A Tauberian theorem for the weighted mean method of improper integrals of fuzzy-number-valued functions

Abstract

Let 0 ≠ p (x) be a nondecreasing real valued function on [0, ∞) such that p (0) =0 and $\underset{x\to \infty }{liminf}\frac{p\left(\lambda x\right)}{p\left(x\right)}>1\mathrm{for}\mathrm{every}\lambda >1.$

Given a fuzzy-number-valued continuous function f (x) on [0, ∞), we define $s\left(x\right):={\int }_0^{x}f\left(t\right)\mathit{dt}\mathrm{and}\sigma \left(x\right):=\frac{1}{p\left(x\right)}{\int }_0^{x}s\left(t\right)\mathit{dp}\left(t\right),x>0.$

It is known that the limit $\underset{x\to \infty }{lim}s\left(x\right)=\mu$ exists, then the limit $\underset{x\to \infty }{lim}\sigma \left(x\right)=\mu$ also exists. But the converse of this implication need not be satisfied in general.

In this paper, our goal is to find a condition under which the existence of $\underset{x\to \infty }{lim}\sigma \left(x\right)=\mu$ follows from that of $\underset{x\to \infty }{lim}s\left(x\right)=\mu$ .

As special cases, we obtain some Tauberian conditions of slowly decreasing type and Landau type for the Cesàro summability method of improper integrals of fuzzy-number-valued functions.