Quartic Functional Equation Research Articles

Stability of Generalized Quartic Functional Equation in Random Normed Spaces

Stability of Generalized Quartic Functional Equation in Random Normed Spaces

Fuzzy stability of mixed type functional equations in Modular spaces

In this article, our aim is to find some stability results for mixed type cubic and quartic functional equations in fuzzy modular spaces using the fundamental results of fixed-point theory. The fixed point method provides one of the effective techniques that can be used to investigate the fuzzy stability of a mixed type cubic and quartic functional equations.

Stability results of mixed type functional equations in modular spaces and 2-Banach spaces ∗

In this paper, we investigate the generalized Hyers-Ulam-Stability of additive and quartic functional equations in modular spaces with and without the △2-condition using the direct method and also in 2-Banach Spaces.

Fixed point approach to the stability of a cubic and quartic mixed type functional equation in non-archimedean spaces

Fixed point approach to the stability of a cubic and quartic mixed type functional equation in non-archimedean spaces

On Direct Fuzzy Stability of General Quartic Functional Equation

The goal of this paper is to investigate the Generalized Hyers - Ulam - Rassias (HUR) stability of generalquartic functional equation (GQFE) fq(kr1 + (k -1)r2) + fq(kr1 - (k -1)r2) = 2k4fq(r1) + 2(k -1)4fq(r2) +6k2(k -1)2[fq(r1 + r2) + fq(r1 - r2)] - 12k2(k - 1)2[fq(r1) + fq(r2)] in fuzzy normed spaces (F.N. spaces). The stability of the equation is proved by using direct method. The stability in sense of Hyers - Ulam and Ulam- Gavruta - Rassias is also studied.

Stability Results of the Additive and Quartic Functional Equations in Random p-Normed Spaces

Stability Results of the Additive and Quartic Functional Equations in Random p-Normed Spaces

Ulam Stability Results of Functional Equations in Modular Spaces and 2-Banach Spaces

In this work, we investigate the refined stability of the additive, quartic, and quintic functional equations in modular spaces with and without the Δ2-condition using the direct method (Hyers method). We also examine Ulam stability in 2-Banach space using the direct method. Additionally, using a suitable counterexample, we eventually demonstrate that the stability of these equations fails in a certain case.

Refined stability of the additive, quartic and sextic functional equations with counter-examples

<abstract><p>In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.</p></abstract>

Functional equations with applications in image security system

This paper deals with the investigation of general solution and stabilities of the generalized ‐cubic, ‐quartic functional equations in modular spaces using fixed point method. The main purpose of this research is not limited to introducing generalized functional equations and investigating its stability but to propose a new method in image security system for our main focusing on the new results encryption and decryption using our introduced functional equation.

General system of cubic–quartic functional equations in quasi-β-normed spaces

In the current article, we define the multicubic–quartic mappings and describe them as an equation. We also study n-variable mappings, which are mixed type cubic–quartic in each variable and then give a characterization of such mappings. Indeed, we unify the general system of cubic–quartic functional equations defining a multimixed cubic–quartic mapping to a single equation, say, the multimixed cubic–quartic functional equation. Furthermore, we show under what conditions every multimixed cubic–quartic mapping can be multicubic, multiquartic and multicubic–quartic. In addition, by means of a known fixed-point result, we prove the Găvrua stability of multimixed cubic–quartic functional equations in the setting of quasi-β-normed spaces. One of the important results is that every multimixed cubic–quartic functional equation on a quasi-β-normed space is the Hyers–Ulam stable. Lastly, we investigate the hyperstability of multicubic -derivations on -algebras.

Stability of Quartic Functional Equation in Modular Spaces via Hyers and Fixed-Point Methods

In this work, we introduce a new type of generalised quartic functional equation and obtain the general solution. We then investigate the stability results by using the Hyers method in modular space for quartic functional equations without using the Fatou property, without using the Δb-condition and without using both the Δb-condition and the Fatou property. Moreover, we investigate the stability results for this functional equation with the help of a fixed-point technique involving the idea of the Fatou property in modular spaces. Furthermore, a suitable counter example is also demonstrated to prove the non-stability of a singular case.

Solution and Stability of Quartic Functional Equations in Modular Spaces by Using Fatou Property

We propose a novel generalized quartic functional equation and investigate its Hyers–Ulam stability in modular spaces using a fixed point technique and the Fatou property in this paper.

Quartic Functional Equation: Ulam‐Type Stability in (β, p)‐Banach Space and Non‐Archimedean β‐Normed Space

In this study, we use the alternative fixed‐point approach and the direct method to examine the generalized Hyers–Ulam stability of the quartic functional equation g(u + mv) + g(u − mv) = 2(m − 1)(7m − 9)g(u) + 2(m2 − 1)m2g(v) − (m − 1)2g(2u) + m2{g(u + v) + g(u − v)}, with a fixed positive integer m/ge2 in the context of (β, p)‐Banach space. In non‐Archimedean β‐normed space, we also verify Hyers–Ulam stability for the quartic functional equation stated. Many of the findings in the literature are improved and generalized by our findings.

The Stability Analysis of A-Quartic Functional Equation

In this paper, we study the general solution of the functional equation, which is derived from additive–quartic mappings. In addition, we establish the generalized Hyers–Ulam stability of the additive–quartic functional equation in Banach spaces by using direct and fixed point methods.

Nonlinear approximation of multiplicative inverse quartic functional equation in intuitionistic fuzzy normed spaces

In this paper, authors investigate the stability results of multiplicative inverse quartic functional (MIQF) equation associating a constant, sum of powers of norms, general control function, mixed product-sum of powers of norm and product of different powers of norms pertinent to the results established by Gavruta and Rassias in intuitionistic fuzzy normed spaces. After that, we proved the stability results of MIQF equation through the fixed point alternative.

Stabilities and instabilities of an advanced quartic functional equation in various normed spaces

In this paper, a new and advanced quartic functional equation is introduced, and its classical stability results in various normed spaces are discussed. An appropriate illustration is also presented to invalidate the stability result for a very critical instance. At the end of this paper, we have concluded that the stability results are valid in various normed spaces.

New Stability Results of Multiplicative Inverse Quartic Functional Equations

The purpose of this investigation is to introduce different forms of multiplicative inverse functional equations, to solve them and to establish the stability results of them in the framework of matrix normed spaces. A suitable counter-example is also provided to prove the instability of the results for a singular case. The applications of the equations dealt in this study are discussed with the fluid resistances of blood vessels and also an important concept in Raman spectroscopy.
 2010 Mathematics Subject Classification. 39B82, 39B72.

Orthogonal Stability and Nonstability of a Generalized Quartic Functional Equation in Quasi-β-Normed Spaces

In this work, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation in quasi-β-normed spaces. Moreover, we prove that this functional equation is not stable in a special condition by a counterexample.

Fuzzy Stability Results of Generalized Quartic Functional Equations

In the present paper, we introduce a new type of quartic functional equation and examine the Hyers–Ulam stability in fuzzy normed spaces by employing the direct method and fixed point techniques. We provide some applications in which the stability of this quartic functional equation can be controlled by sums and products of powers of norms. In particular, we show that if the control function is the fuzzy norm of the product of powers of norms, the quartic functional equation is hyperstable.

Hyers-Ulam stability of an <i>n</i>-variable quartic functional equation

In this note we investigate the general solution for the quartic functional equation of the form $ \begin{align*} \left( 3n+4 \right)f\left( \sum\limits_{i = 1}^{n}{{{x}_{i}}} \right)+\sum\limits_{j = 1}^{n}{f\left( -n{{x}_{j}}+\sum\limits_{i = 1,i\ne j}^{n}{{{x}_{i}}} \right)}& = \left( {{n}^{2}}+2n+1 \right)\sum\limits_{i = 1,i\ne j\ne k}^{n}{f\left( {{x}_{i}}+{{x}_{j}}+{{x}_{k}} \right)}\\ &-\frac{1}{2}\left( 3{{n}^{3}}-2{{n}^{2}}-13n-8 \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}+{{x}_{j}} \right)} \\ & +\frac{1}{2}\left( {{n}^{3}}+2{{n}^{2}}+n \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}-{{x}_{j}} \right)}\\ &+\frac{1}{2}\left( 3{{n}^{4}}-5{{n}^{3}}-7{{n}^{2}}+13n+12 \right)\sum\limits_{i = 1}^{n}{f\left( {{x}_{i}} \right)} \\ \end{align*} $ $\left(n\in \mathbb{N}, \, \, n > 4 \right)$ and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces using the direct approach and the fixed point approach.