Generalized orthogonality equations in finite-dimensional normed spaces

Abstract

Let X, Y be real normed spaces and let rho '_+, rho '_- be norm derivatives. In this work, we solve a system of functional equations ρ+′(f(x),f(y))=g(x)ρ+′(x,y),ρ-′(f(x),f(y))=g(x)ρ-′(x,y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll}\\rho '_+(f(x),f(y))=g(x)\\rho '_+(x,y),\\\\ \\rho '_-(f(x),f(y))=g(x)\\rho '_-(x,y), \\end{array}\\right. } \\end{aligned}$$\\end{document}with unknown functions f:X!rightarrow !Y, g:Xrightarrow mathbb {R}. Moreover, we give partial answer to open problem posed in 2010.