Say we begin with a mapping x->f(x) . You want a solution of f(x)=0 (1). Transforming it in a zero problem x-P(x)=0 (2) should yield the solutions of the original problem.this means that the solution set of (2) contains that of (1).

You now consider the solution of the second problem as a fixed point problem.

x = P(x)  ->   x_{k+1} = P(x_k)   

If the solution sets of (1) and (2) are the same, then clearly (2) can have no extraneous fixed points, but its fixed points may not be contracting, hence the difficulty is indeed twofold:

proper problem transformation of (1) into (2) and having the fixed points  of (2) being contractive.

But it is hard to imagine that there exists  no P  such that the solution set of (2) is not larger but contains  that of (1) 

 And if now (2) has additional fixed points as in your cited case, then this means the transformation is useful only if you have sufficiently good initial guesses for the solutions originally sought. For example there are many nice methods to transform a constrained optimization problem with smooth functions into a smooth unconstrained optimization which is formally much easier to solve, but unhappily all these methods exhibit lots of local optimizer which are not satisfying the constraints of the original problem  besides cases with very nice additional properties.

I second Peter's answer: it is often rather easy to transform a problem into a fixed-point one, BUT the properties of the mapping F in the fixed-point problem "F(x) = x" may have weird properties - in particular it may be far from contractive. So I would most often NOT force the original problem to be a fixed-point problem, but to treat it more like what it is when you first formulated in, be it a system of equations, a variational inequality, or indeed an optimization problem with or without constraints.

What says this polnomiographs for p(z)=z^3-1. Please give some valuable comments. 

Thank you.

Dear researchers,

Because of your valuable comments and suggestion, I have publish one paper.

Thanks.