Beschreibung:
<jats:p>Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>-difference equations are largely similar. The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>M</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>M</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a linear polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math> is good linear operator, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> with degree deg <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>P</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math>, both with small meromorphic coefficients, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is a meromorphic function.</jats:p>