<p>We study real-valued solutions of the equation <Equation ID="Equ34"> <EquationSource Format="TEX">\( \varphi (x)=\sum ^n_{j=1}p_j(x)\varphi \left( f_j(x)\right) \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>p</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>φ</mi> <mfenced close=")" open="("> <msub> <mi>f</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </math></EquationSource> </Equation>as well as its multiplicative version <Equation ID="Equ35"> <EquationSource Format="TEX">\( \psi (x)=\prod ^n_{j=1}\psi \left( f_j(x)\right) ^{p_j(x)}. \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mi>ψ</mi> <msup> <mfenced close=")" open="("> <msub> <mi>f</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <msub> <mi>p</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Both find interesting applications in probability theory, mainly in various characterization problems. As a corollary we obtain a new characterization of the exponential function. The results generalize a number of theorems proved by M. Kuczma, B. Choczewski and R. Ger, J.A. Baker, M.C. Zdun also a previous one proved by the author and W. Jarczyk.</p>