If [mathematical expression not reproducible] is a sequence of nested soft

closed intervals satisfying the properties [I.sub.1] [contains or equal to] [I.sub.2] [contains or equal to] ...

Kosol, "Rate of convergence of S-iteration and Ishikawa iteration for continuous functions on

closed intervals," Thai Journal of Mathematics, vol.

In [35], the bounds are obtained by the maximin theorem of continuous function on

closed intervals for the finite element model.

Choose the

closed intervals J and I such that J [subset] I = [a, b] and that for every [t.sub.0] [member of] J, the unique solution x(-; [t.sub.0], 0) with the initial condition ([t.sub.0], 0), exists on some open interval [mathematical expression not reproducible].

We denote the set of all real valued

closed intervals by IR.

However, in spherical circle planes we may have two

closed intervals of such touching circles in [K.sub.x], one consisting of separating circles and the other of nonseparating ones; we do not know whether even more involved situations may occur.

Nevertheless, the abovementioned operators are straightforward extensions of their respective proposals for the case of HFEs; they only focus on the endpoints of the

closed intervals of interval-valued hesitant fuzzy elements (IVHFEs) on the basis of the characteristics of interval numbers and therefore are not rich enough to capture all the information contained in IVHFEs.

Our "augmentation" terminology does not mean to suggest that the intervals themselves are larger, just that the top and bottom elements of the corresponding

closed intervals are longer.

Let [bar.x] = [[x.sup.-],[x.sup.+]] and [bar.y] = [[y.sup.-],[y.sup.+]] be two

closed intervals in R.

defines a binary operation on the set of

closed intervals. In case of division, it is assumed that 0 [member of] B.

A bi-interval representation of a bigraph G = (U,V; E) is a pair ([I.sub.U], [I.sub.V]) of sets of

closed intervals such that [I.sub.U] = {[[l.sub.u], [r.sub.u]]|u [member of] U} and [I.sub.V] = {[[l.sub.v], [r.sub.v]]|v [member of] V}, and {u, v} [member of] E (G) if and only if [[l.sub.u], [r.sub.u]] [intersection] [[l.sub.v], [r.sub.v]] = 0 for u [member of] U and v [member of] V.

Because the stress S and strength R are functions of these interval variables respectively, they will vary within some

closed intervals [S.sup.I] and [R.sup.I] .