Quadratic Functional 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.

Relaxation and optimal finiteness domain for degenerate quadratic functionals. One-dimensional case

The aim of this paper is the study, in the one-dimensional case, of the relaxation of a quadratic functional admitting a very degenerate weight w, which may not satisfy both the doubling condition and the classical Poincaré inequality. The main result deals with the relaxation on the greatest ambient space L0(Ω) of measurable functions endowed with the topology of convergence in measure w dx. Here w is an auxiliary weight fitting the degenerations of the original weight w. Also the relaxation w.r.t. the L2(Ω, w˜)-convergence is studied. The crucial tool of the proof is a Poincaré type inequality, involving the weights w and w, on the greatest finiteness domain Dw of the relaxed functionals.

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.

An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L1[0,T] and, secondly, the existence of nonincreasing solutions in unbounded interval L1(R+). Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ∈L1[0,T] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ∈L1(R+) on some parameters. Finally, some examples are provided to verify our investigation.

Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

Given a bounded open set Ω⊂Rn, we study sequences of quadratic functionals on the Sobolev space H01(Ω), perturbed by sequences of bounded linear functionals. We prove that their Γ-limits, in the weak topology of H01(Ω), can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of G- and H-convergence completely characterises the quadratic and linear parts of the Γ-limit and shows that their coefficients do not depend on Ω. The constant, which instead depends on Ω and will be denoted by −ν(Ω), plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that ν coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets Ω. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in Γ-convergence cannot be easily improved.

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.

Dynamic Response of Tunnel Structures in Inhomogeneous Medium Under SH Wave: Shear Modulus in Quadratic Functional Form

Dynamic Response of Tunnel Structures in Inhomogeneous Medium Under SH Wave: Shear Modulus in Quadratic Functional Form

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.

Restart of Accelerated First-Order Methods With Linear Convergence Under a Quadratic Functional Growth Condition

Accelerated first-order methods, also called fast gradient methods, are popular optimization methods in the field of convex optimization. However, they are prone to suffer from oscillatory behavior that slows their convergence when medium to high accuracy is desired. In order to address this, restart schemes have been proposed in the literature, which seek to improve the practical convergence by suppressing the oscillatory behavior. This article presents a restart scheme for accelerated first-order methods, for which we show linear convergence under the satisfaction of a quadratic functional growth condition, thus encompassing a broad class of non-necessarily strongly convex optimization problems. Moreover, the worst-case convergence rate is comparable to the one obtained using an (generally nonimplementable) optimal fixed-rate restart strategy. We compare the proposed algorithm with other restart schemes by applying them to a model predictive control case study.

Four-dimensional closed manifolds admit a weak harmonic Weyl metric

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

Generalized Ulam-Hyers Stability Results of a Quadratic Functional Equation in Felbin’s Type Fuzzy Normed Linear Spaces

This paper presents the generalized Ulam-Hyers stability of the following quadratic functional equationF((x+y)/2 – z) + f(y+z)/2 – x) + f((z+x)/2 – y) = 3/4(f(z−x) + f(z−y) + f(x−y))in Felbin’s type fuzzy normed linear spaces (f-NLS) using direct and fixed point methods.

Stability of a Quadratic Functional Equation

Stability of a Quadratic Functional Equation

Critical metrics and massive gravity solutions on three-dimensional Brinkmann waves* *Supported by projects PID2019-105138GB-C21(AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain).

Three-dimensional Brinkmann waves which are critical for quadratic curvature functionals are determined. Generically, if the metric is critical for some functional then it is critical for all of them. In contrast, there are four special functionals that do not share critical metrics with any other quadratic functional. It is also shown that these metrics provide explicit solutions for different massive gravity models.

Ulam Stabilities and Instabilities of Euler–Lagrange-Rassias Quadratic Functional Equation in Non-Archimedean IFN Spaces

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

Inner product spaces and quadratic functional equations

In this paper, we introduce the functional equations f(2x−y)+f(x+2y)=5[f(x)+f(y)],f(2x−y)+f(x+2y)=5f(x)+4f(y)+f(−y),f(2x−y)+f(x+2y)=5f(x)+f(2y)+f(−y),f(2x−y)+f(x+2y)=4[f(x)+f(y)]+[f(−x)+f(−y)].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} f(2x-y)+f(x+2y)&=5\\bigl[f(x)+f(y)\\bigr], \\\\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=4\\bigl[f(x)+f(y)\\bigr]+\\bigl[f(-x)+f(-y)\\bigr]. \\end{aligned}$$ \\end{document} We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

Group inference in high dimensions with applications to hierarchical testing

High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local heritability. Group inference in regression models can be measured with respect to a weighted quadratic functional of the regression sub-vector corresponding to the group. Asymptotically unbiased estimators of these weighted quadratic functionals are constructed and a novel procedure using these estimators for inference is proposed. We derive its asymptotic Gaussian distribution which enables the construction of asymptotically valid confidence intervals and tests which perform well in terms of length or power. The proposed test is computationally efficient even for a large group, statistically valid for any group size and achieving good power performance for testing large groups with many small regression coefficients. We apply the methodology to several interesting statistical problems and demonstrate its strength and usefulness on simulated and real data.

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design, which is a combination of Willmore functional and total Gauss curvature. First, we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y = f(x) about the x-axis. Then, choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional, we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals. Our results not only provide a strictly mathematical proof for numerical methods, but also give a more reasonable and more extensive choice for the background surfaces.

Stability results of the functional equation deriving from quadratic function in random normed spaces

<abstract> <p>The authors introduce a finite variable quadratic functional equation and derive its solution. Also, authors examine its Hyers-Ulam stability in Random Normed space(RN-space) by means of two different methods.</p> </abstract>

Limit Theorems for Tapered Toeplitz Quadratic Functionals of Continuous-time Gaussian Stationary Processes

Let {X(t), t ∈ ℝ} be a centered real-valued stationary Gaussian process with spectral density f. The paper considers a question concerning asymptotic distribution of tapered Toeplitz type quadratic functional $$Q_T^h$$ of the process X(t), generated by an integrable even function g and a taper function h. Sufficient conditions in terms of functions f, g and h ensuring central limit theorems for standard normalized quadratic functionals $$Q_T^h$$ are obtained, extending the results of Ginovyan and Sahakyan (Probability Theory and Related Fields 138, 551–579, 2007) to the tapered case and sharpening the results of Ginovyan and Sahakyan (Electronic Journal of Statistics 13, 255–283, 2019) for the Gaussian case.

Alienation of the Quadratic and Additive Functional Equations

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation (*) $$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$ resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.