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On the stability of the linear functional equation in a single variable on complete metric groups
Soon-Mo Jung, Dorian Popa, Michael Th. Rassias
TL;DR
The paper addresses Hyers-Ulam stability for a single-variable linear functional equation on complete metric groups. It studies generalized stability under the stated group and metric assumptions and establishes existence and uniqueness of an exact solution near each approximate solution.
Problem
The paper investigates Hyers-Ulam stability of the linear functional equation in a single variable for functions taking values in complete metric groups.
Method
The paper studies generalized Hyers-Ulam stability for the homogeneous equation under complete metric-group assumptions and analyzes a Cauchy sequence in the group.
Results
For every function satisfying the stated inequality, the theorem gives a unique exact solution of the functional equation satisfying the stated stability bound.
Takeaways & Limitations
The result extends stability analysis of the single-variable linear functional equation to functions with values in complete metric groups.
Takeaways & Limitations
The setting assumes a nonempty domain, a complete metric group, and a metric invariant under left translations.
Abstract
from arXiv · showhide
In this paper we obtain a result on Hyers-Ulam stability of the linear functional equation in a single variable $f(\varphi(x)) = g(x) \cdot f(x)$ on a complete metric group.
1 Introduction
The introduction situates Hyers-Ulam stability within functional-equation theory and presents this paper’s focus on the homogeneous linear functional equation over complete metric groups.
- 1 Introduction: The paper studies Hyers-Ulam stability, where approximate solutions are compared with nearby exact solutions.The introduction connects this stability concept with optimization theory and applications in economics.
- 1 Introduction: The introduction places the subject in a history beginning with Ulam’s perturbation question and Hyers’s result for the Cauchy equation on Banach spaces.It also notes earlier stability work involving the Cauchy equation on positive integers.
- 1 Introduction: The linear functional equation in a single variable uses unknown f and given functions g, h, and ϕ, with gamma and digamma equations as particular cases.The displayed equation is identified as a source of several classical functional equations.
- 1 Introduction: Prior work established stability results for equation (1.1), including generalized stability for gamma-type equations and mappings into Banach spaces.The introduction also cites subsequent studies of stability and nonstability for higher-order equations in one variable.
- 1 Introduction: This paper extends the study to the homogeneous equation for functions from an arbitrary nonempty set into a complete metric group.The group operation and inverse are required to be continuous under the relevant product and metric topologies.
2 Stability of linear functional equation
The paper establishes generalized Hyers-Ulam stability for a linear functional equation on complete metric groups, constructing a unique exact solution near every approximate solution under summability conditions.
- Stability framework: The framework applies to generalized stability, where the perturbation is controlled by a function ε rather than a constant error.Constant ε recovers ordinary Hyers-Ulam stability.
- Main stability theorem: Theorem 2.2 gives a unique exact solution f0 for every function f satisfying the prescribed inequality, with an explicit closeness bound.The theorem is the section’s central result and provides both existence and uniqueness.
- Existence proof: The proof constructs iterated approximations, shows they form a Cauchy sequence, and uses completeness of the metric group to define f0.The remainder of the associated series tends to zero, yielding convergence.
- Existence proof: The limiting function satisfies f0(ϕ(x)) = g(x) · f0(x), so it solves the target linear functional equation.The functional equation follows by passing to the limit in the recurrence for the approximations.
- Uniqueness proof: Uniqueness follows because the error bound tends to zero along iterates of ϕ, forcing any two admissible exact solutions to coincide.The proof compares two solutions and uses lim n→∞ Φ(ϕn(x)) = 0.
- Digamma application: The theorem is applied to the Digamma functional equation, yielding generalized Hyers-Ulam stability and a unique nearby solution on the positive reals.The application uses ϕ(x) = x + 1, G = R, and g(x) = 1/x.