Completely continuous functions in intuitionistic fuzzy topological spaces.

T : K [right arrow] K is a

completely continuous operator.

and then T : P [right arrow] P is

completely continuous.

From a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we deduce that the operator A : C [right arrow] C is

completely continuous.

The same argument shows that W is compact if C is

completely continuous with [bar.D(T)] [subset or equal to] D(C).

Patients such as the one reported here in whom the styloid process was thick and

completely continuous with the hyoid bone are rare, with only three previous cases reported [7-9].

Thus, the operator T:P [right arrow] Pis

completely continuous.

Therefore, by the Arzela-Ascoli theorem, the operator P : C [right arrow] C is

completely continuous.

In Theorem 10 we suppose that f is

completely continuous, which allows us to prove that the associated fixed point operator is

completely continuous required by a Leray-Schauder approach.

Assume [[OMEGA].sub.1], [[OMEGA].sub.2] are bounded open subsets of E with 0 [member of] [[OMEGA].sub.1], [[bar.[OMEGA].sub.1]] [subset] [[OMEGA].sub.2], and let S : P [intersection] ([[OMEGA].sub.2]\[[OMEGA].sub.1]) [right arrow] P be a continuous and

completely continuous operator such that, either

Suppose that A : P [intersection] ([[OMEGA].sub.2] \ [[OMEGA].sub.1]) [right arrow] P is

completely continuous. If either [mathematical expression not reproducible] hold, then A has at least one fixed point in P [intersection]([[OMEGA].sub.2]\[[OMEGA].sub.1]).

Hanafy,

Completely continuous functions in intuitionistic fuzzy topological spaces, Czechoslovak Mathematical Journal, 53(4) (2003), 793-803