Our paper focuses on exploring the existence and characteristics of solutions for various complex difference equations of Fermat-type such as \( f(z)^2 + \alpha (z)^2 (e^{P(z)})^2 f(z+c)^2 = Q(z) e^{2\beta (z)} \) and \( f(z)^2 + \alpha (z)^2 (e^{P(z)})^2 (\Delta _c f(z))^2 = Q(z) e^{2\beta (z)} \) , where \( \alpha (z) \) , \( \beta (z) \) , \( P(z) \) , and \( Q(z) \) are non-zero polynomials in \( {\mathbb {C}} \) and \( c \in {\mathbb {C}} {\setminus } \{0\} \) . Our findings represent significant advancements over previous theorems established by Long Jian-ren and Qin Da-zhuan (Appl Math J Chin Univ 39:69-88, 2024), particularly in terms of both the existence and explicit forms of solutions. Moreover, some examples are provided to strengthen our results.
