Continuation Theorem Research Articles

Large time behavior of nonautonomous differential systems modeling antibiotic‐resistant bacteria in rivers

In this paper, we study a general nonautonomous model for bacterial dynamics in rivers. The mathematical model is represented by a nonautonomous system of nonlinear ordinary differential equations. We show the existence of a bounded positive invariant and attracting set. By using the Lyapunov function method, we establish global stability of steady‐state solutions of the associated autonomous system. Second, the existence of positive periodic solutions of the nonautonomous system is proven using a continuation theorem based on coincidence degree theory.

Solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian at resonance

This article investigates the existence of solutions of mixed Hilfer fractional differential equations with p-Laplacian under the functional boundary conditions at resonance. By defining Banach spaces with appropriate norms, constructing suitable operators, and using the extension of the continuity theorem, some of the current results are extended to the nonlinear situation, and some new existence results of the problem are obtained. Finally, an example is given to verify our main results.

Continuity of roots for polynomials over valued fields

We study connections between polynomials which are close to each other, i.e., whose respective coefficients are close in the topology induced by a valuation. This paper consists of both an overview on known root continuity theorems and new results on the subject. We present theorems which have been published over the years, correcting and improving some of their formulations and proofs. We also give a sketch of several approaches to root continuity, such as employing the Newton Polygon and induction on the degree of the polynomial. We study the behavior of the irreducible factors of a given polynomial, the extensions generated by its roots and invariants connected to that polynomial under transition to a second polynomial which is sufficiently close. Further, we present applications of root continuity to the study of valued field extensions and ramification theory, including the theory of the defect.

Unified Proofs of Three Fundamental Properties of Continuous Functions

Summary We provide a unified approach to three fundamental properties of continuous functions on closed and bounded intervals: the intermediate value theorem, and the uniform continuity theorem. We prove all three using the same building block, only making use of the least upper bound axiom and the definition of continuity.

Existence of positive periodic solutions of several types of biological models with periodic coefficients

This paper investigates the existence of positive periodic solutions for several types of important biological models with periodic coefficients, including the HIV model with multiple infection stages and general incidence, the epidemic models (such as the SIRS model, the SIRI model, the SEIRS model, etc.) with general incidence. By means of the continuation theorem in the coincidence degree, and the combination of analytical techniques such as constructing appropriate auxiliary functions, we give several classes of explicit sufficient conditions for the existence of positive periodic solutions for these biological models. When these models with periodic coefficients degenerate to the corresponding autonomous cases, our conditions all naturally degenerate to the basic reproduction number is greater than 1. Our results greatly improve and expand the studies in the existing literatures.

Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies

For an exchange economy with a continuum of traders and a finite-dimensional commodity space under some standard assumptions, Aumann showed that the core and the set of competitive allocations (two most important solutions) coincide and Mas-Colell proved that the bargaining set and the set of competitive allocations coincide. However, in the case of exchange economies with a finite number of traders, it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we establish the equivalence of the fuzzy core, the fuzzy bargaining set (or Aubin bargaining set), and the set of competitive allocations in a finite coalition production economy with an infinite-dimensional commodity space under some standard assumptions. We first derive a continuous equivalence theorem and then discretize it to obtain the desired equivalence in finite economies.

Differential equations involving homeomorphism with nonlinear boundary conditions

We establish existence results for differential equations of the form: where is continuous and is a homeomorphism, with nonlinear boundary conditions: where are continuous and are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators.

Dynamic simulation model for three-wheel air-cycle refrigeration systems in civil aircrafts

Dynamic simulation model of air-cycle refrigeration systems in civil aircrafts is essential for revealing the operation mechanism during flight process. In the present study, a dynamic model of three-wheel air-cycle refrigeration system was developed. A near-logarithm heat transfer temperature difference function is utilized to improve the stability of the heat transfer solution. A dynamic solution method based on continuity equation and theorem of momentum was presented as the framework of the dynamic mass transfer solution. A pressure-flow decoupling method is presented to solve the dynamic pressure in the high-pressure zone, resolving the absence of key pressure parameters. The model has been validated based on experimental data, with a pack discharge temperature deviation of 0.05K. Dynamic simulations have been conducted with actual flight data to obtain transient performance under airborne condition. In the takeoff and landing phase, the drastic change in the environment results in low discharge temperature, endangering the condenser with icing risk, which can be avoided by enabling the TCV. In the cruise phase, the discharge temperature slightly exceeds the upper limit for a couple of minutes. The optimization by slightly extending the length of heat exchanger channels succeeded in eliminating the excessive discharge temperature with minimal side effects.

Neural Networks of the Rational r-th Powers of the Multivariate Bernstein Operators

In this study, a novel neural network for the multivariance Bernstein operators' rational powers was developed. A positive integer is required by these networks. In the space of all real-valued continuous functions, the pointwise and uniform approximation theorems are introduced and examined first. After that, the Lipschitz space is used to study two key theorems. Additionally, some numerical examples are provided to demonstrate how well these neural networks approximate two test functions. The numerical outcomes demonstrate that as input grows, the neural network provides a better approximation. Finally, the graphs used to represent these neural network approximations show the average error between the approximation and the test function.

Dynamics of periodic solution to a electrostatic Micro-Electro-Mechanical system

A canonical mass–spring model of electrostatically actuated Micro-Electro-Mechanical System (MEMS) is studied. We investigate the existence and bifurcations of the periodic solution by means of the continuation theorem of Mańasevich and Mawhin with techniques of a priori estimate and bifurcation theory. The main results answer the open problem proposed by P. Torres in the known literature. It also reveals that the system undergoes saddle node and period doubling bifurcation leading to the multiplicity of periodic solution. Moreover, bistability of periodic solution generated by the combination of these bifurcations is detected for the first time, which provides some new insight into the so called pull-in instability in the system. At last, periodic orbits with different stability and corresponding time series are given to illustrate the results.

Existence of Periodic Solutions for a Class of Fourth-Order Difference Equation

We apply the continuation theorem of Mawhin to ensure that a fourth-order nonlinear difference equation of the form Δ 4 u k − 2 − a k u α k + b k u β k = 0 with periodic boundary conditions possesses at least one nontrivial positive solution, where Δ u k = u k + 1 − u k is the forward difference operator, α > 0 , β > 0 and α ≠ β , a k , b k are T -periodic functions and a k b k > 0 . As applications, we give some examples to illustrate the application of these theorems.

EXISTENCE RESULTS OF GENERALIZED PROPORTIONAL FRACTIONAL DIFFERENTIAL EQUATIONS AT RESONANCE CASE BY THE TOPOLOGICAL DEGREE THEORY

In this paper we study the existence of solutions to multi-point boundary value problem of fractional differential equations at resonance, involving the Generalized Proportional Fractional derivative(GPF derivatives). the concerned results are obtained via extansion of Mawhin's continuation theorem. An illustrative example is presented.

Existencia de solución de un problema distribucional para una ecuación de Schrödinger generalizada

In this article, we prove the existence and uniqueness of the solution of the homogeneous generalized Schrödinger equation of order m in the periodic distributional space P0, where m is an even number not a multiple of four. Furthermore, we prove that the solution depends continuously respect to the initial data in P0. Introducing a family of weakly continuous operators, we prove that this family is a group in P0. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we give the conclusions and remarks derived from this study.

Existence of periodic orbits for planar differential systems with delay angle

By using the Mawhin’s continuation theorem, this paper establishes a criterion for the existence of periodic orbits of the planar differential systems with delay angle. An example is provided to illustrate the application of the result.

Clifford-valued linear canonical transform: Convolution and uncertainty principles

Linear canonical transform is a three-parameter class of linear integral transformation, which has found many applications in optics, filter design, speech, image, video, and signal processing due to its flexibility in representing information within the linear canonical domain. In this article, we extend the linear canonical transform to the class of Clifford-valued signals and introduce a novel transform coined as Clifford-valued linear canonical transform. The primary analysis encompasses the derivation of fundamental properties of the proposed transform such as translation and scaling covariance, inner product relation and inversion formula. Subsequently, we investigate the convolution, continuity, and differentiation theorems for the Clifford-valued linear canonical transform. A non-homogeneous partial differential equation has also been solved in the Clifford framework as an application of the differentiation theorem. Finally, we culminate our investigation by deriving several uncertainty principles for the proposed transform.

A General 3D Non-Stationary 6G Channel Model With Time-Space Consistency

This paper proposes a novel general sixth-generation (6G) irregular-shaped geometry-based stochastic model (IS-GBSM) for millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) wireless communication systems. The proposed IS-GBSM can model spherical wavefront propagation, high delay resolution, and three-dimensional (3D) space-time-frequency non-stationarity of channels with time-space consistency, and thus can be applied to integrated communication and sensing systems. To capture the 3D non-stationarity with channel consistency, a new hybrid method combining geometric and parametric methods is developed. Based on the geometric method, i.e., visibility region (VR), the smooth and consistent appearance and disappearance of clusters during array-time evolution are modeled. With the help of the parametric method, a frequency-dependent path gain is introduced, and a visible factor based on uniform continuity theorem and bounded variation function is further proposed to capture the soft cluster power handover during cluster array-time evolution. Key channel statistics of the proposed IS-GBSM are derived and investigated. Simulation results demonstrate that the space-time-frequency non-stationarity is modeled and time-space consistency is captured. Finally, the utility of the proposed IS-GBSM is verified by the excellent agreement between simulation results and measurement.

Enriched Pro-Categories and Shapes

Given a category C and a directed partially ordered set J, a certain category proJ-C on inverse systems in C is constructed such that the ordinary pro-category pro-C is the most special case of a singleton J ≡ {1}. Further, the known pro*-category pro*-C becomes proN-C. Moreover, given a pro-reflective category pair (C, D), the J-shape category ShJ(C, D) and the corresponding J-shape functor SJ are constructed which, in mentioned special cases, become the well known ones. Among several important properties, the continuity theorem for a J-shape category is established. It implies the “J-shape theory” is a genuine one such that the shape and the coarse shape theory are its very special instances.

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

Existence of positive periodic solutions of a delayed periodic Microcystins degradation model with nonlinear functional responses

This paper proposes a delayed periodic Microcystins degradation model with nonlinear functional responses. By using the continuation theorem in the coincidence degree, we obtain two classes of explicit sufficient conditions for the existence of positive periodic solutions of the model.

Stress analysis of an inclusion layer bonded to an irregularly shaped pore under an edge dislocation or a concentrated load

Abstract Pores play an important role in the failure analysis of metal castings. During the solidification process, slag inclusions such as oxides, nitrides and sulfides may form around the pores. This paper provides an analytical solution for an inclusion layer bonded to a square pore under edge dislocation or a concentrated load. Based on the mapping method and analytical continuation theorem, both sliding and climbing forces as well as interfacial stresses induced by a dislocation or concentrated load are obtained in a closed form. The results indicate that an inclusion layer with a larger (or smaller) shape factor would result in a stable equilibrium position near (or far from) the interface. When the shape factor decreases, the stable equilibrium position moves away from the interface if the stiffness of the inclusion layer increases. For a concentrated load, the interfacial stresses increased with the elastic mismatch and corner sharpness. These findings could improve the compatibility between the matrix and inclusion layer in metallurgical manufacturing systems.