Notes on Differential Equations and Differential Inequalities

The following Lemma gives conditions for the existence of a solution of a differential equation which is bounded on the domain R. Lemma 1.1. Given real numbers a < b, if f : (a, b) → R is a continuous, nonvanishing function, and there are some constants C1 > 0, C2 > 0, δ1 ∈ (0, b − a), δ2 ∈ (0, b − a) so that |f(t)| ≤ C1(t − a) for a < t < a + δ1 and |f(t)| ≤ C2(b − t) for b − δ2 < t < b, then there exists a one-to-one, onto function g : R → (a, b) so that y = g(t) is a solution of the equation dy dt = f(y). Proof. 1 f(x) is continuous on (a, b), so the function