50 Results Found

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characteri...

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points hav...

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solut...

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated...

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain $\mathcal{D}\subset {\mathbb{R}}^N$ ($N\ge 3$) with homog...

This paper discusses the asymptotic controllability of fractional-order Sobolev-type perturbed stochastic control systems with Brownian motion and nonlocal fractional-order Sobolev stochastic conditions. A new set of sufficient conditions is establis...

We establish the existence of at least three weak solutions for a Neumann $p\left(x\right)$-Laplacian-like problem with two control parameters. Our main result is due to the critical theorem of Bonanno and Marano.

In this paper, we will study a singular problem involving the fractional $(q_1(x,.)$-$q_2(x,.))$-Laplacian operator in the whole space ${\mathbb{R}}^N,(N\ge 2)$. More precisely, we combine the variational method with monotonicity arguments to prove that the associated f...

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to fin...

We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus...

We consider Fourier multiplier systems on ${\mathbb{R}}^n$ with components belonging to the standard Hörmander class $S_{1,0}^m\left({\mathbb{R}}^n\right)$, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector $\mathrm{\Lambda }\subset \mathbb{C}$ (introduced by Denk,...

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials ${\left\{K_n\right\}}_{n=0}^{\infty }$—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space $H^1[-1,1]$ with respect to the (positive-...

In this article, we examine variational inequalities of the form $\left\{\begin{array}{c}⟨\mathcal{A}\left(u\right),v-u⟩+⟨\mathcal{F}\left(u\right),v-u⟩\ge 0,\forall v\in K\hfill \\ u\in K,\hfill \end{array}\right.$, where $\mathcal{A}$ is a generalized fractional $\mathsf{\Phi }$-Laplace operator, K is a closed co...

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the p...

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-d...

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spac...

This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-S...

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus o...

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or...

In this paper, we investigate the controllability of a class of impulsive $\psi$-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calcu...

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equ...

We investigate the commutativity of Toeplitz operators with trigonometric polynomial symbols on the Beurling subspaces $z^k{H}_s^2$ of the Hardy–Sobolev spaces. An explicit formula for the commutator of two Toeplitz operators is first derived in terms...

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorizati...

In this paper, we investigate the properties of the boundedness of fractional integral operators $K_{\alpha }$ defined on general measure metric spaces. We study their action in Lebesgue spaces $L^p\left(Y\right)$, Morrey spaces $L_{\phi }^p\left(Y\right)$, and extend our analysis to f...

In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential opera...

In this paper, we study the properties of trajectories of systems of ordinary differential equations generated by the velocity field of a moving incompressible viscoelastic fluid with memory along the trajectories in a domain with multiple boundary c...

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order $1<\delta <2$ with infinite delay. The analysis is carried out in an abstract Hilbert s...

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set Ɛ...

Fractional differential equations are commonly employed to characterize the long-term interactions of nonlinear systems, although they complicate inverse problems and numerical treatment. This article extends projection methods in order to numericall...

This paper investigates a multi-objective optimization framework for the simultaneous parameter identification and optimal vaccination control of a generalized non-linear SIR epidemic model. In contrast to traditional single-objective approaches, we...

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence a...

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize feature...

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fr...

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also con...

We study a Dirichlet $\mu$-parametric differential problem driven by a variable competing exponent operator, given by the sum of a negative p-Laplace differential operator and a positive q-Laplace differential operator, with a multivalued reaction ter...

The Steklov–Zaremba problem for the Laplace operator in a bounded domain with a strictly Lipschitz boundary is considered. A homogeneous Dirichlet condition is specified on the closed part of the boundary of the domain, and the Steklov boundary...

Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from $S^3$ embedded in ${\mathbb{R}}^4$. The presence of the cut introduces singular pertur...

In this article, we introduce a new mathematical functional whose minimization determines the quality of the solution for the exemplar-based inpainting-by-patch problem. The new functional expression includes finite difference terms in a similar fash...

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo conf...

The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals ${\int }_0^{2\pi }{e}^{\mathrm{i}\omega x}\phi \left(x\right)dx$ in the Sobolev space ${\tilde{H}}_2^m$. Here, ${\tilde{H}}_2^m$ is the Hilbert space of periodic and com...

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents...

This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier’s slip conditions on the channel walls. The important feature of o...

The present paper investigates the following inhomogeneous generalized Hartree equation $i\dot{u}+\Delta u=\pm {\left|u\right|}^{p-2}{\left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)u$, where the wave function is $u:=u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}$, with $N\ge 2$. In addition, th...

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...

We consider a problem of high-resolution array radar/SAR imaging formalized in terms of a nonlinear ill-posed inverse problem of nonparametric estimation of the power spatial spectrum pattern (SSP) of the random wavefield scattered from a remotely se...

Time-fractional diffusion equations with a Caputo derivative of order $\alpha \in (0,1)$ provide effective models for anomalous diffusion processes with memory effects, while the weakly singular fractional kernel and rough source terms generally lead...

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...

This paper presents a linear mathematical model of bass guitar string vibration with experimentally identified, frequency- and fret-dependent modal damping, aimed at high-fidelity generative sound synthesis. To identify the string damping parameters...

Time-fractional generalizations of the Lindblad master equation describe open quantum systems whose coupling to the environment exhibits power-law memory. We develop the optimal-control theory of such systems and analyse the consistency of the adjoin...

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain $\Omega \subset {\mathbb{R}}^n$ in the unknown $u=u(x,t)$, ${\mathbf{D}}_t^{{\nu }_0}\left({\varrho }_{0}u\right)-{\mathbf{D}}_t^{{\nu }_1}({\varrho }_{1}u$...