Difference inequalities, comparison tests, and some consequences

Abstract

Difference inequalities, comparison tests, and some consequences We study the behavior of nonnegative sequences which satisfy certain difference inequalities. Several comparison tests involving difference inequalities are developed for nonnegative sequences. Using the aforementioned comparison tests, it is possible to determine the global stability and boundedness character for nonnegative solutions of particular rational difference equations in a range of their parameters.

On the characterization of rational difference equations

“…First, we shall remove the assumption gcd is a solution of Equation (6). So that no confusion arises, we label the sets of indices for Equation (6) I * b and I * B and reserve the usual notation for our original difference equation.…”

Unboundedness results for fourth order rational difference equations

“…[7]. We set z m ¼ x mpþb for m [ N. As we have just shown, {z m } satisfies the following difference inequality,…”

“…See Refs [3,7] for a discussion on the technique of boundedness by iteration. In the following theorem, we use the technique of iteration in order to obtain bounds for certain subsequences of the solutions which we study.…”

“…The idea for Theorem 2 came from attempts to combine the methods used in Ref. [2] with the technique of iteration developed in Refs [3,7]. Furthermore, Theorem 3 is a direct generalization of methods used in Refs [4,5].…”

Unboundedness results for systems

“…We now rename this subsequence and apply the methods used in [4]. We set = L for ∈ N. As we have just shown { } satisfies the following difference inequality,…”