Abstract
This article is concerned with the defect indices of singular symmetric linear difference equations of order 2n with complex coefficients and one singular endpoint. We first show that the positive and negative defect indices d+ and d- of a class of singular symmetric linear difference equations of order 2n with complex coefficients satisfy the inequalities n ≤ d+ = d- ≤ 2n and all values of this range are realized. This extends the result for difference equations with real coefficients. In addition, some sufficient conditions for the limit point and the strong limit point cases are given. AMS Classification: 39A70; 34B20.
Highlights
Atkinson first studied the number of linearly independent square summable solutions of second-order symmetric linear difference equations with real coefficients [9]
We study the positive and negative defect indices of Equation (1.1)
We introduce the following space: l2w(I) := y = {y(t)}+t=∞−n ⊂ C : w(t) y(t) 2 < +∞
Summary
1 Introduction In this article, we are interested in the positive and negative defect indices of the following singular symmetric linear difference equation with complex coefficients: n n Atkinson first studied the number of linearly independent square summable solutions of second-order symmetric linear difference equations with real coefficients [9]. It has been shown that the positive and negative defect indices of second-order symmetric difference equations with complex coefficients are still equal; that is (d+, d-) = (1, 1) or (2, 2).
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