In order words, [??] is continuous if and only if it is a component-wise

continuous operator.

T : K [right arrow] K is a completely

continuous operator.

We say that the

continuous operator F : [H.sup.n]([D.sub.V]) [right arrow] H([D.sub.V]) belongs to the class [U.sub.n,V] if, for every polynomial p = p(s),

This equality by (2.18) proves the existence of the

continuous operator [(P - [lambda]I).sup.-1], which satisfies

Then, for any 0 < a < b < +[infinity], T : P [subset] ([bar.[Q.sub.b]]\[[OMEGA].sub.a]) [right arrow] P is a completely

continuous operator.

This proves that [F.sub.i] is a

continuous operator on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all i = 1,2.

iii) f : C(I, [R.sup.n]) [right arrow] C(I, [R.sup.n]) is a

continuous operator for which there exist [[beta].sub.0] > 0, a nondecreasing function [beta] : [0, [infinity]) [right arrow] (0, [infinity]) and a bounded function [w.sub.1]: [0, [infinity]) [right arrow] [0, [infinity]) with [lim.sub.t[right arrow]0]+[w.sub.1](t) = 0 such that |(fx)(t)| [less than or equal to] [[beta].sub.0]|x|c [for all]x [member of] C(I, [R.sup.n]), t [member of] I and

They do not need initial or

continuous operator training, operator intervention, or management of the process.

Assume [[OMEGA].sub.1], [[OMEGA].sub.2] are bounded open subsets of E with de [[OMEGA].sub.1], [[bar.[OMEGA]].sub.1] [subset] [[OMEGA].sub.2], and let T: K [intersection] ([[bar.[OMEGA]].sub.2]\[[OMEGA].sub.1]) [right arrow] K be a completely

continuous operator such that one of the following holds:

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the Lipschitz

continuous operator with constant [delta] = r/r - r'.

Keywords: Time scale, delta and nabla derivatives and integrals, Green's function, completely

continuous operator, eigenfunction expansion.