Maximality Theorems on the Sum of Two Maximal Monotone Operators and Applications 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) ≠ 0 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) ≠ 0, 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.