On the orthogonal stability of the pexiderized quadratic equations in modular spaces (Q2878084)

Let \(X\) be a vector space. A function \(\rho:X\to [0,\infty]\) is called a modular if for each \(x, y\in X\) (i) \(\rho(x)=0\) if and only if \(x=0\); (ii) \(\rho (\alpha x)=\rho (x)\) for each scaler \(\alpha\) with \(|\alpha |=1\); (iii) \(\rho(\alpha x+\beta y)\leq \rho(x)+ \rho (y)\) if and only if \(\alpha+ \beta =1\) for \(\alpha, \beta \geq 0\). A modular \(\rho\) defines the modular space \(X_{\rho}=\{x\in X:\rho(\lambda x)\to 0\text{ as }\lambda \to 0\}\). In this paper the author proves the stability of orthogonal pexiderized quadratic functional equation \(f(x+y)+(x-y)=2g(x)+2h(y)\) \((x \perp y),\) in the spirit of Hyers-Ulam in modular spaces. Recall that \(\perp\) is the orthogonality in the sense of Rätz.