= Solution
Work in the <Banach space> $C([0,1])$ with its <supremum norm>, and take the nonempty closed bounded convex set $M=\{u:100\leq u(t)\leq121\text{ for all }t\}$. Define
$$
(Tu)(t)=101+\int_0^t\bigl(\sqrt{|u(s)|}+1\bigr)\,ds.
$$
For $u\in M$ the absolute value is unnecessary, and $\boxed{101\leq Tu(t)\leq101+12=113\leq121}$. Thus $T:M\to M$. Also, for $a,b\geq100$,
$$
|\sqrt a-\sqrt b|=\frac{|a-b|}{\sqrt a+\sqrt b}\leq\frac1{20}|a-b|,
$$
so $\|Tu-Tv\|_\infty\leq\tfrac1{20}\|u-v\|_\infty$ and in particular $T$ is continuous.
The image is uniformly bounded, and $|Tu(t)-Tu(s)|\leq12|t-s|$, so it is <equicontinuous>. The <Arzelà-Ascoli theorem> says that a uniformly bounded equicontinuous family in $C([0,1])$ has <relatively compact> <closure> in the uniform norm. Hence $T$ is compact, and the <Schauder fixed-point theorem> supplies $\boxed{u=Tu}$, the required solution. In fact the displayed contraction estimate also lets the <contraction mapping theorem> give uniqueness within $M$ and convergence of the corresponding <Picard iteration>. The initial constant here is the printed integer $101$.
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