Proving the existence of a unique solution
Can somebody help me out here?
Consider y'(t) = y(t)[a(t) - b(t)y(t)] where a,b : (-infinite, +infinite) maps to (0, +infinity)
and there exists M>0 such that: (1/M) less than or equal to a(t), b(t) less than or equal to M, for all t in the reals
Claim: There exists a unique positive solution Phi(t) defined for all t in the reals in which there exists an m>0 such that: (1/m) less than or equal to Phi(t) less than or equal to m. (i.e. Phi(t) is bounded and bounded away from zero)
note that a(t) and b(t) are not necessarily periodic.
Prove the claim true or false.
Any help would be tremendously appreciated. I'm practically dying over this one. (Angry)