Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Teffera M Asfaw

doi:10.1155/2016/7826475

Abstract

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎andA:X⊇D(A)→2X⁎be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved forT+Aunder weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used conditionD(T)∘∩D(A)≠∅and Browder and Hess who used the quasiboundedness ofTand condition0∈D(T)∩D(A). In particular, the maximality ofT+∂ϕis proved provided thatD(T)∘∩D(ϕ)≠∅, whereϕ:X→(-∞,∞]is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.

Highlights

Is called the “generalized duality mapping” associated with φ
These theorems improved the well-known maximality results of Rockafellar who used condition D(T) ∩ D(A) ≠ 0 and Browder and Hess who used the quasiboundedness of T and condition 0 ∈ D(T) ∩ D(A)
If φ(t) = t for all t ≥ 0, Jφ is denoted by J and is called “the normalized duality mapping.”

Summary

Introduction

Is called the “generalized duality mapping” associated with φ. Theorem 13 provides a new maximality result for T + ∂φ, where φ : X → As a consequence of this maximality result, an existence theorem for solvability of variational inequality problem involving operators of the type T + S with respect to a closed convex subset K of X and the function φ is included in Theorem 16, where S : K → 2X∗ is a bounded pseudomonotone operator.

Results

Conclusion