Stability of a sum form functional equation on open domain (Q2714167)

The functional equation NEWLINE\[NEWLINE \sum^n_{i=1}\varphi (p_i)=d \tag{1} NEWLINE\]NEWLINE where \(n\geq 3\) is fixed integer, \(d\in \mathbb R\) (the set of real numbers), \(\varphi:]0,1[\to\mathbb R\), and (1) holds for all \((p_1\dots p_n)\) with NEWLINE\[NEWLINE 0<p_i,\qquad (i=1,\dots n), \quad \sum^n_{i=1} p_i=1, \tag{2} NEWLINE\]NEWLINE is solved by \textit{L. Losonczi} [Publ. Math. 32, 57-71 (1985; Zbl 0588.39005)] by proving that NEWLINE\[NEWLINE \varphi (p)=A(p)-\frac {A(1)-d}n, \quad p\in ]0,1[ NEWLINE\]NEWLINE with \(A:\mathbb R\to \mathbb R\) satisfying the Cauchy functional equation NEWLINE\[NEWLINE A(x+y)=A(x)+A(y), \qquad x,y\in \mathbb R. \tag{3} NEWLINE\]NEWLINE In the present paper the author proves that equation (1) is stable, that is, supposing that the inequality NEWLINE\[NEWLINE \left|\sum^n_{i=1}\varphi(p_i)-d\right|\leq \varepsilon NEWLINE\]NEWLINE holds for some integer \(n\geq 3\) and \(d,\varepsilon \in \mathbb R\), and for all \((p_1\dots p_n)\) with property (2) then there exists a \(K\in \mathbb R\) such that NEWLINE\[NEWLINE \left|\varphi (p)-A(p)+\frac {A(1)-d}n \right|\leq K\varepsilon , \qquad p\in ]0,1[ NEWLINE\]NEWLINE for some function \(A:\mathbb R\to \mathbb R\) satisfying equation (3).