On the Stability of a Functional Equation Associated with the Fibonacci Numbers

Abstract

We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>and h are given functions.

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