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Additive \(( \rho_1, \rho_2)\)-functional inequalities in complex Banach spaces. (English) Zbl 07225654
Daras, Nicholas J. (ed.) et al., Computational mathematics and variational analysis. Cham: Springer. Springer Optim. Appl. 159, 227-245 (2020).
Summary: In this paper, we introduce and solve the following additive \(( \rho_1, \rho_2)\)-functional inequalities: \[ \begin{split} \|f(x-y) - f(x)+ f(y)\| \ge \|\rho_1 (f(x+y)-f(x)-f(y))\| \\ + \|\rho_2 (f(y-x)-f(y)+f(x))\|, \end{split} \tag{1} \] where \(\rho_1\) and \(\rho_2\) are fixed complex numbers with \(| \rho_1| + | \rho_2| > 1\), and \[ \begin{split} \|f(x+y) - f(x)- f(y)\| \ge \|\rho_1 (f(x-y)-f(x)+f(y))\| \\ + \|\rho_2 (f(y-x)-f(y)+f(x))\|, \end{split}\tag{2} \] where \(\rho_1\) and \(\rho_2\) are fixed complex numbers with \(1 + | \rho_1| > | \rho_2| > 1\). Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive \(( \rho_1, \rho_2)\)-functional inequalities (2) and (1) in complex Banach spaces.
For the entire collection see [Zbl 1446.65002].
MSC:
65Jxx Numerical analysis in abstract spaces
49Jxx Existence theories in calculus of variations and optimal control
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