Nontrivial Positive Solutions Research Articles

Two b–repunits in the Fibonacci sequence

In this article, we find all integers b≥2 such that the Diophantine equation Fn=(bm−1)/(b−1) has at least two nontrivial positive integer solutions (n,m), where Fn is the nth Fibonacci number.

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearityΔmu=±uαin Rn with n⩾1, m⩾1, and α∈R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α. Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in Rn. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere.

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

Nonhomogeneous systems involving critical or subcritical nonlinearities

This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.

Cone expansion and cone compression fixed point theorems for sum of two operators and their applications

In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.

Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. The results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.

Elliptic variational problems with mixed nonlinearities

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in RN, when N ≥ 2, as Here, h(x),k(x)>0 are Lebesgue measurable functions, 1<p<q<∞, if p<N while p<r<q if p ≥ N, and λ>0 is a parameter.

Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds

In this paper, we study two Liouville theorems for nonlinear boundary value problems on extended Grushin manifolds. These problems are closely related to nonlinear fractional Grushin equations. We will prove that these problems do not possess nontrivial positive solution under some assumptions on the nonlinear terms. We use the moving plane method in an integral form to prove our results.

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay

In this paper, we investigate the existence and nonexistence of traveling wave solutions in a nonlocal dispersal epidemic model with spatio-temporal delay. It is shown that this model admits a nontrivial positive traveling wave solution when the basic reproduction number $ R_0>1 $ and the wave speed $ c\geq c^* $ ($ c^* $ is the critical speed) and this model has no traveling wave solutions when $ R_0\leq1 $ or $ c<c^* $. This indicates that $ c^* $ is the minimal wave speed.

PERIODIC SOLUTION OF A STOCHASTIC SIQR EPIDEMIC MODEL INCORPORATING MEDIA COVERAGE

In this paper, we propose a stochastic SIQR epidemic model with periodic parameters and media coverage. Firstly, we study that the stochastic non-autonomous periodic system has a unique global positive solution. Secondly, by using the Khasminskii's theory, we prove that this stochastic periodic system has a nontrivial positive periodic solution. Then, we obtain the sufficient condition for extinction of the disease. Finally, numerical simulations are employed to illustrate our theoretical analysis.

Dynamics of a stochastic glucose-insulin model with impulsive injection of insulin

In this paper, the dynamics of a stochastic glucose-insulin model with impulsive injection of insulin are investigated analytically and numerically. Firstly, we show that the system admits unique positive global solution starting from the positive initial value, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, according to the theory of Khasminskii, we show that there exists at least one nontrivial positive periodic solution. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (i) The presence of environmental noises is capable of supporting the irregular oscillation of blood glucose level, and the average level of the glucose always increases with the increase in noise intensity. (ii) The higher the volatility of the environmental noises, the more difficult the prediction of the peak size of blood glucose level.

Hesitant intuitionistic fuzzy soft b-ideals of BCK-algebras

This paper is concerned with the existence of two nontrivial positive solutions to a class of boundary value problems involving a p-Laplacian of the form where Φp(x) = |x| p−2x is a one dimensional p-Laplacian operator with p > 1, a and b are real constants, and α and β are given by the Riemann-Stieltjes integrals with A and B functions of bounded variation. The approach used is based on fixed point index theory. The results obtained in this paper are new in the literature.

Positive Solutions to a Derivative Dependent p-Laplacian Equation with Riemann-Stieltjes Integral Boundary Conditions

This paper is concerned with the existence of two nontrivial positive solutions to a class of boundary value problems involving a p-Laplacian of the form where Φp(x) = |x| p−2x is a one dimensional p-Laplacian operator with p > 1, a and b are real constants, and α and β are given by the Riemann-Stieltjes integrals with A and B functions of bounded variation. The approach used is based on fixed point index theory. The results obtained in this paper are new in the literature.

Nontrivial periodic solution of a stochastic seasonal rabies epidemic model

In this paper, we consider a stochastic nonautonomous rabies epidemic model. By using Khasminskii’s theory of periodic solutions and constructing a suitable stochastic Lyapunov function, we show that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T-periodic solution under certain condition.

Progressive dynamics of a stochastic epidemic model with logistic growth and saturated treatment

In this paper, a stochastic epidemic model with logistic growth and saturated treatment is formulated to probe the effect of white noise on population. We show that the proposed autonomous stochastic model possess a unique and non-negative solution. We obtain the sufficient conditions for extinction of the infectious disease and persistent in the mean of the stochastic autonomous epidemic model with probability one. That is, if ℛ0˜<1, under some parametric conditions then the infection goes to extinction with probability one and if ℛ0˜>1, under some parametric conditions then the infection persist with probability one. By using Has’minskii’s theory of periodic solutions, we show that the stochastic non-autonomous epidemic model has at least one nontrivial positive θ-periodic solution. Finally, the theoretical results are illustrated by numerical simulations which obtains some additional interesting phenomena.

A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions.

Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

Stationary distribution and periodic solution of stochastic chemostat models with single-species growth on two nutrients

In this paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has’minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results.

Compact Sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on RN, when N≥2,−Δpu=λK(x)ur−1−V(x)uq−1. Here, K(x),V(x)>0 are suitable potentials, 1<p<r<q<∞, and λ>0 is a parameter. To study this problem, some compact embedding results regarding MVq,p(RN)↪LKr(RN) are proved that unify and extend some recent results of the author [12–14].

Asymptotic Analysis of Impulsive Dispersal Predator-Prey Systems with Markov Switching on Finite-State Space

In this paper, we investigate the stochastic dynamics of two dispersal predator-prey systems perturbed by white noise, impulsive effect, and regime switching. For the system just interrupted by white noise, we first prove that the stochastic impulsive system has a nontrivial positive periodic solution. Then the sufficient conditions for persistence in mean and extinction of the system are obtained. For the system with Markov regime switching, we verify it is ergodic and has a stationary distribution. And conditions for extinction of the prey species are established. Finally, we provide a series of numerical simulations to illustrate the theoretical analysis.