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

Teffera M. Asfaw

Abstract and Applied Analysis, 2016, vol. 2016, issue 1

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.

Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2016:y:2016:i:1:n:7826475

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