Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the equality problem of generalized Bajraktarevi\'c means   The purpose of this paper is to investigate the equality problem of
generalized Bajraktarevi\'c means, i.e., to solve the functional equation
\begin{equation}\label{E0}\tag{*}
  f^{(-1)}\bigg(\frac{p_1(x_1)f(x_1)+\dots+p_n(x_n)f(x_n)}{p_1(x_1)+\dots+p_n(x_n)}\bigg)=g^{(-1)}\bigg(\frac{q_1(x_1)g(x_1)+\dots+q_n(x_n)g(x_n)}{q_1(x_1)+\dots+q_n(x_n)}\bigg),
\end{equation} which holds for all $x=(x_1,\dots,x_n)\in I^n$, where $n\geq 2$,
$I$ is a nonempty open real interval, the unknown functions
$f,g:I\to\mathbb{R}$ are strictly monotone, $f^{(-1)}$ and $g^{(-1)}$ denote
their generalized left inverses, respectively, and
$p=(p_1,\dots,p_n):I\to\mathbb{R}_{+}^n$ and
$q=(q_1,\dots,q_n):I\to\mathbb{R}_{+}^n$ are also unknown functions. This
equality problem in the symmetric two-variable (i.e., when $n=2$) case was
already investigated and solved under sixth-order regularity assumptions by
Losonczi in 1999. In the nonsymmetric two-variable case, assuming three times
differentiability of $f$, $g$ and the existence of $i\in\{1,2\}$ such that
either $p_i$ is twice continuously differentiable and $p_{3-i}$ is continuous
on $I$, or $p_i$ is twice differentiable and $p_{3-i}$ is once differentiable
on $I$, we prove that \eqref{E0} holds if and only if there exist four
constants $a,b,c,d\in\mathbb{R}$ with $ad\neq bc$ such that \begin{equation*}
  cf+d>0,\qquad
  g=\frac{af+b}{cf+d},\qquad\mbox{and}\qquad q_\ell=(cf+d)p_\ell\qquad
(\ell\in\{1,\dots,n\}). \end{equation*} In the case $n\geq 3$, we obtain the
same conclusion with weaker regularity assumptions. Namely, we suppose that $f$
and $g$ are three times differentiable, $p$ is continuous and there exist
$i,j,k\in\{1,\dots,n\}$ with $i\neq j\neq k\neq i$ such that $p_i,p_j,p_k$ are
differentiable.

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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol