Our study looks into the non-existence of transcendental entire solutions for partial differential-difference equation of Fermat typef(z)∂f(z)∂z1+∂f(z)∂z2+ .... +∂f(z)∂znm+ f(z + c)l = 1,where l, m are positive integers and c = (c1, c2, ...., cn) ∈ Cn\{0} with finite order constraintin Cn. Additionally, using the Nevanlinna theory of meromorphic functions in various complex variables, this article examines the existence and an explicit form of solutions to several Fermat-type partial differential-difference equations in Cn. We obtain conclusions about the existence and form of transcendental entire solutions for specific class of functional equation of the type of Fermat with more general form ofαf(z) + β1∂f(z)∂z1+ β2∂f(z)∂z2+ .... + βn∂f(z)∂zn2+ f(z + c)2 = eg(z),where α, βj (j = 1, 2, ..., n) are constants in C and g(z) form a polynomial in Cn with a finite order constraint. The conclusions offered as solutions to these equations have significantly improved the theorems previously proposed by Xu, Cao, Liu, Wang, Zhang, and Zheng [17, 32, 33, 34, 35]. The method used in this article is different from the one used in previous ones. The existence conditions and forms of transcendental entire solutions with a finite order of such equations are demonstrated by providing several examples.
2010 Mathematics Subject Classification. 30D35.