The existence and forms of solutions for several systems of the Fermat-type difference-differential equations

Abstract

The sets of functional equations

$$\left\{\begin{array}{ccc}\hfill f^{m_1}+g^{n_1}& =R_1,\hfill & \hfill \\ \hfill f^{m_2}+g^{n_2}& =R_2,\hfill \end{array}\right.$$

can be regarded as the Fermat-type system, where $m_1,m_2,n_1,n_2$ are positive integers, $R_1,R_2$ are nonzero polynomials. The purpose of this paper is to study the existence of solutions for several Fermat-type systems of the difference-differential equations

$$\left\{\begin{array}{ccc}\hfill {\left[f_1^{\prime }\left(z\right)\right]}^2+f_2{\left(z+c\right)}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2^{\prime }\left(z\right)\right]}^2+f_1{\left(z+c\right)}^2& =R_2\left(z\right),\hfill \end{array}\right.\left\{\begin{array}{ccc}\hfill {\left[f_1^{\prime }\left(z\right)\right]}^2+{\left[f_2\left(z+c\right)-f_1\left(z\right)\right]}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2^{\prime }\left(z\right)\right]}^2+{\left[f_1\left(z+c\right)-f_2\left(z\right)\right]}^2& =R_2\left(z\right),\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{ccc}\hfill {\left[f_1\left(z\right)\right]}^2+{\left[f_2\left(z+c\right)-f_1\left(z\right)\right]}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2\left(z\right)\right]}^2+{\left[f_1\left(z+c\right)-f_2\left(z\right)\right]}^2& =R_2\left(z\right).\hfill \end{array}\right.$$

Our results about the existence and the forms of solutions for these Fermat types systems are some improvements of the previous theorems given by Gao, Liu, Yang, Cao, Qi. Meanwhile, we give some examples to utilize the existence and forms of solutions for such systems in each case.