  We establish that mass conserving single terminal-linkage networks of
chemical reactions admit positive steady states regardless of network
deficiency and the choice of reaction rate constants. This result holds for
closed systems without material exchange across the boundary, as well as for
open systems with material exchange at rates that satisfy a simple sufficient
and necessary condition. Our proof uses a fixed point of a novel convex
optimization formulation to find the steady state behavior of chemical reaction
networks that satisfy the law of mass-action kinetics. A fixed point iteration
can be used to compute these steady states, and we show that it converges for
weakly reversible homogeneous systems. We report the results of our algorithm
on numerical experiments.
