Entire and meromorphic solutions for systems of the differential difference equations

Abstract

Abstract With the help of the Nevanlinna theory of meromorphic functions, the purpose of this article is to describe the existence and the forms of transcendental entire and meromorphic solutions for several systems of the quadratic trinomial functional equations: f ( z ) 2 + 2 α f ( z ) g ( z + c ) + g ( z + c ) 2 = 1 , g ( z ) 2 + 2 α g ( z ) f ( z + c ) + f ( z + c ) 2 = 1 , \left\{\begin{array}{l}f{\left(z)}^{2}+2\alpha f\left(z)g\left(z+c)+g{\left(z+c)}^{2}=1,\\ g{\left(z)}^{2}+2\alpha g\left(z)f\left(z+c)+f{\left(z+c)}^{2}=1,\end{array}\right. f ( z + c ) 2 + 2 α f ( z + c ) g ′ ( z ) + g ′ ( z ) 2 = 1 , g ( z + c ) 2 + 2 α g ( z + c ) f ′ ( z ) + f ′ ( z ) 2 = 1 , \left\{\begin{array}{l}f{\left(z+c)}^{2}+2\alpha f\left(z+c)g^{\prime} \left(z)+g^{\prime} {\left(z)}^{2}=1,\\ g{\left(z+c)}^{2}+2\alpha g\left(z+c)f^{\prime} \left(z)+f^{\prime} {\left(z)}^{2}=1,\end{array}\right. and f ( z + c ) 2 + 2 α f ( z + c ) g ″ ( z ) + g ″ ( z ) 2 = 1 , g ( z + c ) 2 + 2 α g ( z + c ) f ″ ( z ) + f ″ ( z ) 2 = 1 . \left\{\begin{array}{l}f{\left(z+c)}^{2}+2\alpha f\left(z+c){g}^{^{\prime\prime} }\left(z)+{g}^{^{\prime\prime} }{\left(z)}^{2}=1,\\ g{\left(z+c)}^{2}+2\alpha g\left(z+c){f}^{^{\prime\prime} }\left(z)+{f}^{^{\prime\prime} }{\left(z)}^{2}=1.\end{array}\right. We obtain a series of results on the forms of the entire solutions with finite order for such systems, which are some improvements and generalizations of the previous theorems given by Gao et al. Moreover, we provide some examples to explain the existence and forms of solutions for such systems in each case.