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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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