Topological Degree Research Articles

Existence results for Schrödinger type double phase variable exponent problems with convection term in $ \mathbb R^{N} $

<abstract><p>This paper was concerned with a new class of Schrödinger equations involving double phase operators with variable exponent in $ \mathbb R^{N} $. We gave the corresponding Musielak-Orlicz Sobolev spaces and proved certain properties of the double phase operator. Moreover, our main tools were the topological degree theory and Galerkin method, since the equation contained a convection term. By using these methods, we derived the existence of weak solution for the above problems. Our result extended some recent work in the literature.</p></abstract>

Existence and uniqueness of solutions for the $p$-Lamé Dirichlet problem by topological degree

Existence and uniqueness of solutions for the $p$-Lamé Dirichlet problem by topological degree

TOPOLOGICAL DEGREE METHOD FOR A CLASS OF Ψ-CAPUTO FRACTIONAL DIFFERENTIAL LANGEVIN EQUATION

This paper deals with the existence and uniqueness of solution for a new class of Ψ-Caputo fractional differential Langevin equation. The suggested study is based on some basic definitions of topological degree theory and fractional calculus. We established the existence result by using the topological degree method for condensing maps, and by means of Banach’s fixed point theorem we obtained the uniqueness result. As application, we give an illustrative example to demonstrate our theoretical result.

Multiple positive radial solutions for Dirichlet problem of the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime

In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.

EXISTENCE OF SOLUTIONS FOR A SECOND ORDER THREE POINT BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON A HALF LINE

In this paper, we consider the second order three point boundary value problem with integral bounday conditions on a half-line of the form u”(t) + q(t)f(t, u(t), u'(t)) = 0, t∈ (0,+∞), u(0) − au’ (0) = λη 0 u(s)ds, lim t→+∞ u (t) = u (+∞) = λ η0 u(s)ds, where λ > 0, 0 < λη < 1, a > 0, q : (0,+∞) → (0,+∞), f : [0,+∞) × R × R → R is continuous and satisfies Nagumo’s condition. We employ the Schauder’s fixed point theorem, the upper and lower solution method and topological degree theories to show the existence of at least one solution and then existence of three solutions of the above boundary value problem. Finally, we have shown the importance of our results through an examples.

Nonlinear elliptic equations by topological degree in Musielak-Orlicz-Sobolev spaces

"We prove by using the topological degree theory the existence of at least one weak solution for the nonlinear elliptic equation − div a1(x, ∇u) + a0(x, u) = f (x, u, ∇u) with homogeneous Dirichlet boundary condition in Musielak-Orlicz-Sobolev spaces. Keywords: Nonlinear elliptic equation, Musielak-Orlicz-Sobolev space, topological degree."

Bifurcation from interval and multiple positive solutions for a nonlinear fractional differential equations with integral boundary conditions

This paper studies the bifurcation from interval and Multiple positive solutions for a boundary-value problem of nonlinear fractional differential equations with integral boundary conditions. Using the bifurcation technique and the topological degree theory, the existence of Multiple positive solutions is investigated and some sufficient conditions are obtained.

P(x)-Kirchhoff-type problem with no-flux boundary conditions and convection

Abstract This article establishes the existence of a weak solution for a class of p ( x ) p\left(x) -Kirchhoff-type problem under no-flux boundary conditions with a reaction term depending also on the gradient convection. The proof of the main result is constructed by utilizing the theory of topological degree and the theory of generalized Sobolev spaces.

A weak solution to a Kirkoff type problem with discontinuous nonlinearities

This paper is devoted to the study of a class of Kirchhoff-type problems with discontinuous nonlinearities with Neumann boundary data. Here,
 by employing the topological degree methods for the abstract Hammerstein equation, we establish the existence of at least one solution.

The existence problem of feedback control for one fractional Voigt model

In this paper, we study the feedback control problem for a mathematical model that describes the motion of a viscoelastic fluid with memory along velocity eld trajectories. We prove the existence of an optimal control that gives a minimum to a given bounded and semi-continuous from below quality functional. The proof uses the approximation-topological approach, the theory of regular Lagrangian flows, and the theory of topological degree for multivalued vector elds.

Exploration and validation of key genes associated with early lymph node metastasis in thyroid carcinoma using weighted gene co-expression network analysis and machine learning.

Thyroid carcinoma (THCA), the most common endocrine neoplasm, typically exhibits an indolent behavior. However, in some instances, lymph node metastasis (LNM) may occur in the early stages, with the underlying mechanisms not yet fully understood. LNM potential was defined as the tumor's capability to metastasize to lymph nodes at an early stage, even when the tumor volume is small. We performed differential expression analysis using the 'Limma' R package and conducted enrichment analyses using the Metascape tool. Co-expression networks were established using the 'WGCNA' R package, with the soft threshold power determined by the 'pickSoftThreshold' algorithm. For unsupervised clustering, we utilized the 'ConsensusCluster Plus' R package. To determine the topological features and degree centralities of each node (protein) within the Protein-Protein Interaction (PPI) network, we used the CytoNCA plugin integrated with the Cytoscape tool. Immune cell infiltration was assessed using the Immune Cell Abundance Identifier (ImmuCellAI) database. We applied the Least Absolute Shrinkage and Selection Operator (LASSO), Support Vector Machine (SVM), and Random Forest (RF) algorithms individually, with the 'glmnet,' 'e1071,' and 'randomForest' R packages, respectively. Ridge regression was performed using the 'oncoPredict' algorithm, and all the predictions were based on data from the Genomics of Drug Sensitivity in Cancer (GDSC) database. To ascertain the protein expression levels and subcellular localization of genes, we consulted the Human Protein Atlas (HPA) database. Molecular docking was carried out using the mcule 1-click Docking server online. Experimental validation of gene and protein expression levels was conducted through Real-Time Quantitative PCR (RT-qPCR) and immunohistochemistry (IHC) assays. Through WGCNA and PPI network analysis, we identified twelve hub genes as the most relevant to LNM potential from these two modules. These 12 hub genes displayed differential expression in THCA and exhibited significant correlations with the downregulation of neutrophil infiltration, as well as the upregulation of dendritic cell and macrophage infiltration, along with activation of the EMT pathway in THCA. We propose a novel molecular classification approach and provide an online web-based nomogram for evaluating the LNM potential of THCA (http://www.empowerstats.net/pmodel/?m=17617_LNM). Machine learning algorithms have identified ERBB3 as the most critical gene associated with LNM potential in THCA. ERBB3 exhibits high expression in patients with THCA who have experienced LNM or have advanced-stage disease. The differential methylation levels partially explain this differential expression of ERBB3. ROC analysis has identified ERBB3 as a diagnostic marker for THCA (AUC=0.89), THCA with high LNM potential (AUC=0.75), and lymph nodes with tumor metastasis (AUC=0.86). We have presented a comprehensive review of endocrine disruptor chemical (EDC) exposures, environmental toxins, and pharmacological agents that may potentially impact LNM potential. Molecular docking revealed a docking score of -10.1 kcal/mol for Lapatinib and ERBB3, indicating a strong binding affinity. In conclusion, our study, utilizing bioinformatics analysis techniques, identified gene modules and hub genes influencing LNM potential in THCA patients. ERBB3 was identified as a key gene with therapeutic implications. We have also developed a novel molecular classification approach and a user-friendly web-based nomogram tool for assessing LNM potential. These findings pave the way for investigations into the mechanisms underlying differences in LNM potential and provide guidance for personalized clinical treatment plans.

Subsonic steady-states for bipolar hydrodynamic model for semiconductors

In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ=∞, namely, the pure Euler-Poisson case; the other case is jτ≪1, which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy.

Fractional Sobolev Spaces with Kernel Function on Compact Riemannian Manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p(·,·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\cdot , \\cdot )$$\\end{document}-Laplacian operator. As one of the main tools, topological degree theory is applied.

Identification of potential molecular targets and repurposed drugs for tuberculosis using network-based screening approach, molecular docking, and simulation

The spread of drug-resistant strains of tuberculosis has hampered efforts to control the disease worldwide. The Mycobacterium tuberculosis cell wall envelope is dynamic, with complex features that protect it from the host immunological response. As a result, the bacterial cell wall components represent a potential target for drug discovery. Protein-protein interaction networks (PPIN) are critical for understanding disease conditions and identifying precise therapeutic targets. We used a rational theoretical approach by constructing a PPIN with the proteins involved in cell wall biosynthesis. The PPIN was constructed through the STRING database and embB was identified as a key protein by using four topological measures, betweenness, closeness, degree, and eigenvector, in the CytoNCA tool in Cytoscape. The ‘Drug repurposing’ approach was employed to find suitable inhibitors against embB. We used the Schrödinger suites for molecular docking, molecular dynamics simulation, and binding free energy calculations to validate the binding of protein with the ligand. FDA-approved drugs from the ZINC database and DrugBank were screened against embB (PDB ID: 7BVF) using high-throughput virtual screening, standard precision, and extra precision docking. The drugs were screened based on the XP docking score of the standard drug ethambutol. Accordingly, from the top five hits, azilsartan and dihydroergotamine were selected based on the binding free energy values and were further subjected to Molecular Dynamics Simulation studies for 100 ns. Our study confirms that Azilsartan and Dihydroergotamine form stable complexes with embB and can be used as potential lead molecules based on further in vitro and in vivo experimental validation. Communicated by Ramaswamy H. Sarma

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

Abstract In this paper we study a Neumann boundary value problem of a new p(x)-Kirchhoff type problems driven by p(x)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (S+) type,weprove theexistenceofaweak solutionsof this problem. Our results are a natural generalisation of some existing ones in the context of p(x)-Kirchhoff type problems.

Existence and multiplicity of solutions for the Cauchy problem of a fractional Lorentz force equation

This paper aims to deal with the Cauchy problem of a fractional Lorentz force equation. By the methods of reducing and topological degree in cone, the existence and multiplicity of solutions to the problem were obtained, which extend and enrich some previous results.

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

Dissipationless layertronics in axion insulator MnBi2Te4.

Surface electrons in axion insulators are endowed with a topological layer degree of freedom followed by exotic transport phenomena, e.g., the layer Hall effect. Here, we propose that such a layer degree of freedom can be manipulated in a dissipationless way based on the antiferromagnetic [Formula: see text] with tailored domain structure. This makes [Formula: see text] a versatile platform to exploit the 'layertronics' to encode, process and store information. Importantly, the layer filter, layer valve and layer reverser devices can be achieved using the layer-locked chiral domain wall modes. The dissipationless nature of the domain wall modes makes the performance of the layertronic devices superior to those in spintronics and valleytronics. Specifically, the layer reverser, a layer version of the Datta-Das transistor, also fills up the blank in designing the valley reverser in valleytronics. Our work sheds light on constructing new generation electronic devices with high performance and low-energy consumption in the framework of layertronics.

On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

AbstractIn this paper we consider semilinear equations $$-\Delta u=f(u)$$ - Δ u = f ( u ) with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution u has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument.

Dynamics of pumping effect in 1 pipe-2 tanks flow configuration with periodic excitation

This paper is dedicated to analyzing the pumping effect in rigid 1 pipe-2 tanks flow configuration proposed by G. Propst. Our main purpose is to study the dynamic behaviors of the periodic solutions of the governing differential equation, as the periodic solution will result in the periodically forced pump in the configuration. We establish a sufficient condition for the existence of the harmonic solution by applying the topological degree theory. Additionally, we utilize the regular perturbation method and averaging method to examine the stability of the harmonic solution, as well as the existence and stability of the second-order subharmonic solution of the equation. Through numerical bifurcation analysis, we present the bifurcation diagrams of the periodic solutions, which reveal that higher-order subharmonic solutions with varying stability can be generated through period-doubling bifurcation of the periodic solutions.