Equations In Modular Spaces Research Articles

A Fixed Point Approach to the Stability of a Quadratic Functional Equation in Modular Spaces Without Δ2-Conditions

Abstract In this paper, we investigate the Hyers-Ulam-Rassias stability property of a quadratic functional equation. The even and odd cases for the corresponding function are treated separately before combining them into a single stability result. The study is undertaken in a relatively new structure of modular spaces. The theorems are deduced without using the familiar Δ2-property of that space. This complicated the proofs. In the proofs, a fixed point methodology is used for which a modular space version of Banach contraction mapping principle is utilized. Several corollaries and an illustrative example are provided.

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.

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.

Ulam Stability of a General Linear Functional Equation in Modular Spaces

Using the direct method, we prove the Ulam stability results for the general linear functional equation of the form ∑i=1mAi(fφi(x¯))=D(x¯) for all x¯∈Xn, where f is the unknown mapping from a linear space X over a field K∈{R,C} into a linear space Y over field K; n and m are positive integers; φ1,…,φm are linear mappings from Xn to X; A1,…,Am are continuous endomorphisms of Y; and D:Xn→Y is fixed. In this paper, the stability inequality is considered with regard to a convex modular on Y, which is lower semicontinuous and satisfies an additional condition (the Δ2-condition). Our main result generalizes many similar stability outcomes published so far for modular space. It also shows that there is some kind of symmetry between the stability results for equations in modular spaces and those in classical normed spaces.

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.

A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

Abstract In this paper, we consider pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.

Stability of additive-quadratic 3D functional equation in modular spaces by direct method

This paper deals with the Ulam–Hyers stability of the following additive-quadratic mixed type functional equation: [Formula: see text] [Formula: see text] [Formula: see text] in modular spaces using the direct method.

Stability of an additive-quartic functional equation in modular spaces

In this paper, we prove the Ulam-Hyers stability of the following additive-quartic functional equation \[ f\left(\frac{u+v}{2}-w\right) +f\left(\frac{v+w}{2}-u\right)+f\left(\frac{w+u}{2}-v\right) =\frac{25}{32}\left(f(u-v)+f(v-w)+f(w-u)\right)-\frac{7}{32}\left(f(v-u)+f(w-v)+f(u-w)\right) \] in modular spaces by using the direct method.

Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach

Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach

Stability of the Apollonius Type Additive Functional Equation in Modular Spaces and Fuzzy Banach Spaces

In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types.

Refined stability of additive and quadratic functional equations in modular spaces

The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces.

On the generalized Ulam-Hyers-Rassias stability for quartic functional equation in modular spaces

On the generalized Ulam-Hyers-Rassias stability for quartic functional equation in modular spaces

Stability of general A-cubic functional equations in modular spaces

In this paper, by using fixed point theory, we investigate the generalized Hyers–Ulam stability of an \(\alpha \)-cubic functional equation in modular spaces.

A Fixed Point Approach to the Stability of Quadratic Functional Equations in Modular Spaces

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation f(kx + y) + f(kx – y) = 2k2f(x) + 2f(y) for any fixed positive integers k ∈ Ζ+ in modular spaces by using fixed point method.

On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces

Abstract In this paper, we establish the Hyers-Ulam-Rassias stability of the quadratic functional equation of Pexiderized type f(x + y)+ f(x - y) = 2g(x)+ 2h(y), x⊥y in which ⊥ is the orthogonality in the sense of Rätz in modular spaces.

On the generalized orthogonal stability of mixed type additive-cubic functional equations in modular spaces

In this paper, we establish the Hyers-Ulam-Rassias stability of the mixed type additive-cubic functional equation $$f(2x+y)+f(2x-y)-f(4x)=2[f(x+y)+f(x-y)]-8f(2x)+10f(x)-2f(-x),$$ with $x\bot y,$ where $\bot $ is the orthogonality in the sense of Ratz in modular spaces.