The purpose of this paper is to collect some tools of Nevanlinna theory, which are used in the study of meromorphic solutions of differential, difference, and \(q\)-difference equations, in a common toolbox for the study of general classes of functional equations by introducing the notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. Let \(\mathcal{M}\) be the field of meromorphic functions in the complex plane, and let \(\mathcal{N}\subset\mathcal{M}\). A linear operator \(L:\mathcal{N}\rightarrow\mathcal{N}\) is a good linear operator for \(\mathcal{N}\) with the exceptional set property \(\mathbb{P}\) if the following two properties hold:
(1) for any \(f\in\mathcal{N}\), \(\displaystyle m\left(r,\frac{L(f)}f\right)=o(T(r,f)) \) as \(r\to\infty\) outside of an exceptional set \(E_f\) with the property \(\mathbb{P}\);
(2) the counting functions \(N(r,f)\) and \(N(r,L(f))\) are asymptotically equivalent.
The authors apply these methods to study the growth of meromorphic solutions of the functional equation \(M(z, f) + P(z, f) = h(z)\), where \(M(z, f)\) is a linear polynomial in f and \(L(f)\), where \(L\) is good linear operator, \(P(z, f)\) is a polynomial in \(f\) with degree deg \(P\geq2\), both with small meromorphic coefficients, and \(h(z)\) is a meromorphic function. For example, let, for any \(f\in\mathcal{N}\), \(N(r,f)=o(T(r,f))\) as \(r\to\infty\) outside of an exceptional set \(E_f\) with the property \(\mathbb{P}\), and let \(\{L_k,k\in J\}\) be a finite collection of good linear operators for \(\mathcal{N}\) with the exceptional set property \(\mathbb{P}\). If \(f_1,f_2\in\mathcal{N}\) are any two meromorphic solutions of the equation \[ M(z,f)+P(z,f)=h(z)\,, \] where \(P(z,f)=b_2(z)f^2+\dots+b_n(z)f^n\) is a polynomial in \(f\) with small meromorphic coefficients, \(h\in\mathcal{M}\), and \(M(z,f)\) is a linear polynomial in \(f\) and \(L_k(f)\), \(k\in J\), with small meromorphic coefficients, then \[ T(r,f_2)=O(T(r,f_1))+o(T(r,f_1)), \] where \(r\to\infty\) outside of an exceptional set \(E\) with the property \(\mathbb{P}\).