Dynamical systems proof that $ f(t)$ is less than or equal to $g(t)$

Question

Suppose that $f'(t) \le F(f(t),t)$ where $F$ is continuously differentiable. If $g(t)$ is a solution to the equation $g'(t) = F(g(t),t)$ and $g(a) = f(a)$, then prove that $f(t) \le g(t)$ for all $t \ge a$

Answer 1

This is a very or most general form of the Grönwall lemma/theorem.

The ODE $y'(t)=F(y(t),t)$ has a parametrized general solution or flow $\phi(t;t_0,x_0)$, so that $x(t)=\phi(t;t_0,x_0)$ is the solution to the IVP with $x(t_0)=x_0$.

The exact solution of the claim is thus obtained as $g(t)=\phi(t;a,f(a))$. The differential equation for the flow is thus $$ \partial_1\phi(t;t_0,x_0)=F(\phi(t;t_0,x_0),t), $$ where $\partial_k\phi$ is the partial derivative for the $k$th argument, $\partial_1=\frac\partial{\partial t}$, $\partial_2=\frac\partial{\partial t_0}$, $\partial_3=\frac\partial{\partial x_0}$.

If one uses the value of the solution at any other point $s$ as initial condition, one gets the same solution, thus the composition property of the flow, $$ \phi(t;t_0,x_0)=\phi(t;s,\phi(s;t_0,x_0)). $$ An interesting relation of the derivatives at the initial point $(t_0,x_0)$ results from taking the derivative for this middle time $s$ $$ 0=\frac{d}{ds}\phi(t;t_0,x_0)=\partial_2\phi(t;s,\phi(s;t_0,x_0))+\partial_3\phi(t;s,\phi(s;t_0,x_0))F(\phi(s;t_0,x_0),s), $$ and evaluated at $s=t_0$, $$ \partial_2\phi(t;t_0,x_0)=-\partial_3\phi(t;t_0,x_0)F(x_0,t_0). $$

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Having established these technicalities, start to prove the claim.

Lemma: $\phi$ is monotonously increasing in its third argument, if $x_0<y_0$ then $\phi(t;t_0,x_0)<\phi(t;t_0,y_0)$ for all times.

Prf.: If solutions start at different values at the same time, then their curves can not cross, as at the crossing point they would be solutions of the same IVP and thus identical by the uniqueness theorem.

Cor.: Thus $\partial_3\phi(t;t_0,x_0)\ge 0$.

Now consider $$ h(t)=\phi(a;t,f(t)) $$ which projects back the graph of $f$ along the flow trajectories of $F$ to the initial values at $t=a$, or in the other direction, $h(t)$ is the initial value to get $f(t)$ at time $t$ with the flow, $$ f(t)=\phi(t;a,h(t)) $$ Then the dynamic for these initial values is obtained as \begin{align} h'(t)&=\partial_2\phi(a;t,f(t))+\partial_3\phi(a;t,f(t))f'(t) \\[.3em] &=\partial_3\phi(a;t,f(t))[-F(f(t),t)+f'(t)] \\[.3em] &\le 0 \end{align}

Thus $h$ is falling and $h(t)\le h(a)=f(a)$ for $t\ge a$. As the monotonicity of the initial values results in the same monotonicity in the flow lines, this proves the claim $$ f(t)=\phi(t;a,h(t))\le \phi(t;a,f(a))=\phi(t;a,g(a))=g(t). $$