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Let $\alpha, \beta:\mathbb R_+\to [0,1]$ be continuous and decreasing functions s.t. $\alpha(0)=1=\beta(0)$ and $\alpha, \beta$ are continuously differentiable on $(0,\infty)$ satisfying for some $c>0$ $$-\frac{c}{\sqrt{t}}\le\alpha'(t), \beta'(t)\le 0,\quad \forall t>0.$$ Let $f,g :\mathbb R_+\to [0,1]$ be continuous s.t. for all $t\ge 0$ \begin{eqnarray} \|\alpha-\beta\|_t &\le& C\sqrt{t} \big(\|\alpha-\beta\|_t+\|f-g\|_t\big) \\ \|f-g\|_t &\le& C\sqrt{t}\left(\|\alpha-\beta\|_t+\|f-g\|_t + \int_0^t|\alpha'(s)-\beta'(s)|ds\right), \end{eqnarray} where $C>0$ is some constant and $\|\cdot\|_t$ denotes the uniform norm of functions on $[0,t]$. Can we prove the existence of $T>0$ s.t. $(\alpha,f)=(\beta,g)$ over the interval $[0,T]$?

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$\newcommand{\al}{\alpha}\newcommand{\be}{\beta}$The answer is no.

E.g., suppose that for all small enough $t>0$ we have \begin{equation*} f(t)=t,\quad g(t)=0,\quad \al'(t)=-\frac1{\sqrt t}\,\Big(1+\sin\frac1t\Big),\quad \be'(t)=-\frac1{\sqrt t}, \end{equation*} with $\al(0)=1=\be(0)$.

Then all your conditions hold. In particular, for $t\downarrow0$, \begin{equation*} |\al(t)-\be(t)|=\Big|\int_{1/t}^\infty\frac{\sin z}{z^{3/2}}\,dz\Big|=O(t^{3/2}) \end{equation*} and \begin{equation*} \int_0^t|\al'(s)-\be'(s)|\,ds=\int_{1/t}^\infty\frac{|\sin z|}{z^{3/2}}\,dz\asymp t^{1/2}, \end{equation*} so that
\begin{equation*} \begin{aligned} \|\al-\be\|_t &\le C\sqrt{t} \big(\|\al-\be\|_t+\|f-g\|_t\big) \\ \|f-g\|_t &\le C\sqrt{t}\left(\|\al-\be\|_t+\|f-g\|_t + \int_0^t|\al'(s)-\be'(s)|ds\right) \end{aligned} \tag{1}\label{1} \end{equation*} for some real $C>0$ and all $t$ in some right neighborhood of $0$. So, inequalities \eqref{1} hold for some real $C>0$ and all real $t\ge0$.

Yet, $f\ne g$ in any right neighborhood of $0$. So, $(\al,f)\ne(\be,g)$ on $[0,T]$, for any real $T>0$.

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  • $\begingroup$ Thank you very much Iosif for your example. Indeed, my family is contaminated by covid and I have to stay at home (with my laptop left in office). So sorry for not replying in time $\endgroup$
    – GJC20
    Commented Jul 12, 2022 at 12:09
  • $\begingroup$ @GJC20 : Thank you for your response. Best wishes to you and your family! $\endgroup$ Commented Jul 12, 2022 at 13:48

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