Abstract
The aim of this paper is to introduce and solve the following pradical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y),x,y ∈R, where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzdȩk’s fixed point theorem [14], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y)+G(x,y)
Highlights
A classical question in the theory of functional equation is the following:”When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of this equation.”If the answer is affirmative, we say that equation is stable
Ulam [37]) asked the following question concerning the stability of group homomorphisms
The main purpose of this paper is to achieve the general solution of the functional equation (1.5) and establish some hyperstability results for the considered equation in non-Archimedean Banach space
Summary
A classical question in the theory of functional equation is the following:. ”When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of this equation.”. In the case p < 0, each f : E1 → E2 satisfying (1.1) must be additive This result is called the hyperstability of Cauchy functional equation. A function || · ||∗ : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: 1. The main purpose of this paper is to achieve the general solution of the functional equation (1.5) and establish some hyperstability results for the considered equation in non-Archimedean Banach space. 2],we state an analogue of the fixed point theorem [13, Theorem 1] in non- Archimedean Banach space We use it to assert the existence of a uniquefixed point of operator T : Y X −→ Y X. There exists a unique fixed point ψ ∈ Y X of T with kφ(x) − ψ(x)k∗ ≤ sup Λnε(x), n∈N0 x ∈ X
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