What to write in a chapter of a book?
Rather than answering the specific question ("What to write in a chapter of a book?") I'm going to suggest that this is an XY Problem. The real first question is whether this is a real book, or if it's a predatory publisher sending out an email blast to random victims. The limited evidence in the question and comments suggests that it may well be a predatory publisher:


The recipient is not a well known researcher 
There isn't a clear and well-described topic for the chapter
The recipient doesn't know the putative professors personally


An edit to the question suggests that the authors may be real people, but with the other red flags that are raised I would still be very suspicious. The professors may not be the actual senders of the letter, or they may in turn be victims of a scam. 

It's not impossible that this is a real book, but the odds are very much against it. Please be cautious before doing something that may damage your reputation and career.

Edit since in the comments someone is skeptical that there is such a thing as predatory book-chapter scams:


Example 1
Example 2
Example 3
Example 4


The claim in the comments is that book chapters can't be predatory because "You can't have a predatory book". These examples show charges for chapters, but also through selling worthless, low-quality books to libraries. 

In general these predatory publishers depend on naive people; that includes people who think "You can't have a predatory book".