Variable Exponent Research Articles

Global nonexistence for viscoelastic equations with distributed delay and variable exponents

In this paper, we consider a viscoelastic equation with distributed delay and variable exponents. We show the global nonexistence of solutions for the equation. Using the energy method, we show that the solutions with not only negative initial energy but also positive initial energy blow up in finite time.

Infinitely many low‐ and high‐energy solutions for double‐phase problems with variable exponent

AbstractThe aim of this paper is the study of double‐phase problems with variable exponent. Using the Clark's theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively.

On the Non-Linearity of S-Type Variable Exponent Absolutely Summable New Difference Sequence Space

This article presents the domain of general quantum difference in Nakano sequence space. Some topological and geometric behavior, the multiplication mappings defined on it, and the spectrum of mapping ideals constructed by this space and s−numbers have been introduced. Existing results are constructed by controlling the general quantum difference and power of this new space, which is a major strength.

Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities

Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities

Existence results for variable-order fractional Kirchhoff equations with variable exponents

This study establishes the existence of two nontrivial weak solutions for a specific class of Kirchhoff-type equations. Utilizing the Mountain Pass Theorem and the Ekeland variational principle, alongside insights from fractional Sobolev spaces with variable orders and exponents, we provide a concise demonstration of our result.

A novel parabolic model driven by double phase flux operator with variable exponents: Application to image decomposition and denoising

This work tackles a novel parabolic equation driven by a nonlinear operator with double variable exponents, aiming to decompose and denoise images. Our primary approach involves enhancing classical models based on variable exponent operators by considering a novel nonlinear operator having a double-phase flux with unbalanced growth. We begin initially by analyzing the theoretical solvability of our model. Employing the Musielak-Orlicz space, we establish a suitable functional framework for investigating the proposed model. Subsequently, we use the Faedo-Galerkin approach to establish the existence and uniqueness of a weak solution for our problem. Furthermore, we provide a demonstration showcasing how our model preserves solution positivity, emphasizing this as a consistent feature of the proposed model. To evaluate our theoretical results, we present various numerical simulations on some grayscale and medical images (Magnetic Resonance Images (MRI)). These simulations covered aspects like decomposition, robustness and behavior sensitivity of model parameters with visual and quantitative comparisons. The obtained numerical results strongly support the efficiency of the proposed model in preserving features, reducing artifacts and deleting noise in comparison to some well-known existing state-of-the-art models. Furthermore, the proposed model quantitatively surpasses the competitive models in terms of several well-known criteria, namely the Peak Signal-to-Noise Ratio (PSNR), the Structural Similarity Index (SSIM) and the Mean-Square Error (MSE).

On a $p(x,.)$-integrodifferential problem with Neumann boundary conditions

This study examines a class of a bi-nonlocal problem involving a generalized integro-differential operator of elliptic type, characterized by a singular kernel, with nonlocal nonlinear Neumann boundary conditions. We establish the existence of infinitely many solutions in a general fractional Sobolev space with variable exponent.

Multilinear Fractional Hardy Operators on $$p$$-Adic Functional Spaces with Variable Exponents

Multilinear Fractional Hardy Operators on $$p$$-Adic Functional Spaces with Variable Exponents

Well-posedness results to parabolic problems involving (p(x),q(x))-growth structure with L1-data

We investigate the well-posedness (existence and uniqueness) of solutions to nonlinear parabolic problems with variable exponents and irregular data. Firstly, we establish the existence and uniqueness of weak solutions to the studied model when the data are regular enough. Secondly, we assume that the data belongs only to L1 and we prove the existence and uniqueness of both renormalized and entropy solutions. Finally, we demonstrate the equivalence between the renormalized and entropy notion of solutions to the considered problems. Our approach is based essentially on the use of truncation methods and involves some new technical estimates.

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

Existence of weak solutions for double phase fractional problems with variable exponents

Abstract In this present paper, we are first of all interested in some continuity and compactness results for the space ψ-fractional 𝕊 𝒜 α , β ; ψ ⁢ ( Λ ) {\mathbb{S}^{\alpha,\beta;\psi}_{\mathcal{A}}(\Lambda)} . In this sense, we investigate the existence of at least two solutions with constant signs using truncation arguments and comparison methods of a new class of fractional differential equations with m ⁢ ( ξ ) {m(\xi)} -Laplacian with double phase.

Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

Abstract The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L 1(Ω). Furthermore the weight function θ(·) is in W̊ 1,p→(·) (Ω), with θ(·) > 0 and connected with the coefficient b(·) ∈ L 1(Ω) of the lower order term.

Design of fixed-time sliding mode control using variable exponents

This paper proposes the design of a robust fixed-time sliding mode controller for a second-order nonlinear system with matched and unmatched uncertainties. First, a fixed-time stable scalar dynamics with a variable exponent is introduced. We establish the global fixed-time stability results for the proposed dynamics using Lyapunov analysis. The maximum convergence time has been derived in terms of the design parameters, which is independent of the initial conditions. These results are then extended to solve the robust fixed-time stabilization for uncertain nonlinear second-order system. Specifically, we proceed with the sliding mode control design, where the sliding surface and the control law are motivated by the proposed scalar dynamics. Moreover, the proposed design is free from singularity and guarantees fixed-time robust stabilization. Simulation results with comparative analysis has been included. Further, experimental validation on a single link manipulator is provided to demonstrate the performance of the proposed approach.

Some existence results for a class of Dirichlet problems with variable exponents

In this paper, we study the existence and multiplicity of weak solutions of the p(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(x)$\\end{document}-Laplacian problems that are subject to Dirichlet boundary conditions, employing variational techniques as our primary analytical framework.

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

Ap weights and an application to Hankel operators on Fock spaces with variable exponents on [formula omitted]

We characterize Ap weights for p>1 and via extrapolation we characterize boundedness and compactness of Hankel operators between Fock spaces of variable exponent and the Lebesgue spaces of variable exponents. We also give some characterizations of the symbol class which is some BMO-type spaces with variable exponent on the complex plane Cn.

Fixed-time attitude control for hypersonic morphing vehicles: A dynamic memory event-triggering approach

This paper focuses on achieving fast attitude tracking and resource preservation for hypersonic morphing vehicles (HMVs) in continuous morphing stages. Firstly, the most remarkable feature of an HMV are the morphing wings, leading to dramatic aerodynamic property changes. As a result, fast tracking and excellent adaptation to various morphing statuses are necessary to guarantee the success of flight missions. Thus, this work employs the fixed-time stability theory to establish a high-performance controller, including variable exponent coefficients. Compared to the traditional constant exponent coefficient-based methods, some singularity problems are avoided, and the global fixed-time stability is ensured. Then, in order to implement the control input in a low-resource occupation, the dynamic event-triggering mechanism is presented to provide a flexible updating policy. The proposed event-triggering technology has a more concise and efficient structure than the traditional methods. In this case, the controller and the dynamic variable of the event-triggering method simultaneously adopt the exponent coefficients. Finally, the closed-loop stability is proven via the Lyapunov thesis, and simulation results show the effectiveness and the advantage of the proposed control strategy.

Intrinsic square function and wavelet characterizations of variable Hardy–Lorentz spaces

Let p(⋅):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition and q∈(0,1]. The goal of this article is to characterize the variable Hardy–Lorentz space Hp(⋅),q(Rn) in terms of various intrinsic square functions including the intrinsic g-function, the intrinsic Lusin area integral and the intrinsic gλ⁎-function. Using the intrinsic square function and the known atomic characterizations of Hp(⋅),q(Rn), and establishing some estimates on a discrete Littlewood–Paley g-function and a Peetre-type maximal function, the authors further give several equivalent characterizations of Hp(⋅),q(Rn) via wavelets. What is different from the existing works is that we merely assume p+=ess supx∈Rnp(x)∈(0,∞). One of the important tools that make it possible is to find a new dual space of Hp(⋅),q(Rn).

Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent

Abstract In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.