Nontrivial Positive Solutions Research Articles

On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$ f ( z ) = f ( x ) f ( y ) = f ( u ) f ( v )

We show that the Diophantine system $$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$ has infinitely many nontrivial positive integer solutions for \(f(X)=X^2-1\), and infinitely many nontrivial rational solutions for \(f(X)=X^2+b\) with nonzero integer b.

Periodic solution for the stochastic chemostat with general response function

This paper addresses a stochastic chemostat model with periodic dilution rate and general class of response functions. The general functional response is assumed to satisfy two classifications of conditions, and these assumptions on the functional response are relative weak that are valid for many forms of growth response. For the chemostat with periodic dilution rate, we derive the sufficient criteria for the existence of the stochastic nontrivial positive periodic solution, by utilizing Khasminskii’s theory on periodic Markov process.

Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses

In this paper, stochastic non-autonomous predator–prey models with and without impulses are investigated. The effects of generalized nonlinear harvesting for prey and predator populations are considered. For the stochastic system without impulses, the existence and uniqueness of the positive solution is proven and sufficient conditions that guarantee the extinction and persistence of the population in the mean are achieved. We show the existence of a nontrivial positive periodic solution by constructing appropriate Lyapunov functions and using Khasminskii’s theory. Moreover, the global attractiveness and stochastic persistence in probability of the stochastic model are discussed. Results show that the stronger noises and nonlinear harvesting component can significantly influence the dynamics of the system and lead to the extinction of the predator population. Additionally, for the stochastic predator–prey system with impulsive effect, we prove that there exists a positive periodic solution. Numerical simulations are conducted to show the effectiveness and feasibility of the obtained results.

Stationary distribution and extinction of the DS-I-A model disease with periodic parameter function and Markovian switching

This paper introduces the DS-I-A model with periodic parameter function and Markovian switching. First, we will prove that the solution of the system is positive and global. Furthermore, we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system. Moreover, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to DS-I-A model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching. Finally, we test our theory conclusion by simulations.

Periodic solution and stationary distribution of stochastic SIR epidemic models with higher order perturbation

In this paper, we investigate two stochastic SIR epidemic models with higher order perturbation. For the nonautonomous periodic case of the model, by using Has’minskii’s theory of periodic solution, we show that the system has at least one nontrivial positive T-periodic solution. For the system disturbed by both the white noise and telephone noise, we establish sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the positive solution.

Liouville type theorem for some Choquard type equation

In this paper, we study the nonexistence result of positive solution for the following Choquard equation: $$-\Delta u=\int_{ \mathbb R^N} \frac{F(u(y))}{|x-y|^{N-\alpha}}dy f(u(x)), x\in \mathbb R^N, $$ where $N\geq 3$ is the dimension of the space. Under some assumptions on $F$ and $f$, we prove that the above equation does not possess nontrivial positive solution. We use the moving plane method in an integral form to prove this result.

Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay

In this paper, we consider two SEIR epidemic models with distributed delay in random environments. First of all, by constructing a suitable stochastic Lyapunov function, we obtain the existence of stationarity of the positive solution to the stochastic autonomous system. Then we establish sufficient conditions for extinction of the disease. Finally, by using Khasminskii's theory of periodic solutions, we prove that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T-periodic solution under a simple condition.

Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity

We consider the question of the existence of the Dirichlet problem for second-order elliptic equations with spectral parameter and a nonlinearity discontinuous with respect to the phase variable. Here it is not assumed that the differential operator is formally selfadjoint. Using the method of upper and lower solutions, we establish results on the existence of nontrivial (positive and negative) solutions under positive values of the spectral parameter for the problems under study.

Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis

We investigate the dynamics of a nonautonomous stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. By constructing suitable stochastic Lyapunov functions and using Has’minskii theory, we prove that there exists at least one nontrivial positive periodic solution of the system. Moreover, the sufficient conditions for extinction of the disease are obtained by using the theory of nonautonomous stochastic differential equations. Finally, numerical simulations are utilized to illustrate our theoretical analysis.

The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

Periodic solution for a stochastic non-autonomous competitive Lotka–Volterra model in a polluted environment

In this paper, we consider a stochastic non-autonomous competitive Lotka–Volterra model in a polluted environment. We derive sufficient criteria for the existence and global attractivity of the boundary periodic solutions. Furthermore, we obtain conditions for the existence and global attractivity of a nontrivial positive periodic solution. Finally we make simulations to illustrate our analytical results.

Periodic solution and stationary distribution of stochastic S-DI-A epidemic models

This paper considers two S-DI-A models in random environments. Firstly, using Has’minskii theory of periodic solution, we show that stochastic periodic S-DI-A model has a nontrivial positive periodic solution if . Then, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to S-DI-A model perturbed by white and telephone noises. Finally, examples are introduced to illustrate the results developed.

A Liouville type theorem for a class of anisotropic equations

AbstractIn this paper we are dealing with entire solutions of a general class of anisotropic equations. Under some appropriate conditions on the data, we show that the corresponding equations cannot have non-trivial positive solutions bounded from above.

Periodic solution for a stochastic nonautonomous SIR epidemic model with logistic growth

Abstract In this paper, we analyze the dynamics of a stochastic nonautonomous SIR epidemic model, in which population growth is subject to logistic growth in absence of disease. For the periodic system, we present sufficient conditions for persistence of the epidemic and in the case of persistence, by constructing some suitable Lyapunov functions, we show that there is at least one nontrivial positive periodic solution. One of the most important findings is that random perturbations may be beneficial to formate the periodic solution to the stochastic nonautonomous SIR epidemic model.

Nontrivial periodic solution of a stochastic non-autonomous SISV epidemic model

Abstract In this paper, we consider a stochastic non-autonomous SISV epidemic model. For the non-autonomous periodic system, firstly, we get the threshold of the system which determines whether the epidemic occurs or not. Then in the case of persistence, we show that there exists at least one nontrivial positive periodic solution of the stochastic system.

Periodic solutions and stationary distribution of mutualism models in random environments

Abstract This paper is concerned with mutualism models in random environments. For the periodic mutualism model disturbed by white noise, using Has’minskii theory of periodic solution, we show that this model admits a nontrivial positive periodic solution. Then sufficient conditions for the existence and global attractivity of the boundary periodic solutions are established. For the mutualism model disturbed by both white noise and telephone noise, sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the solution are established. Finally, examples are introduced to illustrate the results developed.

Systems of ϕ-Laplacian Three-Point Boundary-Value Problems on the Positive Half Line

We study the existence of positive solutions to boundary-value problems for two systems of two secondorder nonlinear three-point ϕ-Laplacian equations defined on the positive half line. The nonlinearities may change sign, exhibit time singularities at the origin, and depend both on the solutions and on their first derivatives. Using the fixed-point theory, we prove some results on the existence of nontrivial positive solutions on appropriate cones in some weighted Banach spaces.

The periodic solutions of a stochastic chemostat model with periodic washout rate

A stochastic chemostat model with periodic washout rate is proposed in this paper. First, sufficient conditions are established for the existence of stochastic nontrivial positive periodic solution for the system, on the basis of Khasminskii’s theory for periodic Markov processes. Furthermore, we observe that there exists a unique boundary periodic solution of the stochastic model which is globally attractive. Numerical simulations are carried out to illustrate our main conclusions.

Iterative solutions for fractional nonlocal boundary value problems involving integral conditions

This paper is concerned with the following nonlinear fractional boundary value problem: $$\left \{ \textstyle\begin{array}{l} D_{0+}^{\alpha}u(t)+f(t,u(t))=0,\quad t\in(0,1), u ( 0 ) =u' ( 0 ) =0,\qquad D_{0+}^{\beta}u(1)=\int _{0}^{1}D_{0+}^{\beta}u(t)\, dA(t), \end{array}\displaystyle \right . $$ where $2<\alpha\leq3$ , $0<\beta\leq1$ are real numbers and $\int _{0}^{1}D_{0+}^{\beta}u(t)\, dA(t)$ denotes a Riemann-Stieltjes integral. By means of monotone iterative technique and some inequalities associated with the Green function, not only the existence of nontrivial solutions or positive solutions is obtained but also iterative schemes for approximating the solutions are established, which start off with simple functions, which are feasible for computational purposes. An example is also included to illustrate the main results.

Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting

In this paper, we investigate the dynamics of the stochastic autonomous and non-autonomous predator-prey systems with nonlinear predator harvesting respectively. For the autonomous system, we first give the existence of the global positive solution. Then, in the case of persistence, we prove that there exists a unique stationary distribution and it has ergodicity by constructing a suitable Lyapunov function. The result shows that, the relatively weaker white noise will strengthen the stability of the system, but the stronger white noise will result in the extinction of one or two species. Particularly, for the non-autonomous periodic system, we show that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, numerical simulations illustrate our theoretical results.