Review Questions
1. If f(t) be a continuous function and v(t) a nonnegative continuous function on the interval $t_0 \leq t \leq t_0+a$. If a continuous function u(t) has property
$$u(t) \leq f(t) + \int_{t_0}^{t} u(s)v(s)ds \quad \text{for } t \in [t_0, t_0 + a],$$
then
$$u(t) \leq f(t) + \int_{t_0}^{t} f(s)v(s) \exp\left[\int_{s}^{t} v(\tau)d\tau\right]ds \quad \text{for } t \in [t_0, t_0 + a].$$
2. If u(t) and v(t) be nonnegative continuous function on some interval $t_0 \leq t \leq t_0+a$. Also, let the function f(t) be positive, continuous, and monotonically non decreasing on $t_0 \leq t \leq t_0+a$ and satisfy the inequality If a continuous function u(t) has property
$$u(t) \leq f(t) + \int_{t_0}^{t} u(s)v(s)ds \quad \text{for } t \in [t_0, t_0 + a],$$
then, prove that
$$u(t) \leq f(t)\exp\left[\int_{t_0}^{t} v(s)ds\right] \quad \text{for } t \in [t_0, t_0+a].$$
