The notions of statistical limit, limit inferior and limit superior of a measurable function at \documentclass{aastex}
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\(\infty\)
\end{document} were introduced by Móricz. These notions can be considered as the nondiscrete analogues of those introduced for sequences of numbers by H. Fast, J. A. Fridy and C. Orhan. Let \documentclass{aastex}
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\(0 \not \equiv p\: \mathbb{R}_+ \to \mathbb{R}_+\)
\end{document} be a nondecreasing function such that \documentclass{aastex}
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\(p(0)=0\)
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$$\mbox{st-\!}\liminf_{t \to \infty} \frac{p(\lambda t)}{p(t)} >1 \ \text{for every} \lambda >1.$$
\end{document} Given a real- or complex-valued function \documentclass{aastex}
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\(f \in L_{{\rm loc}}^1 (\mathbb{R}_+)\)
\end{document}, we define \documentclass{aastex}
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$$s(x):= \int^x_0 f(u) \, du\ \text{and}\ \sigma(t) := \frac{1}{p(t)} \int^t_0 s(x) d p(x),\quad t>0.$$
\end{document} Our goal is to find necessary and sufficient conditions under which the existence of \documentclass{aastex}
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\(\mbox{st-}\lim s(t)=l\)
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\(\mbox{st-}\lim \sigma(t)=l\)
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\(l\)
\end{document} is a finite number. In the case of real-valued functions we present one-sided Tauberian conditions, while in the case of complex-valued functions we present two-sided Tauberian conditions.
Pointwise stabilization of the Poisson integral for the diffusion type equations with inertia
