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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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