On a class of operator equations in locally convex spaces

We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. Research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.