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A new class of monotone mappings and a new class of variational inclusions in Banach spaces. (English) Zbl 1263.49014

Let E be a Banach space, E * its topological dual, and CB(E) denote the collection of all nonempty closed bounded subsets of E. For k3, let A:EE * , p, f 1 ,f 2 ,,f k :EE, F:E k E * be nonlinear maps and T 1 ,T 2 ,,T k :ECB(E), M:E k E * multimaps. Under some monotonicity assumptions on M, the author studies the following problem: for a given aE * , find xE, t 1 T 1 (x),t 2 T 2 (x),,t k T k (x) such that

aA(x-p(x))+M(f 1 (x),f 2 (x),,f k (x))-F(t 1 ,t 2 ,,t k )·

By using the technique of proximal maps, the author reduces this problem to a fixed point problem and suggests an iterative algorithm for its solving.

MSC:
49J53Set-valued and variational analysis
47H04Set-valued operators
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
References:
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