Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number

• Дата: 2021
• Автори: Önder, Z., Çanak, İ., Önder, Zerrin, Çanak, İbrahim
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2021
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/584
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and$$P_n:=\sum_{k=0}^{n}p_k\to\infty\qquad \text{as}\qquad n\to\infty.$$Let $(u_n)$ be a sequence of fuzzy numbers.The weighted mean of $(u_n)$ is defined by$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_ku_k\qquad \text{for}\qquad n =0,1,2,\ldots \,. $$It is known that the existence of the limit $\lim u_n=\mu_{0}$ implies that of $\lim t_n=\mu_{0}.$ For the the existence of the limit $st$-$\lim t_n=\mu_{0},$ we require the boundedness of $(u_n)$ in addition to the existence of the limit $\lim u_n=\mu_{0}.$ But, in general, the converse of this implication is not true. In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=\mu_{0}$ follows from that of $\lim t_n=\mu_{0}$ or $st$-$\lim t_n=\mu_{0}.$ These Tauberian conditions are satisfied if $(u_n)$ satisfies the two-sided condition of Hardy type relative to $(P_n).$