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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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