Hyers–Ulam stability of functional inequalities: a fixed point approach

Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers–Ulam–Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples.

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“…Recently, A. Batool et al [35], proved the Hyers-Ulam stability of the cubic and quartic functional equation…”

Section: Introductionmentioning

confidence: 99%

Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces

et al. 2022

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“…In 2003, Radu [18] introduced a new method, called the fixed point alternative method, to investigate the existence of exact solutions and error estimations and established that a fixed point alternative method is more essential to the solution of the Ulam stability problem for approximate homomorphisms. Subsequently, some authors [19,20] applied the fixed alternative method to investigate the Hyers-Ulam stability of several functional equa-tions in various directions [21,22]. To further explore the oscillation theory of functional differential equations, we refer the readers to [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

The Stability Analysis of A-Quartic Functional Equation

Muthamilarasi¹,

Santra

Balasubramanian³

et al. 2021

Mathematics

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Generalized Hyers–Ulam stability of ρ-functional inequalities

Nawaz,

Bariq,

Batool

et al. 2023

J Inequal Appl

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In our research work generalized Hyers-Ulam stability of the following functional inequalities is analyzed by using fixed point approach: $$\begin{aligned}& \biggl\Vert f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x) \\& \quad {}-\rho \biggl(4f\biggl(x+\frac{y}{2}\biggr)+4\biggl(f\biggl(x- \frac{y}{2}\biggr)-f(x+y)-f(x-y)\biggr)-6f(x),r\biggr) \biggr\Vert \geq \frac{r}{r+\varphi (x, y)} \end{aligned}$$ ∥ f ( 2 x + y ) + f ( 2 x − y ) − 2 f ( x + y ) − 2 f ( x − y ) − 12 f ( x ) − ρ ( 4 f ( x + y 2 ) + 4 ( f ( x − y 2 ) − f ( x + y ) − f ( x − y ) ) − 6 f ( x ) , r ) ∥ ≥ r r + φ ( x , y ) and $$\begin{aligned}& \biggl\Vert f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y) \\& \qquad {}-\rho \biggl(8f\biggl(x+\frac{y}{2}\biggr)+8\biggl(f\biggl(x- \frac{y}{2}\biggr)-2f(x+y)-2f(x-y)\biggr)-12f(x)+3f(y),r\biggr) \biggr\Vert \\& \quad \geq \frac{r}{r+\varphi (x, y)} \end{aligned}$$ ∥ f ( 2 x + y ) + f ( 2 x − y ) − 4 f ( x + y ) − 4 f ( x − y ) − 24 f ( x ) + 6 f ( y ) − ρ ( 8 f ( x + y 2 ) + 8 ( f ( x − y 2 ) − 2 f ( x + y ) − 2 f ( x − y ) ) − 12 f ( x ) + 3 f ( y ) , r ) ∥ ≥ r r + φ ( x , y ) in the setting of fuzzy matrix, where $\rho \neq 2$ ρ ≠ 2 is a real number.We also discussed Hyers-Ulam stability from the application point of view.

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Hyers–Ulam stability of functional inequalities: a fixed point approach

Abstract: Using the fixed point method, we prove the Hyers–Ulam stability of a cubic and quartic functional equation and of an additive and quartic functional equation in matrix Banach algebras.

Cited by 6 publications

References 25 publications

Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces

Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces

The Stability Analysis of A-Quartic Functional Equation

Generalized Hyers–Ulam stability of ρ-functional inequalities

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