On dichotomy and well conditioning in BVP

We investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand. It is shown that if these stability bounds are moderate (i.e. if the problem is well conditioned) then the homogeneous solution space is dichotomic, which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes. This is done by carefully examining the Green’s functions. If these exhibit an exponential behaviour then the solution space is also exponentially dichotomic. On the other hand, we also show that (exponential) dichotomy implies moderate stability constants, i.e. well conditioning. From this it follows that both concepts are more or less equivalent