Fan-Hemicontinuity for the gradient of the norm in Hilbert space

Abstract It was claimed in [Sadeqi, I.—Salehi Paydar, M.: A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results, J. Optim. Theory Appl. 165(2) (2015), 344–358] that the gradient of a convex Gâteaux differentiable function is Fanhemicontinuous. The aim of the present paper is to correct this implication by exemplifying for ∇‖⋅‖ $\nabla \|\cdot \|$in a Hilbert space. This goal was achieved by considering its domain to be scalarly-positive (see Definition 3). For a perturbation of a Fan-hemicontinuous operator with ∇‖⋅‖, $\nabla \|\cdot \|,$weak compactness for the set of solutions of the variational inequality problem is obtained.