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A sharp oscillation criterion for a difference equation with constant delay
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DOI：10.1186/s13662-020-03016-x.png)
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Abstract
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It is known that all solutions of the difference equation Delta x(n)+p(n)x(n-k)=0,n >= 0, where {p(n)}n=0 infinity is a nonnegative sequence of reals and k is a natural number, oscillate if liminfn -> infinity Sigma i=n-kn-1p(i)>(kk+1)k+1. In the case that Sigma i=n-kn-1p(i) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with limsupn -> infinity Sigma i=n-kn-1p(i)>(kk+1)k+1. An example illustrating the applicability and importance of the result is presented.
Keywords:
Oscillation
Delay
Difference equations
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