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Operator and differential equations in ordered spaces. (English) Zbl 0936.47042

The authors develop a generalized iteration method for operator equations of the form \(Lu=Nu\) where \(L:D(L)\rightarrow X\) and \(N:D(N)\rightarrow X\) are arbitrary mappings from sets \(D(L)\) and \(D(N)\) with non empty intersection to a partially ordered set \(X\). No algebraic or topological structure is assumed, hence the results are applicable to problems of discontinuous and implicit type. The existence of extremal solutions and the dependence on data is studied as well. Applications are given to initial and boundary value problems for implicit Carathéodory type ordinary and partial differential equations.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
35H99 Close-to-elliptic equations
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