= Stage solvability of an implicit Runge-Kutta method
For a vector field uniformly <Lipschitz continuous> in the state with constant $L$, the stage fixed-point map of an <implicit Runge-Kutta method> is a contraction in the maximum stage norm if $hL\max_i\sum_j|a_{ij}|<1$. The <contraction mapping theorem> then gives unique stages. This is a sufficient small-step condition; a <dissipative vector field> may permit solvability for larger steps. An <algebraic stability> or <B-stability> estimate compares existing stage solutions and should not be mistaken for an unqualified stage-existence theorem.
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