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Bregman \(f\)-projection operator with applications to variational inequalities in Banach spaces. (English) Zbl 07022677
Summary: Using Bregman functions, we introduce the new concept of Bregman generalized \(f\)-projection operator \(\text{Proj}_C^{f, g} : E^* \rightarrow C\), where \(E\) is a reflexive Banach space with dual space \(E^*; f : E \rightarrow \mathbb{R} \cup \left\{+ \infty\right\}\) is a proper, convex, lower semicontinuous and bounded from below function; \(g : E \rightarrow \mathbb{R}\) is a strictly convex and Gâteaux differentiable function; and \(C\) is a nonempty, closed, and convex subset of \(E\). The existence of a solution for a class of variational inequalities in Banach spaces is presented.

MSC:
49 Calculus of variations and optimal control; optimization
47 Operator theory
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