Approximation of Mixed Euler-Lagrange σ -Cubic-Quartic Functional Equation in Felbin ’ s Type f-NLS

In this research paper, the authors present a new mixed Euler-Lagrange σ -cubic-quartic functional equation. For this introduced mixed type functional equation, the authors obtain general solution and investigate the various stabilities related to the Ulam problem in Felbin ’ s type of fuzzy normed linear space (f-NLS) with suitable counterexamples. This approach leads us to approximate the Euler-Lagrange σ -cubic-quartic functional equation with better estimation.


Introduction
One of the famous questions concerning the stability of homomorphisms was raised by Ulam [1] in 1940. The author Hyers [2] provided a partial answer to Ulam's question in 1941, and then, a generalized solution to Ulam's question was given by Rassias [3] in 1978, which is called Hyers-Ulam-Rassias stability or generalized Hyers-Ulam stability. The generalization of Hyers stability result by Rassias [4] is called Ulam-Gavruta-Rassias stability. Later, Ravi et al. [5] investigated the stability using mixed powers of norms which is called Rassias stability.
Motivated from the above historical developments in the field of FEs, the authors introduce a new mixed Euler-Lagrange σ-cubic-quartic functional equation (FE) where σ ∈ ℝ − f0,±1g. For this mixed type FE, authors obtain the general solution and investigate the various stabilities related to Ulam problem [1] in Felbin's type f-NLS with suitable counterexamples.

Theorem 14.
Consider the odd mapping π : T ⟶ S for which we can find Φ : T × T ⟶ Λ * ðℝÞ for a linear space T and a fuzzy Banach space (f-BS) S where So, we can find a unique cubic function Θ : Proof. Putting s = 0 in (9) implies that Multiply both sides of equation (12) by 4/σ 2 ðσ 2 − 1Þ, so we get Again multiplying (13) by 1/2 3σ+3 and replacing t by 2 σ t, we obtain and it leads to ∀t ∈ T with σ ≥ l, nonnegative integers. Now, (8) and (15) imply that the sequence fπð2 σ tÞ/2 3σ g is fuzzy Cauchy in S. So, the sequence fπð2 σ tÞ/2 3σ g converges, which let us to define the mapping Θ : Considering l = 0 and allowing σ ⟶ ∞ in (15), we obtain 3 Journal of Function Spaces and it gives (10). Using (8) and (9), we have which implies that Θ : T ⟶ S is cubic. Suppose that Θ′ : T ⟶ S is a cubic mapping satisfying (10) and implies Θ = Θ ′ , which shows the uniqueness of Θ.
Theorem 15. Consider π : T ⟶ S and let there exist a func- for a linear space T and a fuzzy Banach space (f-BS) S. So, we can find a unique cubic mapping Θ : T ⟶ S, such that where The following corollary gives the Hyers-Ulam, Hyers-Ulam-Rassias, and Rassias stabilities of (2).

Corollary 16.
Consider π : T ⟶ S and let there be real numbers δ and ρ such that then there is a unique cubic mapping Q : T ⟶ S such that In the next example, we consider the unstability of FE (2) for p = 3 in Corollary 16.
For this t, we have which contradicts (34). Therefore, the functional equation (2) is not stable in the sense of Ulam, Hyers, and Rassias if ρ = 3.