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An abstract monotone iterative technique. (English) Zbl 0883.47058

On the Hilbert space $H={L}^{2}\left({\Omega }\right)$, where ${\Omega }\subset {ℝ}^{n}$ is open and bounded, the author considers a nonlinear equation (1) $Lu=Nu$ where the linear operator $L:D\left(L\right)\subset H↦H$ satisfies the maximum principle

$u\in D\left(L\right),\phantom{\rule{1.em}{0ex}}Lu+\lambda u\ge 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Omega }\phantom{\rule{4pt}{0ex}}⟹\phantom{\rule{4pt}{0ex}}u\ge 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\lambda \in \rho \left(L\right),$

while, for the nonlinear operator $N:D\left(N\right)\subset H↦H$, the growth condition $Nu-Nv\ge -m\left(u-v\right)$, $m\le \lambda$, holds on an order interval $J=\left\{u\in H:\alpha \le u\le \beta \right\}$ for some lower and upper solutions $\alpha$ and $\beta$ of (1). Then an iterative scheme is shown to produce monotone sequences $\left\{{\alpha }_{n}\right\}↗\phi$, $\left\{{\beta }_{n}\right\}↘\psi$ on $H$ with ${\alpha }_{0}=\alpha$, ${\beta }_{0}=\beta$, ${\alpha }_{n}\le {\beta }_{n}$, $\forall n$, where $\phi$ and $\psi$ are the minimal and maximal solutions of (1) in $J$, respectively. Some examples are given involving ODEs, PDEs, as well as integro-ODEs, and integro-PDEs.

##### MSC:
 47H07 Monotone and positive operators on ordered topological linear spaces 47J25 Iterative procedures (nonlinear operator equations)