A real sequence {xk} is said to be (∗)-monotone with respect to a sequence {pk} and a positive integer σ if xk > 0 and (−1)∆(xk − pkxk−σ) ≥ 0 for n ≥ 0. This paper is concerned with the existence of (∗)-monotone solutions of a neutral difference equation. Existence criteria are derived by means of a comparison theorem and by establishing explicit existence criteria for positive and/or bounded s...

A dynamical system is called positive if any solution of the system starting from nonnegative states maintains nonnegative states forever. In many applications where variables represent nonnegative quantities we often encounter positive dynamical systems as mathematical models see 1, 2 , and many researches for positive systems have been done actively; for recent developments see, for example, ...

For a class of Lucas sequences {xn}, we show that if n is a positive integer then xn has a primitive prime factor which divides xn to an odd power, except perhaps when n = 1, 2, 3 or 6. This has several desirable consequences.

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers {Fn}, where F1 = 1, F2 = 2 and Fn+1 = Fn + Fn−1. For general recurrences {Gn} with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uni...