This paper by R. T. Rockafellar explores the maximality of sums of nonlinear monotone operators in a real Banach space. A monotone operator \( T \) from \( X \) to \( X^* \) is defined as a mapping such that for any \( x, y \in X \) and \( x^*, y^* \in T(x), T(y) \), respectively, the inequality \( \langle x - y, x^* - y^* \rangle \geq 0 \) holds. An operator \( T \) is maximal if its graph is not properly contained in the graph of any other monotone operator. The paper focuses on conditions under which the sum of two maximal monotone operators \( T_1 \) and \( T_2 \) remains maximal. Key results include: 1. **Theorem 1**: If \( X \) is reflexive, \( T_1 \) and \( T_2 \) are maximal monotone operators, and either \( D(T_1) \cap \text{int } D(T_2) \neq \varnothing \) or there exists an \( x \in \text{cl } D(T_1) \cap \text{cl } D(T_2) \) such that \( T_2 \) is locally bounded at \( x \), then \( T_1 + T_2 \) is maximal. 2. **Theorem 2**: If \( X \) is finite-dimensional, \( T_1 \) and \( T_2 \) are maximal monotone operators, and \( \text{ri } D(T_1) \cap \text{ri } D(T_2) \neq \varnothing \), then \( T_1 + T_2 \) is maximal. 3. **Theorem 3**: If \( K \) is a nonempty closed convex subset of \( X \), \( T_1 \) is the normality operator for \( K \), and \( T_2 \) is any single-valued monotone operator with \( D(T_2) \supseteq K \) and \( T_2 \) is hemicontinuous on \( K \), then \( T_1 + T_2 \) is maximal. The paper also discusses applications of these theorems to variational inequalities and provides conditions for the existence of solutions to such inequalities. The proofs rely on the theory of monotone operators, duality mappings, and properties of maximal monotone operators.This paper by R. T. Rockafellar explores the maximality of sums of nonlinear monotone operators in a real Banach space. A monotone operator \( T \) from \( X \) to \( X^* \) is defined as a mapping such that for any \( x, y \in X \) and \( x^*, y^* \in T(x), T(y) \), respectively, the inequality \( \langle x - y, x^* - y^* \rangle \geq 0 \) holds. An operator \( T \) is maximal if its graph is not properly contained in the graph of any other monotone operator. The paper focuses on conditions under which the sum of two maximal monotone operators \( T_1 \) and \( T_2 \) remains maximal. Key results include: 1. **Theorem 1**: If \( X \) is reflexive, \( T_1 \) and \( T_2 \) are maximal monotone operators, and either \( D(T_1) \cap \text{int } D(T_2) \neq \varnothing \) or there exists an \( x \in \text{cl } D(T_1) \cap \text{cl } D(T_2) \) such that \( T_2 \) is locally bounded at \( x \), then \( T_1 + T_2 \) is maximal. 2. **Theorem 2**: If \( X \) is finite-dimensional, \( T_1 \) and \( T_2 \) are maximal monotone operators, and \( \text{ri } D(T_1) \cap \text{ri } D(T_2) \neq \varnothing \), then \( T_1 + T_2 \) is maximal. 3. **Theorem 3**: If \( K \) is a nonempty closed convex subset of \( X \), \( T_1 \) is the normality operator for \( K \), and \( T_2 \) is any single-valued monotone operator with \( D(T_2) \supseteq K \) and \( T_2 \) is hemicontinuous on \( K \), then \( T_1 + T_2 \) is maximal. The paper also discusses applications of these theorems to variational inequalities and provides conditions for the existence of solutions to such inequalities. The proofs rely on the theory of monotone operators, duality mappings, and properties of maximal monotone operators.