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$\lambda=a_1 (n-2)(n-1)+\gamma_1 (m-2)(m-1)+\beta_1 (n-1)(m-1)+\delta_1 (n-1)+\epsilon_1 (m-1),$

where $n=1,2,...,N$, $m=1,2,...,M$ is a necessary and sufficient condition for the linear partial differential equation

$(a_1x^2+a_2x+a_3)u_{x x}+(\beta_1xy+\beta_2x+\beta_3y+\beta_4)u_{x y} $

$+(\gamma_1y^2+\gamma_2y+\gamma_3)u_{y y}+(\delta_1x+\delta_2)u_x+(\epsilon_1y+\epsilon_2)u_y=\lambda u, $

where $a_i$, $\beta_j$, $\gamma_i$, $\delta_s$, $\epsilon_s$, $i=1,2,3$, $j=1,2,3,4$, $s=1,2$ are real or complex constants, to have polynomial solutions of the form

$u(x,y)=\sum_{n=1}^N\sum_{m=1}^Mu_{n m}x^{n-1}y^{m-1}.$

The proof of this result is obtained using a functional analytic method which reduces the problem of polynomial solutions of such partial differential equations to an eigenvalue problem of a specific linear operator in an abstract Hilbert space. The main result of this paper generalizes previously obtained results by other researchers.

The problem of the existence of complex $\ell_1$ solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and $z-$ transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.

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