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

Jung Rye Lee (), Choonkil Park () and Themistocles M. Rassias ()
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Jung Rye Lee: Daejin University
Choonkil Park: Hanyang University
Themistocles M. Rassias: National Technical University of Athens

A chapter in Computational Mathematics and Variational Analysis, 2020, pp 227-245 from Springer

Abstract: Abstract In this paper, we introduce and solve the following additive (ρ 1, ρ 2)-functional inequalities: 1 f x − y − f ( x ) + f ( y ) ≥ ∥ ρ 1 ( f ( x + y ) − f ( x ) − f ( y ) ) ∥ + ρ 2 f ( y − x ) − f ( y ) + f ( x ) , $$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x-y\right) - f(x )+ f(y)\right\| &\displaystyle \ge &\displaystyle \|\rho_1 (f(x+y)-f(x)-f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\|, {} \end{array} \end{aligned} $$ where ρ 1 and ρ 2 are fixed complex numbers with |ρ 1| + |ρ 2| > 1, and 2 f x + y − f ( x ) − f ( y ) ≥ ∥ ρ 1 ( f ( x − y ) − f ( x ) + f ( y ) ) ∥ + ρ 2 f ( y − x ) − f ( y ) + f ( x ) , $$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x+y\right) - f(x )- f(y)\right\|&\displaystyle \ge &\displaystyle \|\rho_1 (f(x-y)-f(x)+f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\| ,{} \end{array} \end{aligned} $$ 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.

Date: 2020
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DOI: 10.1007/978-3-030-44625-3_13

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