Nonnegative Solutions Research Articles

Three nonnegative solutions for Sturm-Liouville BVP and application to the complete Sturm-Liouville equations

<abstract><p>The main purpose of this manuscript is to investigate the Sturm-Liouville BVP for non-autonomous Lagrangian systems. Under the suitable assumptions, we establish an existence theorem for three nonnegative solutions via Bonanno-Candito's three critical point theory. As an application in the complete Sturm-Liouville equations with Sturm-Liouville BVC, we get an existence theorem of three nonnegative solutions. Meanwhile, we give three examples to show the correctness of our results.</p></abstract>

Mathematical Modelling of the Dynamics and Control of Communicable Diseases with Emphasis on Cholera Epidemics

Cholera remains a disease of public health importance, associated with high morbidity and mortality. This research provides a mathematical approach on the effective control and possible eradication of Vibrio cholera causing menace. This was done using a deterministic SPIRB mathematical model, were we defined S-susceptible, P-protected, I-infected, R-recovered and B-bacteria. We carried out mathematical analyses on the model, such as the invariant region analysis, non-negativity of solutions, equilibrium points, basic reproduction number, stability analysis, sensitivity analysis and bifurcation analysis. All solutions of the SPIRB model are positive and there exists an invariant region for the model. Although, the disease-endemic equilibrium of the model is globally asymptomatically stable, the disease-free equilibrium of the model is not globally stable but locally asymptomatically stable. It was also revealed that the recruitment rate into the susceptible class, the probability that each contact is effective enough to cause infection, the contact rate with contaminated environment, the average contribution of each infected individual to the pathogen population of Vibrio cholera, and the progression rate of protected individuals to the susceptible class are the most sensitive parameters. We showed via the bifurcation analysis that if the progression rate of protected individuals to the susceptible class can be kept very close to zero, say between zero and unity, then the disease-free equilibrium can be kept stable. Numerical simulations were conducted to show the effects of some parameters on the classes.

The fractal-fractional Atangana-Baleanu operator for pneumonia disease: stability, statistical and numerical analyses

<abstract><p>The present paper studies pneumonia transmission dynamics by using fractal-fractional operators in the Atangana-Baleanu sense. Our model predicts pneumonia transmission dynamically. Our goal is to generalize five ODEs of the first order under the assumption of five unknowns (susceptible, vaccinated, carriers, infected, and recovered). The Atangana-Baleanu operator is used in addition to analysing existence, uniqueness, and non-negativity of solutions, local and global stability, Hyers-Ulam stability, and sensitivity analysis. As long as the basic reproduction number $ \mathscr{R}_{0} $ is less than one, the free equilibrium point is local, asymptotic, or otherwise global. Our sensitivity statistical analysis shows that $ \mathscr{R}_{0} $ is most sensitive to pneumonia disease density. Further, we compute a numerical solution for the model by using fractal-fractional. Graphs of the results are presented for demonstration of our proposed method. The results of the Atangana-Baleanu fractal-fractional scheme is in excellent agreement with the actual data.</p></abstract>

A fractional numerical study on a plant disease model with replanting and preventive treatment

Food security has become a significant issue due to the growing human population. In this case, a significant role is played by agriculture. The essential foods are obtained mainly from plants. Plant diseases can, however, decrease both food production and its quality. Therefore, it is substantial to comprehend the dynamics of plant diseases as they can provide insightful information about the dispersal of plant diseases. In order to investigate the dynamics of plant disease and analyze the effects of strategies of disease control, a mathematical model can be applied. We show that this model provides the non-negative solutions that population dynamics requires. The model was investigated by using the Atangana-Baleanu in Caputo sense (ABC) operator which is symmetrical to the Caputo-Fabrizio (CF) operator with a different function. Whereas the ABC operator uses the generalized Mittag-Leffler function while the CF operator employs the exponential kernel. For the proposed model, we have displayed the local and global stability of a nonendemic and an endemic equilibrium, existence and uniqueness theorems. By applying the fractional Adams-Bashforth-Moulton method, we have implemented numerical solutions to illustrate the theoretical analysis.

Global dynamics of deterministic-stochastic dengue infection model including multi specific receptors via crossover effects

<abstract><p>Dengue viruses have distinct viral regularities due to the their serotypes. Dengue can be aggravated from a simple fever in an acute infection to a presumably fatal secondary pathogen. This article investigates a deterministic-stochastic secondary dengue viral infection (SDVI) model including logistic growth and a nonlinear incidence rate through the use of piecewise fractional differential equations. This framework accounts for the fact that the dengue virus can penetrate various kinds of specific receptors. Because of the supplementary infection, the system comprises both heterologous and homologous antibody. For the deterministic case, we determine the invariant region and threshold for the aforesaid model. Besides that, we demonstrate that the suggested stochastic SDVI model yields a global and non-negative solution. Taking into consideration effective Lyapunov candidates, the sufficient requirements for the presence of an ergodic stationary distribution of the solution to the stochastic SDVI model are generated. This report basically utilizes a novel idea of piecewise differentiation and integration. This method aids in the acquisition of mechanisms, including crossover impacts. Graphical illustrations of piecewise modeling techniques for chaos challenges are demonstrated. A piecewise numerical scheme is addressed. For various cases, numerical simulations are presented.</p></abstract>

On the Diophantine Equation 15x – 13y = z2

In this article, we prove that the Diophantine equation 15x – 13y = z2 has nonnegative integer solution. The result reveals that the solution (x, y, z) = (0, 0, 0). Keywords: Diophantine equation; factoring method; modular arithmetic method

Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

<p style='text-indent:20px;'>In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.</p>

Dynamics analysis on a chronic epidemic model with diffusion

In this paper, we propose a reaction-diffusion model of chronic infectious diseases with a special nonlinear incidence rate describing the mechanism of some chronic infections diseases. We first prove that the model has a unique nonnegative global classical solution, and admits a connected global attractor. Further, the threshold dynamics such as the uniform persistence of the model solution and the global stability of the steady-state solution are studied. The theory is verified by numerical simulation. Our results show that chronic infectious diseases have a strong ability to survive, and it is difficult to eradicate chronic infectious diseases completely.

Global stability of a tridiagonal competition model with seasonal succession.

In this paper, we investigate a tridiagonal three-species competition model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. Together with the Brouwer degree theory, sufficient conditions for existence and uniqueness of the positive periodic solution are given. We further obtain the global dynamics of coexistence and extinction for three competing species in this periodically forced environment. Finally, some numerical examples are presented to illustrate the effectiveness of our theoretical results.

Existence of solutions for impulsive wave equations

<abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.</p></abstract>

Existence of periodic solution for double-phase parabolic problems with strongly nonlinear source

The aim of this paper is to study a degenerate double-phase parabolic problem with strongly nonlinear source under Dirichlet boundary conditions, proving the existence of a non-negative periodic weak solution. Our proof is based on the Leray-Schauder topological degree, which poses many problems for this type of equations, but has been overcome by using various techniques or well-known theorems. The system considered is a possible model for problems where the studied entity has different growth coefficients, p and q in our case, in different domains.

SOME FUNCTION SPACES AND THEIR APPLICATIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented.

Modeling the adaptive behavior of an agricultural pest population

In this work, we formulate a model describing the growth of a pest population with seasonal diapause at the larval stage. The model includes the insect resistance to chemical treatments and their adaptation against a hostile environment. It consists on the description of three classes: the immature stage that includes eggs, larvae and pupae, and two mature stages corresponding to the vulnerable adult stage and the insecticide resistant adult stage. The main result consists in an analytical approach for the existence of a nonnegative periodic solution. The proof uses comparison results and Kamke's Theorem for cooperative systems. As an important illustration, a threshold type result on the global dynamics of the pest population is given in terms of an index R. When R≤1, the trivial solution is globally asymptotically stable. When R>1, the positive periodic solution is globally asymptotically stable. Numerical simulations confirm the analytical results.

A Short Note on Two Diophantine Equations 9 x – 3y = z2 and 13x – 7y = z2

In this short note, we show that the Diophantine equation 2 9 3 x y − = z has all non-negative integer solutions , , ∈ { , 2 , 0 : ∈ ℕ ∪ {0}} and the Diophantine equation 2 13 7 x y − = z have the unique non-negative integer solution ( , , ) (0,0,0) x y z = .

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

<p style='text-indent:20px;'>Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.</p>

Population dynamical behaviors of jellyfish model with random perturbation

We are interested in modeling the influence of random fluctuations on population systems. In this paper, we consider the population dynamics of jellyfish stochastic differential equation by introducing the random perturbation. Firstly, we provide the sufficient criteria to study the existence and uniqueness of non-negative solution. We prove that population will be permanence and persistence in mean when the noise is relatively small as well as it is discussed that the population may become extinct with probability one when the noise is too large. Then we show that the system has a stationary distribution under sufficient conditions using Hasminskii's methods and Lyapunov function. Finally, numerical simulations are performed to verify the results attained in this paper.

ALGORITHMS FOR THE RICCATI EQUATION WITH A NONSINGULAR M-MATRIX

We consider different algorithms with linear rate of convergence for computing the minimal nonnegative solution of M-matrix algebraic Riccati equation. The performance of the considered algorithms are illustrated on numerical examples.

Short note on the position of bifurcation points for the limiting system arising from the two competing species model

In this paper, we treat the competition-diffusion system with nonlinear diffusion term, which was proposed by Sigesada, Kawasaki and Teramoto in 1979 to model the segregation of interacting species, and discuss the bifurcation structure of nonnegative solution for the limiting system arising from the competition-diffusion system as the interspecific competition rate tends to $ +\infty $. To do this, we employ the comparison principle and the property of the Bessel function, and study the position of bifurcation points, at which a nonconstant solution bifurcates from the constant solution, for the limiting system.

Cauchy-Dirichlet problems for the porous medium equation

We consider the porous medium equation subject to zero-Dirichlet conditions on a variety of two-dimensional domains, namely strips, slender domains and sectors, allowing us to capture a number of different classes of behaviours. Our focus is on intermediate-asymptotic descriptions, derived by formal arguments and validated against numerical computations. While our emphasis is on non-negative solutions to the slow-diffusion case, we also derive a number of results for sign-change solutions and for fast diffusion. Self-similar solutions of various kinds play a central role, alongside the identification of suitable conserved quantities. The characterisation of domains exhibiting infinite-time hole closure is a particular upshot and we highlight a number of open problems.

Qualitative properties of weighted fourth order elliptic problems

Qualitative properties of nonnegative solutions of weighted fourth order elliptic problems arising from the equations on singular manifolds with conical metrics are studied. The weights may be singular in the domains. We obtain the uniqueness of positive radial solution of the Navier boundary value problem in the ball $ B $ and its exact regularity at the singular point $ x = 0 $, which determines the exact regularity of the unique positive radial solution in $ B $. We also establish the Liouville type results for the equation on $ \mathbb{R}^N \backslash \{0\} $.