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 x ₀ ∈ E, r > 0, λ ₀ > 0 such that inf_{f ∈ Tx}(f, x − x ₀) is lower bounded on each bounded subset of D(T) and, if, for each y ∈ B(x ₀, r), g ∈ E ^∗, x _n  ∈ D(T) and λ _n  ∈ (0, λ ₀) 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.

Dedicated to Haim Brezis and Louis Nirenberg in deep admiration