Degree Theory Research Articles

A Novel High Efficiency Synthetic Fire Strike System

A new type of multi-artillery, multi-functional, multi-purpose synthetically-arranged battle group with high-efficiency fire strike system is constructed. The hierarchical structure and weight relationship of the system are determined by the AHP analytic hierarchy process, and then the inherent ability of each performance index is calculated by the fuzzy membership degree theory. These methods are used to establish the combat effectiveness evaluation model of high-efficiency fire strike system. The analysis shows that, for different damage targets, the deployment method of the efficient fire strike battalion has significantly improved the combat effectiveness of the traditional howitzer battalion. The research results have certain reference value for other arms to improve their weapon combat capabilities, conduct joint weapon operations and evaluate system combat effectiveness.

Robust feasibility of systems of quadratic equations using topological degree theory

Robust feasibility of systems of quadratic equations using topological degree theory

Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results.

Semi-classical states for elliptic system near saddle points of potentials

In this paper, we study the following Hamiltonian elliptic system −ε2Δu+u=Hv(x,u,v) in RN,−ε2Δv+v=Hu(x,u,v) in RN,u(x)→0, v(x)→0 as |x|→∞, where ɛ > 0 is a small parameter, H is a super-quadratic sub-critical Hamiltonian. Our investigation focuses on the cases that and , where K(x) possesses saddle points. By introducing a new penalization associated with the barycenters of functions and applying the Leray–Schauder degree theory, we adopt the local variational arguments in Byeon and Jeanjean (2007 Arch. Ration. Mech. Anal. 185 185–200); Byeon and Tanaka (2013 J. Eur. Math. Soc. 15 1859–99; 2014 Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations (Providence, RI: American Mathematical Society)) and obtain the existence of semi-classical states for the Hamiltonian elliptic system, which concentrate around the saddle points of V as ɛ → 0.

Existence of ground state solutions for weighted biharmonic problem involving non linear exponential growth

In this article, we study the following problem $$\begin{aligned} \Delta (w_{\beta }(x)\Delta u) = \ f(x,u) \quad \text{ in } \quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad \text{ on } \quad \partial B, \end{aligned}$$ where B is the unit ball of $${\mathbb {R}}^{4}$$ and $$ w_{\beta }(x)$$ a singular weight of logarithm type. The reaction source f(x, u) is a radial function with respect to x and it is critical in view of exponential inequality of Adams’ type. The existence result is proved by using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory results.

Comprehensive evaluation of urban talent development environment based on entropy weight-topsis method

For urban, the quantity and quality of talent is often an important measure of their development potential. Based on the theory of talent and environment comfort degree and the theory of urban comfort degree, this paper constructs the Evaluation Index System of urban talent development environment. This paper selects 7 coastal urban in Shandong province as the research objects, and uses entropy Weight-TOPSIS and cluster analysis to measure the talent development environment. The research results show that: (1) The talent development environment of seven coastal urban presents the phenomenon of “siphoning” and “distinctness” of talents. On the whole, Qingdao, Weifang, and Yantai have certain advantages in the talent development environment. (2) Qingdao is the leading city, Weifang, Yantai, Weihai and Dongying are follow-up urban, and Binzhou and Rizhao are backward urban. (3) The environment of the eight first-level indicators forms a “magnetic field” for the development of talents. Only by fully releasing the “magnetic field effect” of the talent development environment can urban ensure that talents are “attracted, retained and used well". This paper puts forward some suggestions to optimize the talent de-velopment environment in coastal urban, which will help to stimulate the vitality and creativity of all kinds of talents in coastal urban.

Dynamics of two-species Holling type-II predator-prey system with cross-diffusion

In this paper, we are devoted to studying a Holling type-II predator-prey system with cross diffusion. First of all, by a series of estimates, we establish the global existence of generalized solutions under some proper assumptions. By analyzing the distribution of eigenvalues, we investigate the local and global stability of the constant solution and also consider the steady state bifurcation and Hopf bifurcation near the constant steady state in details. In addition, the nonexistence of non-constant steady states is also investigated when diffusion rate d is large enough. Finally, sufficient conditions ensuring the existence of non-constant steady states are obtained by using Leray-Schauder degree theory.

Existence of positive periodic solutions for a periodic predator–prey model with fear effect and general functional responses

This paper investigates the existence of positive periodic solutions for a periodic predator-prey model with fear effect and general functional responses. The general functional responses can cover the Holling types II and III functional response, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, the ratio-dependent type with Michaelis–Menten type functional response, etc. Some new sufficient conditions for the existence of positive periodic solutions of the model are obtained by employing the continuation theorem of coincidence degree theory and some ingenious estimation techniques for the upper and lower bounds of the a priori solutions of the corresponding operator equation. Our results considerably improve and extend some known results.

Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory

This article aims to investigate the existence and uniqueness of solutions to impulsive fractional differential equations. Firstly, we show a formula for solutions to an impulsive fractional problem involving a generalization of the classical Caputo derivative with a nonlocal condition in a Banach space. Secondly, some sufficient conditions for the existence of solutions are established by applying fixed point methods. Furthermore, we reconsider some of operators for deriving the existence and uniqueness of the solution to impulsive fractional problem using topological degree techniques. The results support this in one example.

System of fractional boundary value problems at resonance

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at $$t=0$$ and an integral boundary condition at $$t=1$$ . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result.

Stability-Level Evaluation of the Construction Site above the Goaf Based on Combination Weighting and Cloud Model

Mineral resource-based cities have formed a large number of goafs due to the long-term mining of coal. It is of great significance to make full use of the abandoned land resources above the goaf to promote the transformation and development of resource-based cities. In order to avoid the threat of surface residual deformation to the proposed construction project, it is an urgent problem to obtain the stability results of the construction site accurately. First of all, based on the principles of relevance, hierarchy, representativeness and feasibility of index selection, 10 indexes are selected to construct the stability evaluation index system. Then the subjective weight and objective weight of evaluation indexes are determined based on improved AHP, rough set and CRITIC methods, which improves the accuracy of the determination of the index weights. In addition, the membership degree of each index is determined using the cloud model. Finally, the stability grade can be obtained according to the maximum membership degree theory. The above researches are applied to evaluate the stability of the Mianluan expressway construction site, and the results show that the stability level of the study area is not uniform and that there are two states: stable and basically stable. Finally, a sensitivity analysis of the subjective weight of each index is carried out, the index stopping time has the highest sensitivity to weight (12.44%), which is far lower than the corresponding weight change rate of 100%, indicating that the determination of weight is scientific and reasonable. These things considered, the reliability of the evaluation result is indirectly verified according to the field leveling. This research can provide a reference for the effective utilization of land resources above an old goaf.

Periodic solutions to superlinear indefinite planar systems: A topological degree approach

We deal with a planar differential system of the form{u′=h(t,v),v′=−λa(t)g(u), where h is T-periodic in the first variable and strictly increasing in the second variable, λ>0, a is a sign-changing T-periodic weight function and g is superlinear. Based on the coincidence degree theory, in dependence of λ, we prove the existence of T-periodic solutions (u,v) such that u(t)>0 for all t∈R. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or ϕ-Laplacian-type differential operators).

Response solutions of 2-dimensional degenerate systems under quasi-periodic perturbations

This paper considers quasi-periodic perturbations of 2-dimensional degenerate systems. It is proved that if the equilibrium point of the unperturbed system is hyperbolic-type degenerate, then the perturbed system has a small response solution. The proof is based on the topological degree theory and some KAM techniques developed in [3].

On Nonlinear Periodic Problems with Caputo's Exponential Fractional Derivative

In this article, we employ Mawhin's theory of degree of coincidence to provide an existence result for a class of problems involving non-linear implicit fractional differential equations with the exponentially fractional derivative of Caputo. Two examples are provided to demonstrate the applicability of our results.

Existence of anti-periodic solutions for Ψ-Caputo-type fractional p-Laplacian problems via Leray--Schauder degree theory

Abstract The main crux of this work is to study the existence of solutions for a certain type of nonlinear Ψ-Caputo fractional differential equations with anti-periodic boundary conditions and p-Laplacian operator. The proofs are based on the Leray–Schauder degree theory and some basic concepts of Ψ-Caputo fractional calculus. As an application, our theoretical result has been illustrated by providing a suitable example.

The response solution for a class of nonlinear periodic systems under small perturbations

This paper considers periodic perturbations of a class of nonlinear degenerate systems. By the technique of introducing external parameters, the implicit function theorem and the theory of Brouwer topological degree, it is proved that the perturbed systems have response solutions if the perturbations are sufficiently small.

Perturbation of the spectra for asymptotically constant differential operators and applications

We study the spectra for a class of differential operators with asymptotically constant coefficients. These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We present a unified framework to prove the perturbation results on the related spectra. The proof is based on exponential dichotomies and the Brouwer degree theory. As applications, we employ the developed theory to study the stability of quasi-periodic solutions of the Ginzburg–Landau equation, fold-Hopf bifurcating periodic solutions of reaction–diffusion systems coupled with ordinary differential equations, and periodic annulus of the hyperbolic Burgers–Fisher model.

Existence and Stability of a Nonlinear Distributed Delayed Periodic AG-Ecosystem with Competition on Time Scales

The Ayala-Gilpin (AG) kinetics system is one of the famous mathematical models of ecosystem. This model has been widely concerned and studied since it was proposed. This paper stresses on a nonlinear distributed delayed periodic AG-ecosystem with competition on time scales. In the sense of time scale, our model unifies and generalizes the discrete and continuous cases. Firstly, with the aid of the auxiliary function having only two zeros in the real number field, we apply inequality technique and coincidence degree theory to obtain some sufficient criteria which ensure that this model has periodic solutions on time scales. Meanwhile, the global asymptotic stability of the periodic solution is founded by employing stability theory in the sense of Lyapunov. Eventually, we provide an illustrative example and conduct numerical simulation by means of MATLAB tools.

Combined effect of simplicial complexes and interlayer interaction: An example of information-epidemic dynamics on multiplex networks

In this paper, we investigate the effect of self-awareness (interlayer interaction) for information-epidemic dynamics with simplicial complexes both near and away from the epidemic threshold. It is shown that, contrary to previous views, self-awareness plays a key role near the epidemic threshold. In small homogeneous networks, multiple susceptibility peaks can emerge in the susceptibility of the epidemic layer under the combined effect of simplicial complexes and self-awareness, even two types of multiple susceptibility peaks with completely opposite mechanisms. This means that one needs to be very careful when obtaining epidemic thresholds based on susceptibility. Moreover, the self-awareness can regulate the presence or absence of bistable phenomena both in epidemic prevalence and epidemic threshold. We also found that the time series of disease may be nonmonotonic, with a peak, and that self-awareness is one of the factors controlling the relative height between maximum and steady state. In addition, we modify heterogeneous mean-field theory and partial effective degree theory to accommodate the introduction of simplicial complexes in dynamics. We believe that our study has implications for other dynamics concerning higher-order and interlayer interactions.

Multiple positive bound state solutions for fractional Schrödinger-Poisson system with critical growth

In this paper, we deal with the following critical fractional Schrödinger-Poisson system without subcritical perturbation $$ \begin{cases} (-\Delta)^{s} u + V(x)u + K(x)\phi u=|u|^{2_{s}^{*}-2}u, & x\in\mathbb{R}^{3},\\ (-\Delta)^{t}\phi=K(x)u^{2}, & x\in\mathbb{R}^{3}, \end{cases} $$ where $s\in(\frac{3}{4},1),$ $t\in(0,1)$, $2^{\ast}_{s}=\frac{6}{3-2s}$ is the critical Sobolev exponent. When $V(x)$ is positive bounded from below, combining with variation methods and Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this system. The results obtained in this paper extend and improve some recent works in which potential function $V(x)$ may vanish at the infinity.