The extensibility of the Diophantine triple $\{2, b, c\}$

The aim of this paper is to consider the extensibility of the Diophantine triple $\{2,b,c\}$, where $2<b<c$, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of $c$'s (depending on $b$). As corollary, for example, we prove that for $b/2-1$ prime, all Diophantine quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular.