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An abstract monotone iterative technique. (English) Zbl 0883.47058
On the Hilbert space $$H= L^2(\Omega)$$, where $$\Omega \subset \mathbb{R}^n$$ 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 $u \in D(L), \quad Lu+ \lambda u \geq 0\;\text{ on } \Omega \;\Longrightarrow \;u \geq 0\;\text{ on } \Omega, \quad \text{for some }\lambda \in \rho(L),$ while, for the nonlinear operator $$N: D(N) \subset H \mapsto H$$, the growth condition $$Nu- Nv \geq -m(u- v)$$, $$m \leq \lambda$$, holds on an order interval $$J= \{u \in H : \alpha \leq u \leq \beta\}$$ 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 $$\alpha_0= \alpha$$, $$\beta_0= \beta$$, $$\alpha_n \leq \beta_n$$, $$\forall n$$, 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 Banach spaces or other ordered topological vector spaces 47J25 Iterative procedures involving nonlinear operators
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References:
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