The fixed point operator of the biparametric family (2) is

Applying (32) over f(x) = [x.sup.2], the fixed point operator is

Finally, the application of (32) on f(x) = [x.sup.2] + 1 results in the fixed point operator M = [N.sub.11](x)/[D.sub.10](x), where [N.sub.11](x) and [D.sub.10](x) are polynomials of degrees 11 and 10, respectively.

When (39) is applied over f(x) = [x.sup.2] -1, the fixed point operator is

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.

In Section 3, we formulate the fixed point operator equivalent to problem (1).

Vindel, "Bifurcations of the roots of a 6-degree symmetric polynomial coming from the

fixed point operator of a class of iterative methods," in Proceedings of CMMSE, 2014.

The conjugacy classes of its associated fixed point operator, the stability of the strange fixed points, the analysis of the free critical points, and the analysis of the parameter and dynamical planes are made.

Let [O.sub.p](z) be the fixed point operator of Kim's method on p(z).

From a fixed point operator, that associates a polynomial with an iterative method, the dynamical plane illustrates the basins of attraction of the operator.

(12) \% --it is mandatory the previous execution of (13) \% >> syms x (14) \% bounds: [min (Re (z)) max (Re (z)) min (Im (z)) max (Im (z))] (15) \% test: [I, it]=dynamicalPlane (0, [-1 1-11], 400, 20); (16) \% Values (17) x0=bounds (1); xN=bounds (2); y0=bounds (3); yN=bounds (4); (18) funfun=matlabFunction (fun); (19) (20) \% Fixed Point Operator (21) syms x z (22) \% Kim's operator (23) Op = simple (-x.

He shows that bi-similarity, behavioral, and logical equivalence are the same for general model logics and for continuous time stochastic logic with and without a

fixed point operator. Sections include a tutorial on Polish and analytic spaces, measurable selectors, probability measures and categories, and then material on stochastic relations as monads, Eilenberg-Moore algebras for stochastic relations, the existence of semi-pullbacks and interpreting modal and temporal logics.