Bi-additive s-Functional Inequalities and Quasi-∗-Multipliers on Banach ∗-Algebras

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, Department of Mathematics

Chapter Chapter 10 in Ulam Type Stability, 2019, pp 199-215 from Springer

Abstract: Abstract Park introduced and investigated the following bi-additive s-functional inequalities 10.1 ∥ f ( x + y , z + w ) + f ( x + y , z − w ) + f ( x − y , z + w ) + f ( x − y , z − w ) − 4 f ( x , z ) ∥ ≤ s 4 f x + y 2 , z − w + 4 f x − y 2 , z + w − 4 f ( x , z ) + 4 f ( y , w ) , $$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \| f(x{+}y, z{+}w) {+} f(x{+}y, z{-}w) {+} f(x{-}y, z{+}w) {+} f(x{-}y, z{-}w) {-}4f(x,z)\| \\ &\displaystyle \le \left \|s \left(4f\left(\frac{x{+}y}{2}, z{-}w\right) {+} 4f\left(\frac{x{-}y}{2}, z{+}w\right) {-} 4f(x,z ){+} 4 f(y, w)\right)\right\| , \end{array} \end{aligned} $$ 10.2 4 f x + y 2 , z − w + 4 f x − y 2 , z + w − 4 f ( x , z ) + 4 f ( y , w ) ≤ ∥ s ( f ( x + y , z + w ) + f ( x + y , z − w ) + f ( x − y , z + w ) + f ( x − y , z − w ) − 4 f ( x , z ) ) ∥ , $$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \left\|4f\left(\frac{x+y}{2}, z-w\right) +4 f\left(\frac{x-y}{2}, z+w\right) -4 f(x,z )+4 f(y, w)\right\| \\ &\displaystyle \qquad \le \|s ( f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w)\\ &\displaystyle \qquad + f(x-y, z-w) -4f(x,z) )\| , \end{array} \end{aligned} $$ where s is a fixed nonzero complex number with |s|

Keywords: Quasi-multiplier on C ∗-algebra; Quasi-∗-multiplier on Banach algebra; Hyers-Ulam stability; Direct method; Bi-additive s-functional inequality; Primary 39B52, 39B82, 43A22; Secondary 39B62, 46L05 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/978-3-030-28972-0_10

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