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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

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