On the stability of the linear functional equation in a single variable on complete metric groups

In this paper we obtain a result on Hyers–Ulam stability of the linear functional equation in a single variable <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f(\varphi (x))=g(x) \cdot f(x)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>·</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> on a complete metric group. Copyright Springer Science+Business Media New York 2014