Entire Solutions for the Systems of Second-Order Partial Differential-Difference Fermat-Type Equations in $$\mathbb{C}^{2}$$

The prime intention of this paper is to study the conditions for existence and the forms of entire solutions of the systems of nonlinear difference and second-order partial differential-difference Fermat-type equations in $$\mathbb{C}^{2}$$ . One of our main results extends a result of [15] and other results correspond to some new forms of equations that have never been tackled in the literature. We also illustrate a handful number of examples to show the accuracy of our results.