33 Results Found

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generali...

In this paper, we introduce and analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with variable exponent $L^{p\left(x\right)}$ . We primarily consider the Stepanov and Weyl cl...

This paper studies multiplication operators and their associated strongly continuous semigroups acting on variable exponent Lebesgue spaces. We study the abstract Cauchy problem $\dot{u}\left(t\right)=Au\left(t\right)$, $u\left(0\right)=u_0$, in the space $L^{p\left(x\right)}(0,\mathsf{\ell })$ with $\mathsf{\ell }>$...

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the bou...

In this paper, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the asymptotic behaviour of low energy extremals.

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: ${\mathcal{G}}_{ϵ}^{\&l}$...

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space ($L^{p(·)}$). Both the existence, uniqueness, and the stability in the sense of Ulam–Hye...

This manuscript establishes the boundedness results for a class of singular integral operators to a more general framework by imposing conditions on the Hardy–Littlewood maximal function. We establish the results where we construct a suitable r...

In this paper, we investigate various classes of multi-dimensional Doss $\rho$-almost periodic type functions of the form $F:\mathsf{\Lambda }\times X\to Y,$ where $n\in \mathbb{N},$$\varnothing \ne \mathsf{\Lambda }\subseteq {\mathbb{R}}^n,$ X and Y are complex Banach spaces, and $\rho$ is a binary relation on $Y.$ We work in the general setting of...

We investigate the existence of solutions for the Fredholm integral equation $\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta ,$ for $\vartheta \in [0,1]$, in the setting of the modular function spaces...

This study looks closely into the analysis of the variation of constants formula given by $\Phi \left(t\right)=S\left(t\right)\Phi \left(0\right)+{\int }_0^{t}S(t-\sigma )\mathcal{F}(\sigma ,\Phi \left(\sigma \right))d\sigma ,$ for $t\in [0,T],T>0$, within the context of modular function spaces $L_{\rho }$....

A hybrid inversion scheme for through-the-wall imaging is proposed in this paper. The approach is based on a linearized model of the inverse-scattering problem, employing the Green’s function developed for a layered background. The reconstruction is...

In this paper with 1 < p^- ≤ p⁺ <∞ condition we prove a weak convergence result under pointwise convergence and bounded of the sequence. Our theorem is an extension of classical result to variable exponent setting.

We analyze a Dirichlet $\left(p\right(x),\mu q(x\left)\right)$-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter $\mu$ varies on the non-negativ...

Much has been written on variable exponent spaces in recent years. Most of the literature deals with the normed space structure of such spaces. However, because of the variability of the exponent, the underlying modular structure of these spaces is r...

In this manuscript, we introduce a class of measurable functions $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, which is utilized to construct a generalized Bessel–Riesz kernel and the corresponding generalized Bessel–Riesz operator. We establish sufficient conditions ensuring...

An overview of the recent advancements in the development of microwave imaging procedures based on the exploitation of the regularization theory in Lebesgue spaces is reported in this paper. Such inversion schemes have been found to provide accurate...

In this manuscript, we establish the boundedness of the Bessel–Riesz operator $I_{\alpha ,\gamma }f$ in variable Lebesgue spaces $L^{p(·)}$. We prove that $I_{\alpha ,\gamma }f$ is bounded from $L^{p(·)}$ to $L^{p(·)}$ and from $L^{p(·)}$ to $L^{q(\&}$...

We establish a fixed point property for the Lebesgue spaces with variable exponents $L^{p(·)}$, focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation...

We analyze the modular geometry of the Lebesgue space with variable exponent, $L^{p(·)}$ . Our central result is that $L^{p(·)}$ possesses a modular uniform convexity property. Part of the novelty is that the property h...

We prove that the Lebesgue space of variable exponent $L^{p(·)}\left(\mathsf{\Omega }\right)$ is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in ever...

The Stein–Weiss weighted inequality and the John–Nirenberg inequality are among the most important and widely applied results in harmonic analysis, underlying a broad range of developments in weighted norm estimates and the theory of boun...

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators...

Systemic risk measures are crucial for the stability of financial markets. However, traditional frameworks, such as those relying on fixed exponent spaces, rest on an assumption of globally consistent moment conditions, which fails to capture the tim...

This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maxima...

A microwave characterization technique to inspect subsurface scenarios is proposed and numerically assessed in this paper. The approach is based on a combination of finite element electromagnetic modeling and an inversion procedure in Lebesgue spaces...

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such...

In this research work, we investigate the asymptotic behavior of a nonlinear swelling (also called expansive) soil system with a time delay and nonlinear damping of variable exponents. We should note here that swelling soils contain clay minerals tha...

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 f...

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Unde...

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems inv...

Fractional differential operators equipped with classical and new memory kernels have emerged as indispensable tools for modeling systems with hereditary and non-local effects, offering a flexible framework that extends far beyond the classical integ...

Quantitative inversion of GPR data opens the door to precise characterization of underground environments. However, in order to make the inverse scattering problem solution easier from a computational viewpoint, simplifying assumptions are often appl...