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How to solve the equation $AuBu+Cu=f$. (English) Zbl 1051.47009

The authors discuss the problem of solving the initial value problem

$AuBu+Cu=f,\phantom{\rule{1.em}{0ex}}u\left(0\right)=1\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $f,u\in W\left({\Omega }\right)$, $A,B,C\in L\left(W\left({\Omega }\right)\right)$, where $W\left({\Omega }\right)$ is the reproducing kernel space on the subset ${\Omega }$ of ${ℝ}^{1}$ and $L\left(W\left({\Omega }\right)\right)$ is the space of continuous linear operators from $W\left({\Omega }\right)$ into $W\left({\Omega }\right)$.

The authors use the method to transform a one-dimensional nonlinear operator equation into a two-dimensional linear operator equation. To achieve these results, they firstly discuss the problem how to solve a continuous linear operator equation in a separable Hilbert space. If the solution exists, there is given the representation and approximation of the minimal normal solution of the equation and formula are obtained. Further on, there is given a factorization method and characteristic value method to solve equation (1).

##### MSC:
 47A50 Equations and inequalities involving linear operators, with vector unknowns 65J15 Equations with nonlinear operators (numerical methods)
##### Keywords:
operator equation; reproducing kernel; nonlinear operator