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Additive (ρ1, ρ2)-Functional Inequalities in Complex Banach Spaces

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Computational Mathematics and Variational Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 159))

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Abstract

In this paper, we introduce and solve the following additive (ρ 1, ρ 2)-functional inequalities:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x-y\right) - f(x )+ f(y)\right\| &\displaystyle \ge &\displaystyle \|\rho_1 (f(x+y)-f(x)-f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\|, {} \end{array} \end{aligned} $$
(1)

where ρ 1 and ρ 2 are fixed complex numbers with |ρ 1| + |ρ 2| > 1, and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x+y\right) - f(x )- f(y)\right\|&\displaystyle \ge &\displaystyle \|\rho_1 (f(x-y)-f(x)+f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\| ,{} \end{array} \end{aligned} $$
(2)

where ρ 1 and ρ 2 are fixed complex numbers with 1 + |ρ 1| > |ρ 2| > 1. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of the additive (ρ 1, ρ 2)-functional inequalities (2) and (1) in complex Banach spaces.

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Acknowledgements

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).

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Authors and Affiliations

  1. Department of Mathematics, Daejin University, Pocheon-si, Korea

    Jung Rye Lee

  2. Department of Mathematics, Hanyang University, Seoul, South Korea

    Choonkil Park

  3. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Authors
  1. Jung Rye Lee
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  2. Choonkil Park
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  3. Themistocles M. Rassias
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Correspondence to Themistocles M. Rassias .

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Editors and Affiliations

  1. Department of Mathematics, Hellenic Military Academy, Vari Attikis, Greece

    Nicholas J. Daras

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Lee, J.R., Park, C., Rassias, T.M. (2020). Additive (ρ1, ρ2)-Functional Inequalities in Complex Banach Spaces. In: Daras, N., Rassias, T. (eds) Computational Mathematics and Variational Analysis. Springer Optimization and Its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-030-44625-3_13

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