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

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  • Teffera M. Asfaw

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.

Suggested Citation

  • Teffera M. Asfaw, 2016. "Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2016(1).
    • Handle: RePEc:wly:jnlaaa:v:2016:y:2016:i:1:n:7826475
    DOI: 10.1155/2016/7826475
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    References listed on IDEAS

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    1. Eberhard Zeidler, 1990. "Nonlinear Functional Analysis and its Applications," Springer Books, Springer, number 978-1-4612-0981-2, March.
    2. S. Carl & Vy K. Le & D. Motreanu, 2005. "Existence, comparison, and compactness results for quasilinear variational-hemivariational inequalities," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2005, pages 1-17, January.
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