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In this note we investigate the general solution for the quartic functional equation of the form $\left(n\in \mathbb{N}, \, \, n > 4 \right)$ 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.

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 Δ b -condition and without using both the Δ b -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. [ABSTRACT FROM AUTHOR]

In the current work, we introduce a general form of a mixed additive and quartic functional equation. We determine all solutions of this functional equation. We also establish the generalized Hyers-Ulam stability of this new functional equation in quasi-$\beta$-normed spaces.

The analysis of stability results of different multiplicative inverse functional equations is an emerging trend in the research field of fundamental stability of equations emanated through the query of Ulam. This paper focuses on proving various classical stabilities of a new inverse fourth power functional equation connected with stability problems in the setting of fields that do not satisfy Archimedean property. The solution of the equation dealt in this study is interpreted with the intensity of scattered light and decay of radar energy in physics. [ABSTRACT FROM AUTHOR]

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.

In this paper, we establish some hyperstability results of the following functional equation f(2x + y) + f(2x - y) = 4(f (x + y) + f (x - y)) + 24f (x) - 6f (y) in Banach spaces. [ABSTRACT FROM AUTHOR]

The aim of this article is to extend the applications of a fixed point method to provide the intuitionistic fuzzy versions of Hyers-Ulam stability for the generalized forms of cubic and quartic functional equations. This way shows that the concept of stability is related to some fixed point of a suitable operator. [ABSTRACT FROM AUTHOR]

In this article, we extend general fuzzy normed spaces to fuzzy β-normed spaces and adopt the fixed point and direct methods to prove the Hyers-Ulam-Rassias stability of the quartic functional equation $f(2x+y+z)+f(2x+y-z)+f(2x-y+z)+ f(-2x+y+ z)+f(2y)+f(2z) =8[f(x+y)+f(x-y)+f(x+z)+f(x-z)]+2[f(y+z)+f(y-z)]+32f(x)$ in fuzzy β-normed spaces. [ABSTRACT FROM AUTHOR]

In this paper we approximate the quartic functional equations in Lipschitz spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors established the general solution and generalized Ulam - Hyers stability of the simple additive-quadratic and simple cubicquartic functional equations f(2x) = 3f(x) + f(- x), g(2x) = 12g(x) + 4g(- x), via Banach space using direct and fixed point method. [ABSTRACT FROM AUTHOR]

We investigate some stability results and intuitionistic fuzzy continuities concerning the following Euler-Lagrange type quartic functional equation f (ax + y) + f(x ay) + 1/2a(a - 1)2 f(x - y) = 1/2 a(a + 1)2 f(x + y) (a2 - 1)2 (f(x) f(y)) in intuitionistic fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

Let A be a Banach ternary algebra over a scalar field R or C and X be a ternary Banach A-module. A quartic mapping D : (A, []A) → (X, []x) is called a 4th-order ternary derivation if D([x, y, z]) = [D(x), y4, z4 + [x4 D(y), z4 + [x4, y4, D(z)] for all x, y, z ∈ A. In this paper, we prove Ulam stability generalizations of 4th - order ternary derivations associated to the following JMRassias quartic functional equation on fréchet algebras: f(kx + y) + f(kx - y) = k²[f(x + y) + f(x - y)] + 2k²(k - 1)f(x) - 2(k² - 1)f(y). [ABSTRACT FROM AUTHOR]

We investigate some stability results and intuitionistic fuzzy continuities concerning the following Euler-Lagrange type quartic functional equation f(ax + y) + f(x + ay) + 1/2 a(a - 1)² f(x - y) = 1/2 a (a + 1)² f (x + y) + a² - 1) ² (f(x) + f(y)) in intuitionistic fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

Let M = { 1, 2, . . . , n } and let $${\mathcal {V}=\{\,I \subseteq M: 1 \in I\,\}}$$ , where n is an integer greater than 1. Denote $${M{\setminus}{I}}$$ by I for $${I \in \mathcal {V}.}$$ We investigate the solution of the following generalized quartic functional equationin β-Banach modules on a Banach algebra, where $${a_{1},\ldots, a_{n}\in \mathbb{Z}{\setminus}\{0\}}$$ with a ≠ ±1 for all $${\ell \in \{1 , 2, \ldots ,\,n-1\}}$$ and a = 1. Moreover, using the fixed point method, we prove the generalized Hyers-Ulam stability of the above generalized quartic functional equation. Finally, we give an example that the generalized Hyers-Ulam stability does not work. [ABSTRACT FROM AUTHOR]

Let A, B be two rings. A mapping δ : A → B is called quartic derivation, if δ is a quartic function satisfies δ(ab) = a4δ(b) + δ(a)b4 for all a, b ∈ A. The main purpose of this paper to prove the generalized Hyers-Ulam-Rassias stability of the quartic derivations on Banach algebras. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is first to introduce the notation of intuitionistic random normed spaces, and then by virtue of this notation to study the stability of a quartic functional equation in the setting of these spaces under arbitrary triangle norms. Then we prove the stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional equations are also presented in the paper. [Copyright &y& Elsevier]

We establish the generalized Hyers–Ulam stability of the additive–quartic and quadratic–cubic mappings with two variables on abelian groups. [ABSTRACT FROM AUTHOR]

The main problem analyzed in this paper consists in showing that, under some conditions, every almost quartic mapping from a linear space to a random normed space under the Łukasiewicz t-norm can be suitably approximated by a quartic function, which is unique. [ABSTRACT FROM AUTHOR]

Abstract: We will define a notion for a quasi fuzzy p-normed space, then we use the fixed point alternative theorem to establish Hyers–Ulam–Rassias stability of the quartic functional equation where functions map a linear space into a complete quasi fuzzy p-normed space. Later, we will show that there exists a close relationship between the fuzzy continuity behavior of a fuzzy almost quartic function, control function and the unique quartic mapping which approximates the almost quartic map. Finally, some applications of our results in the stability of quartic mappings from a linear space into a quasi p-norm space will be exhibited. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the generalized Hyers–Ulam stability for the following quartic functional equation [Copyright &y& Elsevier]

Abstract In this article, we prove the nonlinear stability of the quartic functional equation 1 6 f ( x + 4 y ) + f ( 4 x - y ) = 3 0 6 9 f x + y 3 + f ( x + 2 y ) (1) + 1 3 6 f ( x - y ) - 1 3 9 4 f ( x + y ) + 4 2 5 f ( y ) - 1 5 3 0 f ( x ) (2) (3) in the setting of random normed spaces Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean space, the theory of fixed point theory, the theory of intuitionistic spaces and the theory of functional equations are also presented in the article.