Abstract
Using Bregman functions, we introduce the new concept of Bregman generalizedf-projection operatorProjCf, g:E*→C, whereEis a reflexive Banach space with dual spaceE*; f: E→ℝ∪+∞is a proper, convex, lower semicontinuous and bounded from below function;g: E→ℝis a strictly convex and Gâteaux differentiable function; andCis a nonempty, closed, and convex subset ofE. The existence of a solution for a class of variational inequalities in Banach spaces is presented.
Highlights
Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators
Using Bregman functions, we introduce the E is a reflexive Banach space with dual space new concept E∗; f : E →
In this paper, using Bregman functions, we introduce the new concept of Bregman generalized f-projection operator
Summary
Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators. Let E be a reflexive Banach space, let f : E → R ∪ {+∞} be a proper, convex, lower semicontinuous function, let g : E → R be strictly convex and Gateaux differentiable, and let C ⊆ E be nonempty. Let E be a Banach space with dual space E∗, let f : E → R ∪ {+∞} be a proper, convex, lower semicontinuous function, let g : E → R be strictly convex and Gateaux differentiable, and let C be a nonempty, closed subset of E. Is a reflexive Banach space with {+∞} is a proper, convex, lower semicontinuous, and bounded from below function, g : E → R is a strictly convex and Gateaux differentiable function, and C is a nonempty, closed, and convex subset of E. Our results improve and generalize some known results in the current literature; see, for example, [20, 21]
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