Topological Degree Research Articles

Periodic solutions of differential equations with periodic constraints

We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

New Results Regarding Positive Periodic Solutions of Generalized Leslie–Gower-Type Population Models

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results.

The effects of toxin and mutual interference among zooplankton on a diffusive plankton–fish model with Crowley–Martin functional response

A diffusive phytoplankton–zooplankton–fish model with toxin and Crowley–Martin functional response is considered. By the global bifurcation theorem and the fixed point index theory, the existence and multiplicity of coexistence states are discussed. Secondly, we investigate the bifurcation from a double eigenvalue by virtue of Lyapunov–Schmidt procedure and implicit function theorem. Then, the effect of large k, which measures the magnitude of interference among zooplankton, is studied by means of the combination of the perturbation theory and topological degree theory. The results indicate that if k is large enough, this system has only a unique asymptotically stable coexistence state provided that toxin is properly small and the maximal growth rates of phytoplankton and fish are suitably large. Furthermore, the extinction and permanence of the time-dependent system are determined by virtue of the comparison principle. Finally, we make some numerical simulations to validate and complement the theoretical analysis and exhibit the critical role of toxin, spatial diffusion and magnitude of interference among zooplankton in the dynamics. The findings suggest that the spatiotemporal dynamics of the systems with toxin and Crowley–Martin functional response are richer and more complex, and toxin and spatial diffusion have significant effects on the coexistence of phytoplankton–zooplankton–fish species.

Dynamics of converting skyrmion bags with different topological degrees into skyrmions in synthetic antiferromagnetic nanotracks

Skyrmion bags are spin textures with any integer topological degree, which can be driven by spin-polarized currents and generate multiple skyrmions when passing through racetracks with special geometries. We have proposed three nanotrack configurations with different narrow channels on synthetic antiferromagnetic racetracks and investigated the dynamic process of current-induced conversion of skyrmion bags into skyrmions. We have found that when skyrmion bags enter narrow channels, they can be converted into magnetic domains, while when the driving force from spin-transfer torque is strong enough, the magnetic domains can break free from the pinning at the ends of channels and form skyrmions. Both the number of channels and driving current density affect the number of generated skyrmions. As the number of channels rises, magnetic domains split at the junctions of channels, forming more magnetic domains and producing more skyrmions. Furthermore, the number of generated skyrmions is also related to the quantity, arrangement, and interaction forces of inner antiskyrmions. When the number of channels remains constant, the number of antiskyrmions only affects the transition of skyrmion bags to magnetic domains and does not affect the movement of magnetic domains or the transition of magnetic domains to skyrmions. The maximum of generated skyrmions in nanotracks with triple channels reaches 9. Dzyaloshinskii–Moriya interaction and anisotropy may play an important role in the structural stability of skyrmion bags, which can affect the splitting behavior of skyrmion bags. This work is beneficial for the design of artificial synapses and the application of neuromorphic computing based on skyrmion bags.

On a class of nonlinear elliptic problem of convolution type via topological degree theory

On a class of nonlinear elliptic problem of convolution type via topological degree theory

Network-Based Modeling of Lean Implementation Strategies and Planning in Prefabricated Construction

Abstract: Prefabricated construction (PC) is increasingly promoted in the construction sector for its potential benefits, including reduced resource assumption and improved quality. Accordingly, Lean methods are popularly applied to PC projects for optimizing operational processes and enhancing their performance in line with strategic objectives. A key factor in effectively implementing Lean to improve strategic control is developing specific strategies and planning that consider their complex interactions. Thus, this paper aims to propose a quantitative network-based model by integrating Interpretive Structural Modeling (ISM) and Matrix Impact Cross-Reference Multiplication Applied to a Classification (MICMAC) under complex network theory to develop a Lean implementation framework for effective strategy formulation. Specifically, 17 Lean implementation strategies for PC in the context of the Chinese prefabrication industry were identified via an extensive literature review and expert interviews. Then, ISM-MICMAC quantitatively identifies the direct and indirect relationships among strategies, while subsequent analysis of Topological Structure Weight (TSW) and Structural Degree Weight (SDW), as complex network parameters, is used to evaluate the importance of each strategy. The findings show that the strategic planning for Lean implementation in PC consists of four levels, i.e., foundation, organizational, technical, and control. Selecting appropriate Lean tools and technologies is crucial for PC implementation, which must be built on a top-level management team and foster a Lean culture. Moreover, it involves building a standardized system of processes and activities, enhancing both internal and external collaboration, and continuously improving processes in response to changes. On one hand, this in-depth network-based analysis offers practical insights for PC stakeholders, particularly in China, on Lean implementation in line with PC performance and strategic control and objectives. On the other hand, the network-based model can be future-implemented globally. Additionally, this study expands the current body of knowledge on Lean in PC by exploring the interrelationships of Lean implementation strategies.

The Bowen-Franks type theorems for multivalued maps and impulsive differential equations

The aim of the present paper is to extend the well known Bowen-Franks type theorems to interval and circle multivalued maps. In this way, a positive topological entropy, including its lower estimate, can be implied by the existence of periodic orbits whose orders differ from a power of 2, or provided the topological degree of given circle maps is absolutely greater than 1. Simple applications are also given to impulsive differential equations.

Topological degree method for a new class of Φ-Hilfer fractional differential Langevin equation

The aim of this article is to study the existence and uniqueness of the solution of a new class of fractional Langevin Φ-Hilfer differential equations. The proposed study is based on some fundamental definitions of fractional calculus and topological degree theory. We obtained the existence result by employing the topological degree method for condensing maps, and by making use of Banach’s fixed point theorem we deal with the uniqueness result. In order to illustrate our theoretical finding, we provide an example.

Modified sine-Gordon theory with static multikinks

We construct a modified non-Bogomol’nyi-Prasad-Sommerfield sine-Gordon theory which supports stable static kinks of arbitrary topological degree N. We use this toy model to study problems that are interesting for higher-dimensional soliton theories supporting multisolitons. We construct a 2-kink collective coordinate model and use it to generate scattering trajectories, which are compared to full-field dynamics. We find that the approximation works well, but starts to fail as radiation becomes more important, due to our model becoming less Bogomol’nyi-Prasad-Sommerfield or when the initial kink velocities are large. We also construct the quantum 2-kink using various approximations and consider how these quantum corrections affect the binding energy of the 2-kink. Published by the American Physical Society 2024

New attitude on sequential Ψ-Caputo differential equations via concept of measures of noncompactness

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko’s attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples.

A finite volume method for a nonlocal thermistor problem

In this work, we consider a more general version of the nonlocal thermistor problem, which describes the temperature diffusion produced when an electric current passes through a material. We investigate the doubly nonlinear problem where the nonlocal term is present on the right-hand side of the equation that describes the temperature evolution. Specifically, we employ topological degree theory to establish the existence of a solution to the considered problem. Furthermore, we separately address the uniqueness of the obtained solution. Additionally, we establish a priori estimates to demonstrate the convergence of a developed finite volume scheme used for the discretization of the continuous parabolic problem. Finally, to numerically simulate the proposed finite volume scheme, we use the Picard-type iteration process for the fully implicit scheme and approximate the nonlocal term represented by the integral with Simpson's rule to validate the efficiency and robustness of the proposed scheme.

Solvability of functional third-order problems of Ambrosetti–Prodi-type

This work presents an Ambrosetti–Prodi alternative for functional problems composed of a fully third-order differential equation with two types of functional boundary conditions. The discussion of existence and non-existence of solution is obtained in a more general case, and the multiplicity of solution is done with restrictive boundary conditions-The main arguments are based on the lower and upper solutions method, together with the Leray–Schauder topological degree theory. We stress that the multiplicity situation requires different speed growths on the variables.An example illustrates the results’ applicability and shows a technique to estimate the bifurcation values of the parameter.

TOPOLOGICAL DEGREE METHOD FOR A COUPLED SYSTEM OF $\psi$-FRACTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS WITH NON LOCAL CONDITIONS

This paper's primary goal is to build the existence of solutions for non-local coupled semi-linear differential equations involving $\psi$-Caputo differential derivatives of an arbitrary $l\in (0,1).$ We establish it with non-local conditions under which the examined coupled system has at least one solution by utilizing the notion of topological degree. Semilinear equations are an extremely unusual use of the corresponding approach.

Solutions to a generalized Chern–Simons Higgs model on finite graphs by topological degree

Consider a finite connected graph denoted as G = (V, E). This study explores a generalized Chern-Simons Higgs model, characterized by the equation Δu=λeu(eu−1)2p+1+f, where Δ denotes the graph Laplacian, λ is a real number, p is a non-negative integer, and f is a function on V. Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li et al. [Calc. Var. 63, 81 (2024)] and Chao and Hou [J. Math. Anal. Appl. 519, 126787 (2023)].

Existence results for coupled sequential ψ-Hilfer fractional impulsive BVPs: topological degree theory approach

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results.

Topological Degree via a Degree of Nondensifiability and Applications

The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. As applications, we study the existence of mild solution of functional semilinear integro-differential equations.

Existence of weak solutions of the fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problem by topological degree

In this paper, we prove the existence result of weak solutions to a class fractional ‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type . The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results.

Existence of multiple unbounded solutions for a three-point boundary value problems on an infinite time scales

Abstract In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

Study of a class of fractional-order evolution hybrid differential equations using a modified Mittag-Leffler-type derivative

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results.