We note that [mathematical expression not reproducible] is well-defined by Theorems 5 and 10.

We now have the following theorem from Theorems 5,10, 11, and 12.

Karapinar, Coupled fixed point

theorems for nonlinear contractions in cone metric spaces, Comput.

Fujisawa, Variations of mixed Hodge structure and semipositivity

theorems, Publ.

In this section, we present two examples of polynomials in order to compare our

theorems with some of the above stated known

theorems and show that for these polynomials our

theorems give better bounds than those obtainable by these known

theorems.

Here, we established some properties of the operators as consequences of

Theorems 1 and 2.

It turns out that the existence of a linear continuous surjection T from [C.sub.p](Y) onto [C.sub.p](X) in Propositions 4 and 7 is also a sufficiently strong condition as the following easy corollary of Uspenskii's

theorems [17] shows.

Theorem 12 implies the following two universality

theorems for composite functions obtained in [12].

Theorem A.[17,

Theorem 1.2] Let X be a paracompact free [Z.sub.p]-space of ind X [greater than or equal to] n, and f : X [right arrow] M a continuous mapping ofX into an m-dimensional connected manifold M (orientable ifp> 2).

There appeared also many generalizations of fixed point

theorems for Kakutani maps.

The following two

theorems can be proved using a scheme similar to that of the proof of

Theorem 15 and with the aid of formula ([b.sub.4]),

Theorem 7, and formula ([c.sub.4]),

Theorem 8, respectively.

How does the process of summation x + y = z transform into the Pythagorean

theorem? There are several ways this can be done.