On polynomial equations in Banach space, perturbation techniques and applications (Q1822035)

We use perturbation techniques to find solutions of the abstract polynomial equation of degree k, \[ x=P_ k(x)=M_ kx^ k+M_{k- 1}x^{k-1}+...+M_ 1x+M_ 0 \] in a Banach space X over the field F of real or complex numbers. The principal new idea is the introduction of an equation similar to the above, \[ z=F_ k(z)=N_ kz^ k+N_{k-1}z^{k-1}+...+N_ 1z+M_ 0. \] The results are then obtained under suitable choices of the p-linear operators \(N_ p\), \(p=1,2,...,k.\) Our techniques provide more accurate information on the location of solutions and yield existence and uniqueness in cases not covered before. An example is provided to justify our method.