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

Abstract EN

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X ⁎ .

Let T : X ⊇ D ( T ) → 2 X ⁎ and A : X ⊇ D ( A ) → 2 X ⁎ 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.