Maximality theorems on the sum of two maximal monotone operators and application to variational inequality problems. (English) Zbl 1470.47036

Asfaw, Teffera M.

Abstr. Appl. Anal. 2016, Article ID 7826475, 10 p. (2016).

Summary: Let \(X\) be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space \(X^\ast\). Let \(T : X \supseteq D(T) \rightarrow 2^{X^\ast}\) and \(A : X \supseteq D(A) \rightarrow 2^{X^\ast}\) 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 improve the well-known maximality results of R. T. Rockafellar [Trans. Am. Math. Soc. 149, 75–88 (1970; Zbl 0222.47017)] who used condition \(\overset\circ{D(T)} \cap D(A) \neq \emptyset\) and F. E. Browder and P. Hess [J. Funct. Anal. 11, 251–294 (1972; Zbl 0249.47044)] who used the quasiboundedness of \(T\) and condition \(0 \in D(T) \cap D(A)\). In particular, the maximality of \(T + \partial \phi\) is proved provided that \(\overset\circ{D(T)} \cap D(\phi) \neq \emptyset\), where \(\phi : X \rightarrow(- \infty, \infty]\) 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.

MSC:

47H05	Monotone operators and generalizations
49J40	Variational inequalities

Keywords:

maximality theorems; sum of two maximal monotone operators; variational inequalities

Citations:

Zbl 0222.47017; Zbl 0249.47044

References:

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