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

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.