Nonlinear coercive operator ($⟨A(u),u-u_0⟩/\Vert u-u_0\Vert \to +\infty$)

For a map $A:M\to X^{\ast }$ on a subset of a real Banach space, coercivity with respect to $u_0\in M$ means the displayed limit as $u\in M$ and $\Vert u-u_0\Vert \to \infty$. Here the brackets denote the dual pairing. This growth condition is useful in existence theorems for monotone operators. A linear Hilbert-space estimate $⟨Au,u⟩\ge c\Vert u{\Vert }^2$, $c>0$, implies the condition at zero. Monotonicity alone does not: the zero operator is monotone and fails this growth condition on an unbounded space.