Diff Eq. Proving the existence of a unique solution

[JoeyJoeJoeJr]

Can somebody help me out here?

Consider y'(t) = y(t)[a(t) - b(t)y(t)] where a,b : (-infinite, +infinite) → (0, +infinity)
and there exists M>0 such that: (1/M) ≤ a(t), b(t) ≤ 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) ≤ Phi(t) ≤ 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.

 

[lippyka]

False. The claim is false. The given conditions are not sufficient to guarantee the existence of a unique positive solution $\Phi(t)$ for all $t \in \mathbb{R}$ that is bounded and bounded away from zero.