# On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric Spaces

On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric... Let $$\mathbb {R}$$ R be the set of real numbers, $$(G,+)$$ ( G , + ) be a commutative group and d be a complete ultrametric on G that is invariant (i.e., $$d(x + z, y + z)= d(x, y$$ d ( x + z , y + z ) = d ( x , y ) for $$x, y, z \in G$$ x , y , z ∈ G ). Under some weak natural assumptions on the function $$\gamma :{\mathbb {R}}^2\rightarrow [0,\infty )$$ γ : R 2 → [ 0 , ∞ ) , we study the generalised hyperstability results when $$f:\mathbb {R}\rightarrow G$$ f : R → G satisfy the following radical cubic inequality \begin{aligned} d\big (f(\root 3 \of {x^3+y^3}),f(x)+f(y)\big ) \le \gamma (x,y), \quad x,y\in \mathbb {R}{\setminus }\{0\}, \end{aligned} d ( f ( x 3 + y 3 3 ) , f ( x ) + f ( y ) ) ≤ γ ( x , y ) , x , y ∈ R \ { 0 } , with $$x\ne -y$$ x ≠ - y . The method is based on a quite recent fixed point theorem (cf. Brzdęk and Cieplińnski in Nonlinear Anal 74:6861–6867, 2011, Theorem 1) in some functions spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric Spaces

, Volume 72 (2) – Jul 17, 2017
15 pages

Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0716-2
Publisher site
See Article on Publisher Site

### Abstract

Let $$\mathbb {R}$$ R be the set of real numbers, $$(G,+)$$ ( G , + ) be a commutative group and d be a complete ultrametric on G that is invariant (i.e., $$d(x + z, y + z)= d(x, y$$ d ( x + z , y + z ) = d ( x , y ) for $$x, y, z \in G$$ x , y , z ∈ G ). Under some weak natural assumptions on the function $$\gamma :{\mathbb {R}}^2\rightarrow [0,\infty )$$ γ : R 2 → [ 0 , ∞ ) , we study the generalised hyperstability results when $$f:\mathbb {R}\rightarrow G$$ f : R → G satisfy the following radical cubic inequality \begin{aligned} d\big (f(\root 3 \of {x^3+y^3}),f(x)+f(y)\big ) \le \gamma (x,y), \quad x,y\in \mathbb {R}{\setminus }\{0\}, \end{aligned} d ( f ( x 3 + y 3 3 ) , f ( x ) + f ( y ) ) ≤ γ ( x , y ) , x , y ∈ R \ { 0 } , with $$x\ne -y$$ x ≠ - y . The method is based on a quite recent fixed point theorem (cf. Brzdęk and Cieplińnski in Nonlinear Anal 74:6861–6867, 2011, Theorem 1) in some functions spaces.

### Journal

Results in MathematicsSpringer Journals

Published: Jul 17, 2017

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