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Abstract and Applied Analysis
Volume 2016, Article ID 7826475, 10 pages
Research Article

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

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Received 19 May 2016; Accepted 17 July 2016

Academic Editor: Lucas Jodar

Copyright © 2016 Teffera M. Asfaw. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space . Let and 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 under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition and Browder and Hess who used the quasiboundedness of and condition . In particular, the maximality of is proved provided that , where 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.