A contraction type argument

Let be a function. First, we have the following trivial result:

Observation. If a non-negative funtion satisfies the following inequality

for all , then we must have in .

The proof of the above observation depends on the non-negativity of . It is worth noting that we do not require the continuity of . Here in this post, we are interested in the following

Main result. If the non-negative, continuous function satisfies near zero and

for all , then there exists some in such a way that in .

The above result, in a special setting, appears in a recent work of Hyder and Sire. (See this if you cannot access the content.) Now we discuss a proof of the above main result, original due to one of my young colleagues.