to find a summation characterization of this solution, see 8 and related references 9, 10 , which are based on the asymptotic analysis of solutions of 1.1. This approach requires the sign restriction of the sequence ck and additional assumptions on the convergence divergence of certain infinite series involving sequences r and c, see Proposition 2.1 .
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