On a mean method of summability

Let $p(x)$ be a nondecreasing real-valued continuous function
 on $R_+:=[0,\infty)$ such that $p(0)=0$ and $p(x) \to \infty$ as $x \infty$.
 Given real or complex-valued integrable function $f$ in Lebesgue's sense every bounded interval $(0,x)$
 for $x>0$, symbol $f \in L^1_{loc} (R_+)$, we set
 $$
 s(x)=\int _{0}^{x}f(u)du
 and
 \sigma _{p}(s(x))=\frac{1}{p(x)}\int_{0}^{x}s(u)dp(u),\,\,\,\,x>0
 provided $p(x)>0$.
 
 A $s(x)$
 is said to summable $l$ by the weighted mean method determined
 $p(x)$, short, $(\overline{N},p)$ $l$,
 if
 \lim_{x \infty}\sigma _{p}(s(x))=l.
 If limit $\lim _{x \infty} s(x)=l$
 exists, then _{p}(s(x))=l$ also exists. However, converse not true general.
 In this paper, give an alternative proof Tauberian theorem stating convergence follows from summability condition of slowly decreasing type with respect weight due Karamata. These conditions are one-sided two-sided if $f(x)$ function, respectively. Alternative proofs some well-known theorems given several important methods can obtained choosing particular functions.