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Fixed points and approximate solutions for nonlinear operator equations. (English) Zbl 0954.47038

The equation

$Lu=Nu,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $L$ and $N$ are linear and nonlinear operators defined on a Hilbert space, respectively, is considered. If $L$ is invertible, then the equation (1) is equivalent to

$u={L}^{-1}\left(Nu\right)·\phantom{\rule{2.em}{0ex}}\left(2\right)$

Thus the problem of existence of solutions of (1) leads to the problem of finding fixed points for the operator ${L}^{-1}\circ N$.

It may happen that $L$ is not invertible but for some $s\in R$, there exists ${\left(L+sI\right)}^{-1}$. Then the equation (1) is equivalent to

$u={\left(L+sI\right)}^{-1}\left(Nu+su\right)·\phantom{\rule{2.em}{0ex}}\left(3\right)$

To solve such equation one may use the monotone iterative technique for increasing operators on the right-hand side of (3) to find fixed points.

In this paper the authors present some new technique for operators which can be written as difference of monotone operators.

They present the method of finding lower and upper solutions of (1) as well as the criteria for the existence of the unique solution of the equation (1).

The last part of the paper contains applications of the concept of lower and upper solutions to the second-order boundary value problems and to the $n$th-order periodic bundary value problems.

##### MSC:
 47H07 Monotone and positive operators on ordered topological linear spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 34B15 Nonlinear boundary value problems for ODE 47J25 Iterative procedures (nonlinear operator equations)