(vi) For a converging sequence ((omega _{(n)})_{n ge 1}) of points in (Omega _+) with a limit (omega ) in (Omega _+), we have (lim _{n rightarrow infty } (G_{omega _{(n)}}, S_{omega _{(n)}}) = (G_omega, S_omega )) in (mathcal{M }_4).
It is said to be upper semi-continuous if for all t ≥ τ, the mapping U ( t, τ ) is upper semi-continuous from X into, that is to say, given a converging sequence x n → x, for some sequence { y n } such that y n ∈ U ( t, τ ) x n for all n, there exists a subsequence of { y n } converging in X to an element of U ( t, τ ) x. Lemma 2.4 [2.4.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in, the sequence converges to.
Let and be sequences in and let and be sequences in converging to Then, for the -distance on the following conditions hold for every : (a if and for any then in particular, if and then ; (b)if and for any then converges to ; (c)if for any with then is a Cauchy sequence; (d)if for any then is a Cauchy sequence.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in, then the sequence converges to.
for all It follows from (3.34) and (3.36) that the sequence is Cauchy for all Since is complete, the sequence converges for all So one can define the mapping by (3.37).
for all Thus we conclude from (3.67) and (3.69) that the sequence is Cauchy for all Since is complete, the sequence converges for all So one can define the mapping by (3.69).
Since is complete, the sequence converges in for all.
Since is bounded, a sequence converges to for any in whenever converges to.
Now we will show that a sequence converges to for each.
