Iterative Oscillation Criteria in Deviating Difference Equations
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Iterative Oscillation Criteria in Deviating Difference Equations George E. Chatzarakis and Irena Jadlovsk´a Abstract. The oscillation of the first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients is investigated, using an iterative procedure. The conditions obtained by this method achieve a marked improvement on all known conditions in the literature. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the results. Mathematics Subject Classification. 39A10, 39A21. Keywords. Difference equation, non-monotone arguments, oscillatory solutions, nonoscillatory solutions.
1. Introduction Consider the difference equation with several retarded arguments of the form: m pi (n)x(τi (n)) = 0, n ∈ N0 , (ER ) Δx(n) + i=1
where N0 is the set of nonnegative integers, and the (dual) difference equation with several advanced arguments of the form: m qi (n)x(σi (n)) = 0 , n ∈ N, (EA ) ∇x(n) − i=1
where N is the set of positive integers. Equations (ER ) and (EA ) are studied under the following assumptions: everywhere (pi (n))n≥0 , (qi (n))n≥1 , 1 ≤ i ≤ m, are sequences of nonnegative real numbers, (τi (n))n≥0 , (σi (n))n≥1 , 1 ≤ i ≤ m, are sequences of integers, such that: τi (n) ≤ n − 1, ∀n ∈ N0 and lim τi (n) = ∞, 1 ≤ i ≤ m
(1.1)
σi (n) ≥ n + 1, ∀n ∈ N, 1 ≤ i ≤ m,
(1.2)
n→∞
and respectively. 0123456789().: V,-vol
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G. E. Chatzarakis and I. Jadlovská
MJOM
Here, as usual, Δ denotes the forward difference operator Δx(n) = x(n+ 1) − x(n) and ∇ corresponds to the backward difference operator ∇x(n) = x(n) − x(n − 1). Set: v = − min τi (n). n≥0 1≤i≤m
Clearly, v is a finite positive integer if (1.1) holds. By a solution of (ER ), we mean a sequence of real numbers (x(n))n≥−v which satisfies (ER ), for all n ≥ 0. It is clear that, for each choice of real numbers c−v , c−v+1 , ..., c−1 , c0 , there exists a unique solution (x(n))n≥−v of (ER ) which satisfies the initial conditions x(−v) = c−v , x(−v+1) = c−v+1 , ..., x(−1) = c−1 , x(0) = c0 . When the initial data are given, we can obtain a unique solution to (ER ) using the method of steps. By a solution of (EA ), we mean a sequence of real numbers (x(n))n≥0 which satisfies (EA ) for all n ≥ 1. A solution (x(n))n≥−v (or (x(n))n≥0 ) of (ER ) (or (EA )) is called oscillatory if the terms x(n) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory. The last few decades, the oscillatory behavior, stability, and existence of positive solutions of equations (ER ) and (EA ) have been the subject of several studies. See, for example, [1–18] and the references cited therein. Most of these papers though are concerned with the special case where the arguments are nondecreasing, while merely a small number of papers are dealing with the general case where the arguments are not necessarily monotone, see, e.g., [2–5,8]. Therefore, an interesting question arising is whether we can state oscillation criteria
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