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An abstract monotone iterative technique. (English) Zbl 0883.47058
On the Hilbert space $H= L^2(\Omega)$, where $\Omega \subset \bbfR^n$ is open and bounded, the author considers a nonlinear equation (1) $Lu= Nu$ where the linear operator $L: D(L) \subset H \mapsto H$ satisfies the maximum principle $$u \in D(L), \quad Lu+ \lambda u \ge 0\ \text{ on } \Omega \ \Longrightarrow \ u \ge 0\ \text{ on } \Omega, \quad \text{for some }\lambda \in \rho(L),$$ while, for the nonlinear operator $N: D(N) \subset H \mapsto H$, the growth condition $Nu- Nv \ge -m(u- v)$, $m \le \lambda$, holds on an order interval $J= \{u \in H : \alpha \le u \le \beta\}$ for some lower and upper solutions $\alpha$ and $\beta$ of (1). Then an iterative scheme is shown to produce monotone sequences $\{\alpha_n\} \nearrow \phi$, $\{\beta_n\} \searrow \psi$ 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:
47H07Monotone and positive operators on ordered topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
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References:
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