Existence of nonoscillatory solutions to neutral dynamic equations on time scales.(English)Zbl 1128.34043

The authors study the existence of a positive solution for the neutral functional dynamic equation on time scale
$[x(t)+p(t)x(g(t))]^{\Delta}+f(t,x(h(t))=0.\tag{1}$
Here is one of the results of the paper.
Theorem. Equation (1) has an eventually positive solution $$x(t)$$ with $$\lim_{t\rightarrow\infty}x(t)=a>0$$ if and only if there exists a constant $$K>0$$ such that $\int_{t_0}^{\infty}\Delta s <\infty.$

MSC:

 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations 39A10 Additive difference equations
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References:

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