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
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
12 implies the following two universality theorems
for composite functions obtained in .
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
How does the process of summation x + y = z transform into the Pythagorean theorem
? There are several ways this can be done.