Quadratic Functional Equation Research Articles

Stability of Quadratic Mappings in 2-Banach Spaces and Related Topics

Functional analysis is an important branch of mathematics widely used to study the stability of different functional equations. This includes the stability of various quadratic functional equations, which typically involve a specific number of variables to reach results more easily in this field. One of the methods used to study the stability of functional equations is the direct method, which is known for its simplicity in proving the stability of this type of functional equation. In this research paper, we have successfully proven the Hyers-Ulam-Rassias stability of the quadratic functional equation in 2-Banach spaces. Specifically, we have shown that the equation: f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(x+z) holds true within this context. Our approach involved using either the usual or the direct method to establish this stability. Furthermore, we have also demonstrated the generalized Hyers-Ulam stability of the quadratic functional equation in 2-Banach spaces using the usual process by considering various conditions. This has led to many exciting results and revealed many related applications.

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

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.

On entire solutions for several systems of quadratic trinomial Fermat type functional equations in $${\mathbb {C}}^2$$

On entire solutions for several systems of quadratic trinomial Fermat type functional equations in $${\mathbb {C}}^2$$

Stability and Instability of an Apollonius-Type Functional Equation

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces.

Generalized Quadratic Functional Equation and Its Stability over Non-Archimedean Normed Space

Generalized Quadratic Functional Equation and Its Stability over Non-Archimedean Normed Space

An application of decision theory on the approximation of a generalized Apollonius-type quadratic functional equation

To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag(A,B,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{diag}(A, B, C)$\\end{document} (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A∗B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$A\\ast B$\\end{document}. Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation.

Characterization and stability of multi-additive-quadratic-cubic mappings

In this work, we introduce a system of additive, quadratic and cubic functional equations (somewhat different from what has been defined by Bodaghi and Rassias) which defines the multi-additive-quadratic-cubic mappings and then unify and characterize such mappings as a single equation without the quadratic condition. Moreover, by using a fixed point theorem, we establish the Găvruţa stability of the multi-additive-quadratic-cubic mappings in the setting of Banach spaces. As few known results, we prove the Hyers stability for multi-additive, multi-quadratic and multi-cubic mappings.

Numerical and analytical approach to the Chandrasekhar quadratic functional integral equation using Picard and Adomian decomposition methods

<p>This research aimed to find numerical solutions to a type of nonlinear initial value problem (IVP) for hybrid fractional differential equations. Using the Adomian decomposition method (ADM) and the Picard method (PM), we studied the Chandrasekhar quadratic integral equation (QIE). Furthermore, we investigated existence and uniqueness results using measures of weak noncompactness. Through a set of examples and numerical simulations, a comparison was made between the results of the AMD and PM.</p>

Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂ n#

Abstract In this article, we investigate the existence and the precise form of finite-order transcendental entire solutions of some system of Fermat-type quadratic binomial and trinomial shift equations in C n {{\mathbb{C}}}^{n} . Our results are the generalizations of the results of [H. Y. Xu, S. Y. Liu, and Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483 (2020), 123641, 1–22, DOI: https://doi.org/10.1016/j.jmaa.2019.123641.] and [H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9.] to a large extent. Most interestingly, as a consequence of our main result, we have shown that the system of quadratic trinomial shift equation has no solution when it reduces to a system of quadratic trinomial difference equation. In addition, some examples relevant to the content of the article have been exhibited.

Exploiting quadratic \(\varphi(\delta_{1},\delta_{2})-\) function inequalities on fuzzy Banach spaces based on general quadratic equations with \(2k\)-variables

In this manuscript, our primary focus revolves around extending the inequalities associated with the Quadratic \(\varphi(\delta_{1},\delta_{2})\)-function. Our approach involves leveraging the general quadratic functional equation encompassing 2k-variables within the context of the fuzzy Banach space. Our main contribution lies in the expansion of these inequalities, representing a significant result within this study.

Qualitative Aspects of a Fractional-Order Integro-Differential Equation with a Quadratic Functional Integro-Differential Constraint

This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution x∈C[0,T] to the problem. Moreover, we outline the necessary demands for the solution’s uniqueness. In addition, the continuous dependence of the solution and the Hyers–Ulam stability of the problem are analyzed. In order to illustrate our results, we provide some particular cases and instances.

Analysis of a Fractional-Order Quadratic Functional Integro-Differential Equation with Nonlocal Fractional-Order Integro-Differential Condition

Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters will be examined. Finally, the problems of conjugate orders and integer orders, and some other problems and remarks will be discussed and presented.

Experimentation of Think Pair Share learning assisted by GeoGebra software on students' mathematics learning outcomes

This study seeks to identify learning models that influence students' mathematics learning outcomes. Based on the level of absorption, the learning outcomes of the SMK Terpadu Jannatul Firdaus students have not yet reached the KKM and the learning outcomes in mathematics are not good or low, especially in the field of quadratic function equations, due to the not optimal indicators of the learning process at the school. One solution that can be applied is the Think Pair Share (TPS) learning model assisted by GeoGebra Software. This quantitative experimental research used a quasi-experimental design, namely the Posttest Only Control Design with a population of all students of class XI at SMK Terpadu Jannatul Firdaus. The sampling technique used cluster random sampling with three randomly selected samples, two as the experimental class and one as the control class. The first experimental class was taught through the application of the TPS learning model assisted by GeoGebra Software, the second experiment used the TPS learning model only, and the third was used as a control class using the conventional model. The research instrument uses a mathematics learning achievement test instrument. The research hypothesis uses Anova test and post-Anova test. The results showed that students who were taught using the TPS model assisted by the GeoGebra Software obtained a better average and outperformed compared to students who were taught using the conventional model or only the TPS model, as calculated in this study, Fcount was 8.620, which was greater than Ftable, which was 3.096.

Stability of n -dimensional mixed type quadratic and cubic functional equation

Stability of n -dimensional mixed type quadratic and cubic functional equation

Hyer-Ulam-Stability of Generalized Quadratic Functional Equation in Random Normed Spaces

The purpose of this paper to propose a finite variable generalized quadratic functional equation with solution. Also investigate Hyers Ulam stability in Random normed space by direct method.

Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on R+(BC(R+)). The main tool applied in this work is the technique associated with measures of noncompactness in BC(R+) by a given modulus of continuity. Next, we formulate and prove a sufficient condition for the solvability of that coupled system. We, additionally, provide an example and some particular cases to demonstrate the effectiveness and value of our results.

Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers–Ulam–Rassias Stability

In this manuscript, we study the properties and equivalence results for different forms of quadratic functional equations. Using the Brzdȩk and Ciepliński fixed‐point approach, we investigate the generalized Hyers–Ulam–Rassias stability for the 3‐variables quadratic functional equation in the setting of 2‐Banach space. Also, we obtain some hyperstability results for the 3‐variables quadratic functional equation. The results obtained in this paper extend several known results of the literature to the setting of 2‐Banach space.

The exact transcendental entire solutions of complex equations with three quadratic terms

<abstract><p>In this paper, we study the entire solutions of two quadratic functional equations in the complex plane. One consists of three basic terms, $ f(z), f'(z) $ and $ f(z+c) $, and the other one consists of $ f(z), f'(z) $ and $ f(qz) $. These two equations can be transformed into functional equations of Fermat-type. We prove that if these two equations admit finite order transcendental entire solutions, then the solutions of these two equations are both exponential functions, and their exponents are one degree polynomials, whose coefficients of the first degree term are closely related to the coefficients of the functional equation. Moreover, examples are given to show that the theorems are true. The feature of this paper is that the Fermat-type equations contain three quadratic terms, while the equations that have been studied in the previous articles in this field contain only two quadratic terms. The addition of $ f(qz) $ will make the proof methods in this paper very different from those in the existing literature. The proof becomes more difficult, and the number of cases that need to be discussed becomes much larger. In addition, when dealing with the analytical property of $ f $, we also use a different method from the previous literature.</p></abstract>

Stabilities of multiplicative inverse quadratic functional equations arising from Pythagorean means

Stabilities of multiplicative inverse quadratic functional equations arising from Pythagorean means