On the Maximality of the Sum of Two Maximal Monotone Operators

Let E be a real reflexive Banach space, E ∗ be the dual space of E and T : D ( T ) ⊆ E → 2 E ∗ $$T: D(T)\subseteq E\to 2^{E^{*}}$$ , S : D ( S ) ⊆ E → 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 inff ∈ 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 ∈ T x n + S λ n x n + J x n $$g\in Tx_n+ S_{\lambda _n}x_n+Jx_n$$ for each n = 1, 2, ⋯, { R λ n S x n } = 1 ∞ $$\{R_{\lambda _n}^Sx_n\}_{=1}^{\infty }$$ is bounded, then we have inf n ≥ 1 ( S λ n x n , R λ n S x n − y ) > − ∞ , $$\displaystyle \inf _{n\geq 1}(S_{\lambda _n}x_n,R_{\lambda _n}^Sx_n-y)>-\infty , $$ where R λ S $$R_{\lambda }^S$$ 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.