Delta Bihari inequality

Theorem

Suppose that $g$ is continuous and nondecreasing, $p$ is rd-continuous and nonnegative, and $y$ is rd-continuous. Let $w$ be the solution of $$w^{\Delta}=p(t)g(w), \quad w(a)=\beta$$ and suppose there is a bijective function $G$ with $(G \circ w)^{\Delta} = p$. Then $$y(t) \leq \beta + \displaystyle\int_a^t p(\tau)g(y(\tau)) \Delta \tau$$ for all $t \in \mathbb{T}$ implies $$y(t) \leq G^{-1} \left[ G(\beta) + \displaystyle\int_a^t p(\tau) \Delta \tau \right]$$ for all $t \in \mathbb{T}$.