2024/10/21 by Zsolt Páles, Páles, Zsolt, Amr Zakaria +1
Mathematics · #26E60 #39B30 #39B40 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2410.16074
openalex publication_date 2024/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider the equality problem of generalized Bajraktarević means, i.e., we are going to solve the functional equation f(-1)((p1(x1)f(x1)+…+pn(xn)f(xn))/(p1(x1)+…+pn(xn)))=g(-1)((q1(x1)g(x1)+…+qn(xn)g(xn))/(q1(x1)+…+qn(xn))), which holds for all x=(x1,…,xn)∈ In, where n≥ 2, I is a nonempty open real interval, the unknown functions f,g:I→ℝ are strictly monotone, f(-1) and g(-1) denote their generalized left inverses, respectively, and the vector-valued weight functions p=(p1,…,pn):I→ℝ+n and q=(q1,…,qn):I→ℝ+n are also unknown. This equality problem in the symmetric two-variable case (i.e., when n=2 and p1=p2, q1=q2) was solved under sixth-order regularity assumptions by Losonczi in 1999. The authors of this paper improved this result in 2023 by reaching the same conclusion assuming only first-order differentiability. In the nonsymmetric case, assuming third-order differentiability of f, g and the first-order differentiability of at least three of the functions p1,…,pn, Grünwald and Páles proved that \eq0 holds if and only if there exist four constants a,b,c,d∈ℝ with ad≠ bc such that cf+dgt;0, g=(af+b)/(cf+d),\qquadand q_ℓ=(cf+d)p_ℓ (ℓ∈\1,…,n\). The main goal of this paper is to establish the same conclusion under first-order differentiability.