Abstract
AbstractLet "Equation missing" be a real Banach space with the dual space "Equation missing". Let "Equation missing" be a proper functional and let "Equation missing" be a bifunction. In this paper, a new concept of "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing" is introduced. The existence and Lipschitz continuity of the "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing" are proved. By using properties of the "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing", a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space "Equation missing". An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space "Equation missing", and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in "Equation missing" a Hilbert space.
Highlights
Let X be a real Banach space with norm · and let X∗ be its dual space
The value of f ∈ B∗ at x ∈ B will be denoted by f, x
Using the properties of the η-proximal mapping, they proved an existence theorem of solutions for a new class of general mixed variational inequalities in a Banach space and suggested an iterative algorithm for computing approximate solutions
Summary
Let X be a real Banach space with norm · and let X∗ be its dual space. The value of f ∈ B∗ at x ∈ B will be denoted by f, x. Very recently, inspired by the research work going on in this field, Xia and Huang 14 first introduced a new concept of η-proximal mapping for a proper subdifferentiable functional on a Banach space They proved an existence theorem and Lipschitz continuity of the η-proximal mapping. Using the properties of the η-proximal mapping, they proved an existence theorem of solutions for a new class of general mixed variational inequalities in a Banach space and suggested an iterative algorithm for computing approximate solutions. They gave the strong convergence criteria of the iterative sequence generated by this algorithm. Our results are new and represent the improvement, extension, and development of Xia and Huang’s results in 14
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