Nagumo [6] developed a degree theory in a setting of linear convex topological space Y for operators of the type I - B, where B : [bar.G] [right arrow] Y is a

compact operator, I is the identity mapping on Y, and G is a nonempty and open subset of X.

Let K : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] be any

compact operator. Then

Let S : [L.sup.[infinity]] (0, b) [right arrow] C[0, b] be a linear

compact operator. Let [P.sub.N]: C[0,b] [right arrow] [S.sup.(-1).sub.m-1] ([[PI].sub.N]) (N [member of] N) be defined by (5.1).

The perturbation of a polaroid operator by a

compact operator may or may not effect the polaroid property of the operator.

If (C(t))t[greater than or equal to] 0 is a compact cosine family, then P is a continuous and

compact operator.

In this context, recall that for a

compact operator Q [member of] L(H), Fredholm's theorem states that for all [lambda] [member of] R, ([lambda]I - Q)x = y is solvable for y [member of] H if and only if y [perpendicular to] N([lambda]I - Q*), i.e., if ([lambda], v) is an eigenpair of Q* then it must hold that [??]y, v[??] = 0.

We will prove that F is a

compact operator. According to (4.12), F([OMEGA]) | [0T] is relatively compact in C([0,T], E).

In this paper, a generalized viscous Rosenau equation that includes linear and nonlinear advective terms is studied numerically by means of a second-order accurate, linearized Crank-Nicolson method and three-point, fourth-order accurate,

compact operator discretizations for the first-, second-, and fourth-order spatial derivatives.

(ii) Assume that the sequence {[T.sub.n]} of bounded operators satisfies that there is at least one

compact operator within all subsequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some subsequence {[j.sub.n]} [subset] [N.sub.0] for any n [member of] [N.sub.0].

Here we prove a lemma which is similar to Riesz lemma in normed space and (e) using this we list the properties of

compact operator.

Therefore we have to transfer S into another continuous and

compact operator T.