Nontrivial Positive Solutions Research Articles

Dynamics of a ratio-dependent three-species food-chain system in a random environment

We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution.

Nonexistence of nontrivial solutions to Kirchhoff-like equations

Subject to given boundary data, nonexistence of solution to the one-dimensional Kirchhoff-like equation − M ( ( a ∗ | u | q ) ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , 0 > t > 1 \begin{equation*} -M\Big (\big (a*|u|^q\big )(1)\Big )u(t)=\lambda f\big (t,u(t)\big ),\ 0>t>1 \end{equation*} is considered. In particular, a condition is provided on the parameter λ \lambda such that for each λ > λ 0 \lambda >\lambda _0 , where λ 0 \lambda _0 is defined in terms of initial data, the boundary value problem has no nontrivial positive solution.

Singular two-phase problem on a complete manifold: analysis and insights

We focus on two-phase problems with singular and superlinear parametric terms on the right-hand side. Using fibering maps and the Nehari manifold method, we prove that there are at least two non-trivial positive solutions in a geometric setting that is locally similar to Euclidean spaces but has different global properties for all except the smallest values of parameter μ>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu > 0.$$\\end{document} Singularities may appear at discrete locations in the manifold, which is a challenge for the work due to the unpredictable behavior of the solution. The findings presented here generalize some known results.

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

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 and Multiplicity of Solutions for a Bi-Non-Local Problem

The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.

Global existence and finite time blow-up for a stochastic non-local reaction-diffusion equation

In this paper, we investigate the existence and blow-up of positive solutions for a stochastic non-local reaction-diffusion equation. Firstly, we give the conditions for the existence of global positive solutions and estimate the probability of existence of non-trivial positive global solution. Secondly, by choosing some special initial data, we prove that stochastic non-local reaction-diffusion equation blows up in finite time in the point-wise sense with probability 1. The random upper bounds for blow-up times are obtained. Furthermore, by increasing the amplitude of the initial data, we can get a blow-up in any short time with a positive probability. Finally, we prove that the blow-up set is the whole region.

Existence and Uniqueness Results of Positive Solution of a Class of Singular Duffing Oscillators

In this paper, we give existence and uniqueness results of nontrivial positive solution of the singular and non-autonomous kind of Duffing oscillator by using fixed point index theory.

On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value for a general semilinear singularly perturbed Neumann boundary value problem, in this paper, the exact critical perturbation value $\varepsilon_c$ which determines the existence or nonexistence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.

REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS

In this paper, we mechanistically formulate a type of stochastic chemostat model with two complementary nutrients, which is affected by seasonal variations and flocculation effect. The phase transition properties of the model are investigated by theoretical analysis and numerical simulation. The well-posedness of the model is considered. Further, by utilizing Khasminskii’s theory, sufficient conditions for the existence of the stochastic nontrivial positive periodic solution are obtained. The existence of the stochastic nontrivial positive periodic solution implies periodic change of microorganism’s density. Some sufficient conditions for the global attractivity of the boundary periodic solution of the model are also derived. At last, numerical simulations are performed to illustrate our theoretical results. It is found numerically that the stable positive periodic solution and a stable boundary periodic solution of the model may coexist. Especially, for appropriate random perturbations, the population of the microorganisms changes from an endangered state to an oscillatory persistence state in some regions.

Existence of multiple nontrivial solutions of the nonlinear Schrödinger-Korteweg-de Vries type system

Abstract In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

Existence of global solutions of a nonautonomous semilinear equation with varying reaction

Let be a generating function, where are nonnegative measurable functions, and let be a continuous function. We prove that reaction-diffusion equations of the prototype possess nontrivial positive global solutions under suitable assumptions on and

Uniqueness of positive solutions to a Schrödinger system with linear and nonlinear couplings

In this paper, we consider the uniqueness of nontrivial positive radial solutions to the following Schrödinger system with linear and nonlinear couplings:{−△u1+V1(x)u1=μ1u13+βu1u22+κu2inRN,−△u2+V2(x)u2=μ2u23+βu12u2+κu1inRN,u1(x),u2(x)>0inRN(N≤3),u1(x),u2(x)→0 as|x|→∞. By using the a priori estimate technique, we obtain interesting uniqueness results for two cases: (1) sufficiently small β,κ>0, (2) large β>0.

Existence and Uniqueness Results for Singular Boundary Value Problem of Weighted Generalized Fisher Type Differential Equation

In this paper, we give existence and uniqueness results of nontrivial positive solution for the boundary value problem of the weighted and generalized Fisher's type differential equation.

Wave propagation in a diffusive SAIV epidemic model with time delays

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

On the Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle $120^\circ$ or $60^\circ$.

Non-blowup at critical exponent for a semilinear nonlocal diffusion equation

In this work, we investigate the positive solutions of a semilinear nonlocal diffusion equation with power nonlinearity. We have discovered a new phenomenon that the equation at the critical exponent, under a certain condition on the kernel function, admits a global in time solution for a small initial datum. This is in sharp contrast to the corresponding heat equation and the nonlocal diffusion equation with a regular or fractional Laplacian kernel, where every (nontrivial) positive solution always blows up in finite time at the critical exponent.

Degree for weakly upper semicontinuous perturbations of quasi-m-accretive operators.

In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F(x), x ∈ U, where [Formula: see text] is an m-accretive operator in a Banach space E, [Formula: see text] is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E. Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Positive solutions to a semipositone superlinear elastic beam equation

A semipositone fourth-order two-point boundary value problem is considered. In mechanics, the problem describes the deflection of an elastic beam rigidly fastened on the left and simply supported on the right. Under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, the existence of nontrivial solutions and positive solutions to this boundary value problem is obtained. The main results are obtained by using the topological method and the fixed point theory of superlinear operators.

Analysis of Periodic Solution of DNA Catalytic Reaction Model With Random Disturbance

The realization of molecular logic circuit is inseparable from the design and analysis of catalytic reaction chain, and the DNA catalytic gate plays an important role in it. Discuss the nature of the solution to DNA catalytic reaction system, using Khasminskii's periodicity and Lyapunov analysis methods to obtain the existence of non-trivial positive periodic solutions of the system, and the solution is globally attractive. The existence of the solution indicates that according to the mathematical model established by the DNA catalytic reaction system, the system may reach the expected concentration value of an ideal state and obtain better reaction data, which provides a theoretical basis for the realization of the DNA catalytic gate function. Numerical simulation results show that under the influence of random disturbance and periodic parameters, the solution to the random DNA catalytic reaction system exists and is globally attractive, which also reflects that the DNA catalytic reaction system can reach an ideal reaction state. The solution to the DNA catalytic system with random disturbance will converge on a certain value and oscillate periodically between the solution to the deterministic system.