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

Abstract

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ and A: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 for T+A under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition D(T)∘∩D(A)≠∅ and Browder and Hess who used the quasiboundedness of T and condition 0∈D(T)∩D(A). In particular, the maximality of T+∂ϕ is proved provided that D(T)∘∩D(ϕ)≠∅, where ϕ: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.