ABSTRACT
Consider the following class of nonlinear variational inequality (NVI) problems, whose solvability is based on an iterative procedure characterized by a variational inequality: Determine an element https://www.w3.org/1998/Math/MathML"> x * ∈ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12515.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> u * ∈ T x * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12516.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> such that https://www.w3.org/1998/Math/MathML"> 〈 u * , x − x * 〉 ≥ 0 for all x ∈ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12517.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ,
where https://www.w3.org/1998/Math/MathML"> T : K → P ( H ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12518.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a multivalued mapping from a real Hilbert space H into https://www.w3.org/1998/Math/MathML"> P ( H ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12519.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure is characterized as a nonlinear variational inequality, that is, for any arbitrarily chosen initial point https://www.w3.org/1998/Math/MathML"> x 0 ∈ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12520.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> u 0 ∈ T x 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12521.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , we have https://www.w3.org/1998/Math/MathML"> 〈 u k + x k + 1 − x k , x − x k + 1 〉 ≥ 0 for all x ∈ K and , for u k ∈ T ( x k ) and for k ≥ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12522.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
which is equivalent to a projection equation https://www.w3.org/1998/Math/MathML"> x k + 1 = P K x k - u k , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq12523.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where PK denotes the projection of H onto K This extends the existing results to the case of a class of nonlinear variational inequalities involving multivalued α-H-cocoercive mappings in a Hilbert space setting.
