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Variational sum of monotone operators. (English) Zbl 0824.47044

Summary: The sum of (nonlinear) maximal monotone operators is reconsidered from the Yosida approximation and graph-convergence point of view. This leads to a new concept, called variational sum, which coincides with the classical (pointwise) sum when the classical sum happens to be maximal monotone. In the case of subdifferentials of convex lower semicontinuous proper functions, the variational sum is equal to the subdifferential of the sum of the functions. A general feature of the variational sum is to involve not only the values of the two operators at the given point but also their values at nearby points.

MSC:

47H05 Monotone operators and generalizations
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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