On the Maximality of the Sum of Two Maximal Monotone Operators

Abstract

Let E be a real reflexive Banach space, E∗ be the dual space of E and \(T: D(T)\subseteq E\to 2^{E^{*}}\), \(S:D(S)\subseteq E\to 2^{E^*}\) be two maximal monotone operators such that D(T) ∩ D(S) ≠ ∅. Assume that there exist x0 ∈ E, r > 0, λ0 > 0 such that inff ∈ Tx(f, x − x0) is lower bounded on each bounded subset of D(T) and, if, for each y ∈ B(x0, r), g ∈ E∗, x n ∈ D(T) and λ n ∈ (0, λ0) with \(g\in Tx_n+ S_{\lambda _n}x_n+Jx_n\) for each n = 1, 2, ⋯, \(\{R_{\lambda _n}^Sx_n\}_{=1}^{\infty }\) is bounded, then we have $$\displaystyle \inf _{n\geq 1}(S_{\lambda _n}x_n,R_{\lambda _n}^Sx_n-y)>-\infty , $$ where \(R_{\lambda }^S\) is the Yosida resolvent of S, then T + S is maximal monotone. Also, we construct a degree theory for the sum of two maximal monotone operators, where the sum may not be maximal monotone, and the degree theory is also applied to study the operator equation 0 ∈ (T + S)x. Finally, we give some applications of the main results to nonlinear partial differential equations.