Consider the rst-order linear di erential equation with several retarded argumentswhere the functions, τ k (t) < t for t ≥ t and lim t→∞ τ k (t) = ∞, for every k = , , . . . , n.Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.
We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.
It is known that all solutions of the difference equation $$\Delta x(n)+p(n)x(n-k)=0, \quad n\geq0, $$ Δ x ( n ) + p ( n ) x ( n − k ) = 0 , n ≥ 0 , where $\{p(n)\}_{n=0}^{\infty}$ { p ( n ) } n = 0 ∞ is a nonnegative sequence of reals and k is a natural number, oscillate if $\liminf_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac {k}{k+1} ) ^{k+1}$ lim inf n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . In the case that $\sum_{i=n-k}^{n-1}p(i)$ ∑ i = n − k n − 1 p ( i ) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with $\limsup_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac{k}{k+1} ) ^{k+1}$ lim sup n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . An example illustrating the applicability and importance of the result is presented.
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