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

Abstract

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^{⁎}$. Let $T:X\supseteq D(T)\to {\mathrm{2}}^{X^{⁎}}$ and $A:X\supseteq D(A)\to {\mathrm{2}}^{X^{⁎}}$ 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 $\stackrel{\circ }{D(T)}\cap D(A)\ne \mathrm{\varnothing }$ and Browder and Hess who used the quasiboundedness of $T$ and condition $\mathrm{0}\in D(T)\cap D(A)$. In particular, the maximality of $T+\partial \varphi$ is proved provided that $\stackrel{\circ }{D(T)}\cap D(\varphi )\ne \mathrm{\varnothing }$, where $\varphi :X\to (-\mathrm{\infty },\mathrm{\infty }]$ 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.