The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation

which holds for all \((x_1,\dots ,x_n)\in I^n\), where \(n\ge 2\), I is a nonempty open real interval, the unknown functions \(f,g:I\rightarrow {\mathbb {R}}\) are strictly monotone, \(f^{(-1)}\) and \(g^{(-1)}\) denote their generalized left inverses, respectively, and \(p=(p_1,\dots ,p_n):I\rightarrow {\mathbb {R}}_{+}^n\) and \(q=(q_1,\dots ,q_n):I\rightarrow {\mathbb {R}}_{+}^n\) 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 (Aequationes Math 58(3):223–241, 1999). In the nonsymmetric two-variable case, assuming the 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 (*) holds if and only if there exist four constants \(a,b,c,d\in {\mathbb {R}}\) with \(ad\ne bc\) such that

In the case \(n\ge 3\), 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 \(i\ne j\ne k\ne i\) such that \(p_i,p_j,p_k\) are differentiable.

本文的目的是研究广义Bajraktarević均值问题，即求解泛函

它适用于所有\（（x_1，\ dots，x_n）\ in I ^ n \），其中\（n \ ge 2 \），I是一个非空的开放实数区间，未知函数\ {f，g：I \ rightarrow {\ mathbb {R}} \）严格是单调的，\（f ^ {（-1）} \）和\（g ^ {（-1）} \）分别表示它们的广义左逆，而\ （p = {p_1，\ dots，p_n）：I \ rightarrow {\ mathbb {R}} _ {+} ^ n \）和\（q = {q_1，\ dots，q_n）：I \ rightarrow {\ mathbb {R}} _ {+} ^ n \）也是未知函数。对称二元变量中的等式问题（即\（n = 2 \））案例已经由Losonczi在六阶正则性假设下进行了调查和解决（Aequationes Math 58（3）：223-241，1999）。在非对称二变量情况下，假设f，g的三倍可微性和\（i \ in \ {1,2 \} \）的存在，使得\（p_i \）两次连续可微，而\（p p_ {3-i} \）在I上是连续的，或者\（p_i \）在I上是可微的两次，而\（p_ {3-i} \）在I上是可微的，我们证明（*）成立且仅当在{\ mathbb {R}} \中存在四个常量\（a，b，c，d \）与\（ad \ ne bc \） 这样

在\（n \ ge 3 \）的情况下，我们以较弱的规律性假设获得了相同的结论。即，我们假设f和g是可微的三倍，p是连续的，并且存在\（i，j，k \ in \ {1，\ dots，n \} \）与\（i \ ne j \ ne k \ ne i \）使得\（p_i，p_j，p_k \）是可微的。