Let X, Y be real normed spaces and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho _{+}^{'},\rho _{-}^{'}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>+</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mo>,</mo> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>-</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation> be norm derivatives. In this work, we solve a system of functional equations <Equation ID="Equ11"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} \rho _{+}^{'}(f(x),f(y))=g(x)\rho _{+}^{'}(x,y)\\ \rho _{-}^{'}(f(x),f(y))=g(x)\rho _{-}^{'}(x,y) \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>+</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>+</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>-</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mmultiscripts> <mi>ρ</mi> <mrow> <mo>-</mo> </mrow> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where the functions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f:X\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g:X\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> are unknown.

Generalized orthogonality equations in normed spaces

期刊论文
出版模式：

开放获取
发表日期：

2025年12月27日

查看全文

Abayomi Dennis Epebinu,
Radosław Łukasik

摘要

Let X, Y be real normed spaces and let $\rho _{+}^{'},\rho _{-}^{'}$ $ρ_{+}^{^{'}}, ρ_{-}^{^{'}}$ be norm derivatives. In this work, we solve a system of functional equations $\begin{aligned} {\left\{ \begin{array}{ll} \rho _{+}^{'}(f(x),f(y))=g(x)\rho _{+}^{'}(x,y)\\ \rho _{-}^{'}(f(x),f(y))=g(x)\rho _{-}^{'}(x,y) \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} ρ_{+}^{^{'}} (f (x), f (y)) = g (x) ρ_{+}^{^{'}} (x, y) \\ ρ_{-}^{^{'}} (f (x), f (y)) = g (x) ρ_{-}^{^{'}} (x, y) \end{matrix}) \end{matrix}$ where the functions $f:X\rightarrow Y$ $f : X \to Y$ and $g:X\rightarrow \mathbb {R}$ $g : X \to R$ are unknown.