论文标题
从Matkowski的不变性问题得出的功能方程的常规解决方案
Regular solutions of a functional equation derived from the invariance problem of Matkowski means
论文作者
论文摘要
The main result of the present paper is about the solutions of the functional equation \Eq{*}{ F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\qquad x,y\in I, } derived originally, in a natural way, from the invariance problem of generalized weighted quasi arithmetic的手段，其中$ f，f_1，f_2，g_1，g_2：i \ to \ mathbb {r} $和$ g：g_1（i）+g_2（i）+g_2（i）\ to \ mathbb {r} $与$ 0 \ g'___________________________________2设置$ i $代表$ \ mathbb {r} $的非空地开放子室间隔。 除此之外，我们还将涉及解决方案不一定是规律的。更准确地说，我们将求解上述方程式，假设$ f $是$ i $的仿射，而$ g_1 $，而$ g_2 $在同一意义上是严格单调的连续函数，其次，$ g_1 $ and $ g_2 $是具有常见加性零件的不可变化的仿射功能。
The main result of the present paper is about the solutions of the functional equation \Eq{*}{ F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\qquad x,y\in I, } derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where $F,f_1,f_2,g_1,g_2:I\to\mathbb{R}$ and $G:g_1(I)+g_2(I)\to\mathbb{R}$ are the unknown functions assumed to be continuously differentiable with $0\notin g'_1(I)\cup g'_2(I)$, and the set $I$ stands for a nonempty open subinterval of $\mathbb{R}$. In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that $F$ is affine on $I$ and $g_1$ and $g_2$ are continuous functions strictly monotone in the same sense, and secondly that $g_1$ and $g_2$ are invertible affine functions with a common additive part.
