Let $\phi,f:(0,\infty)\to [0,\infty)$ be two functions that are continuous and Lebesgue integrable on $(0,b]$, for any $b>0$. Assume also that $$ \int_0^x\phi(x-y)f(y)\,dy=1,\quad \text{for each }x>0. $$ i) Does $\phi$ being non-increasing imply that $f$ is also non-increasing?
ii) Does $\phi$ being decreasing imply that $f$ is also non-increasing?
$\phi$ is non-increasing if $\phi(x)\ge\phi(y)$ whenever $x<y$, and $\phi$ is decreasing if $\phi(x)>\phi(y)$ whenever $x<y$ and $\phi(x)>0$, and $\phi(y)=0$ for all $x<y$ if $\phi(x)=0$.