ON ASYMPTOTICALLY EQUIVALENCE OF ORDER ? FOR SEQUENCE OF SETS USING ?

Abstract

This paper presents the following definitons which is a natural combination of the definition for asymptotically equivalent, lambda-statistical convergence and sigma-convergence. Two set sequences {A(k)} and {B-k} are said to be S-sigma,lambda(alpha)-asymptotically equivalent (Wijsman sense) of multiple L if for each epsilon > 0, for each x is an element of X, lim(n ->infinity)1/lambda(alpha)(n)vertical bar{k is an element of I-n : vertical bar d (x; A(sigma k(m)), B-sigma k(m) ) - L vertical bar >= epsilon }vertical bar = 0 uniformly in m = 1, 2, 3, ..., (denoted by {A(sigma k(m))} similar to(S sigma,lambda alpha) {B-sigma k(m)}). Also, we introduce the concept of S-sigma,theta(alpha)-asymptotically equivalent (Wijsman sense) of multiple L for the set sequences {A(k)} and {B-k} and give some inclusion relations. Using the definition of S-sigma,theta(alpha)-asymptotically equivalence, we shall prove S-sigma,theta(alpha)-asymptotically equivalent analogues of Fridy and Orhan's theorems in [11] and analogues results of Das and Patel in [8] shall also be presented.