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An abstract monotone iterative technique. (English) Zbl 0883.47058

On the Hilbert space H=L 2 (Ω), where Ω n is open and bounded, the author considers a nonlinear equation (1) Lu=Nu where the linear operator L:D(L)HH satisfies the maximum principle


while, for the nonlinear operator N:D(N)HH, the growth condition Nu-Nv-m(u-v), mλ, holds on an order interval J={uH:αuβ} for some lower and upper solutions α and β of (1). Then an iterative scheme is shown to produce monotone sequences {α n }φ, {β n }ψ on H with α 0 =α, β 0 =β, α n β n , n, where φ and ψ 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.

47H07Monotone and positive operators on ordered topological linear spaces
47J25Iterative procedures (nonlinear operator equations)