The Gronwall inequality

We prove the following version generalization of the Gronwall inequality: Let $\mathbf{X}$ be a Banach space and $U\subset \mathbf{X}$ an open convex set in $\mathbf{X}$. Let $f,g\text{\hspace{0.17em}⁣}:[a,b]\times U\to \mathbf{X}$ be continuous functions and let $y,z\text{\hspace{0.17em}⁣}:[a,b]\to U$ satisfy the initial value problems \begin{align*} y’(t)&=f(t,y(t)),\quad y(a)=y_0,\ z’(t)&=g(t,z(t)),\quad z(a)=z_0. \end{align*} Also assume there is a constant $C\ge 0$ so that $$\mathrm{\parallel }g(t,x_2)-g(t,x_1)\mathrm{\parallel }\le C\mathrm{\parallel }{x}_2-x_1\mathrm{\parallel }$$ and a continuous function $\phi \text{\hspace{0.17em}⁣}:[a,b]\to [0,\mathrm{\infty })$ so that $$\mathrm{\parallel }f(t,y(t))-g(t,y(t))\mathrm{\parallel }\le \phi (t)\mathrm{.}$$ Then for $t\in [a,b]$ $$\mathrm{\parallel }y(t)-z(t)\mathrm{\parallel }\le e^{C\mathrm{\mid }t-a\mathrm{\mid }}\mathrm{\parallel }{y}_0-z_0\mathrm{\parallel }+e^{C\mathrm{\mid }t-a\mathrm{\mid }}{\int }_a^t{e}^{-C\mathrm{\mid }s-a\mathrm{\mid }}\phi (s)ds\mathrm{.}$$