Since f is a modulus function, so equation (3.1) gives d ¯ ( X i u, X i v ) < ε 1 for all u, v ≥ u 0, for some ε 1 such that ε > ε 1 > 0 and for each i = 1, 2, …, m, i.e., ( X i u ) is a Cauchy sequence in L ( R ) for each i = 1, 2, …, m. (3.3).
Similarly, as f is a modulus function, so by choosing suitable ε 2 > 0, equation (3.2) gives d ¯ ( Δ m X k u, Δ m X k v ) < ε 2 for all u, v ≥ u 0 and for each k, i.e., ( Δ m X k u ) is a Cauchy sequence in L ( R ) for each k ∈ N. (3.4).
f (: [ 0,infty ) rightarrow [ 0,infty ) ) is called a modulus function if 1. (f ( x ) =0) if and only if (x=0), 2. (f ( x+y ) leq f ( x)+f y ) ) for every (x,yin mathbb{R}^), 3. f is increasing, 4. f is continuous from the right at 0. .
Theorem 2.1 Let f be any modulus function and let there be a φ-function φ and a generalized three parametric real matrix A; let p = ( p n ) be a sequence of positive real numbers and the sequence θ be given.
In the event that convexity of M is supplanted by (M x+y)leq M (x) + M (y)), it is known as a modulus function, presented by Nakano [10].
