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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 $CB\left(E\right)$ denote the collection of all nonempty closed bounded subsets of $E$. For $k\ge 3,$ let $A:E\to {E}^{*},$ $p,$ ${f}_{1},{f}_{2},\cdots ,{f}_{k}:E\to E,$ $F:{E}^{k}\to {E}^{*}$ be nonlinear maps and ${T}_{1},{T}_{2},\cdots ,{T}_{k}:E\to CB\left(E\right),$ $M:{E}^{k}⊸{E}^{*}$ multimaps. Under some monotonicity assumptions on $M,$ the author studies the following problem: for a given $a\in {E}^{*},$ find $x\in E,$ ${t}_{1}\in {T}_{1}\left(x\right),{t}_{2}\in {T}_{2}\left(x\right),\cdots ,{t}_{k}\in {T}_{k}\left(x\right)$ such that

$a\in A\left(x-p\left(x\right)\right)+M\left({f}_{1}\left(x\right),{f}_{2}\left(x\right),\cdots ,{f}_{k}\left(x\right)\right)-F\left({t}_{1},{t}_{2},\cdots ,{t}_{k}\right)·$

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:
 49J53 Set-valued and variational analysis 47H04 Set-valued operators 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations)
##### References:
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