On entire solutions for several systems of quadratic trinomial Fermat type functional equations in $${\mathbb {C}}^2$$

The precise forms of finite order transcendental entire solutions for three kinds of systems of quadratic trinomial Fermat type difference equations, Fermat type partial differential equations and Fermat type partial differential-difference equations in \({\mathbb {C}}^2\) are described by applying the Nevanlinna theory, which improves and generalizes previous results in Luo et al. (Open Math 19(1):1018–1028, 2021) and Xu and Jiang (Rev Real Acad Cienc Exactas Fís Nat Ser A Mat 116:1–19, 2022). Some examples are given to show these results.