Existence Of Positive Solutions Research Articles

Coexistence states in a cross-diffusion system of a competition model

The main goal of this paper is to study the stationary problem for a Lotka–Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by Kuto and Tsujikawa (2015) and Wang and Yan (2015), where the corresponding Neumann problem is analyzed.

A new fixed point approach for solutions of a $ p $-Laplacian fractional $ q $-difference boundary value problem with an integral boundary condition

<p>We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.</p>

An Erdélyi-Kober fractional coupled system: Existence of positive solutions

<abstract><p>This paper studies an Erdélyi-Kober fractional coupled system where the variable is in an infinite interval, and the existence of positive solutions is considered. We first give proper conditions and then use the Guo-Krasnosel'skii fixed point theorem to discuss our problem in a special Banach space. The monotone iterative technique and the existence results of positive solutions for this system are established naturally. To show the plausibility of our main results, several concrete examples are given at the end.</p></abstract>

The Stationary Distribution and Extinction of a Stochastic Five‐Dimensional COVID‐19 Model

This paper investigates a stochastic SEIAIR epidemic model with nonlinear incidence and contacting distance to describe the transmission dynamics of COVID‐19. Firstly, we show the global existence of positive solution of the system. Then, using the Lyapunov function method and theory of stochastic analysis, we set out the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution to the stochastic model. Furthermore, we obtain the sufficient condition for extinction of the disease. Finally, we go through some numerical simulations to demonstrate the theoretical results. Our study extends and improves the results of previous studies.

Existence of positive solutions of elliptic equations with Hardy term

This paper is devoted to studying the existence of positive solutions of the problem: ( ∗ ) { − Δ u = u p | x | a + h ( x , u , ∇ u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is an open bounded smooth domain with boundary ∂ Ω , and 1 < p < N − a N − 2 , 0 < a < 2 . Under suitable conditions of h ( x , u , ∇ u ) , we get a priori estimates for the positive solutions of problem ( ∗ ) . By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem ( ∗ ) when 1 < p < N − a N − 2 .

Positive solutions for a system of fractional $ q $-difference equations with generalized $ p $-Laplacian operators

<abstract><p>In this paper, we consider the existence of positive solutions for a system of fractional $ q $-difference equations with generalized $ p $-Laplacian operators. By using Guo-Krasnosel'skii fixed point theorem, we obtain some existence results of positive solutions for this system with two parameters under some different combinations of superlinearity and sublinearity of the nonlinear terms. In the end, we give two examples to illustrate our main results.</p></abstract>

Existence of Positive Solutions for a Fourth-Order Differential Equation with p-Laplacian and Riemann-Stieltjes Integral Boundary Conditions

Existence of Positive Solutions for a Fourth-Order Differential Equation with p-Laplacian and Riemann-Stieltjes Integral Boundary Conditions

Existence of solutions for asymptotically periodic quasilinear Schrödinger equations with local nonlinearities

This paper is concerned with the existence of positive solutions for asymptotically periodic quasilinear Schrödinger equations. By using a Nehari-type constraint and Moser iteration, we get the existence results which is a complement to the ones in Chu and Liu [Nonlinear Anal. Real World Appl. 44(2018), 118–127]. Moreover, we consider a new reformative asymptotic processes of the potential function and the nonlinearity term is only locally defined.

Existence of positive solutions for generalized quasilinear Schrödinger equations with Sobolev critical growth

In this paper, we are devoted to establishing that the existence of positive solutions for a class of generalized quasilinear elliptic equations in R N with Sobolev critical growth, which have appeared from plasma physics, as well as high-power ultrashort laser in matter. To begin, by changing the variable, quasilinear equations are transformed into semilinear equations. The positive solutions to semilinear equations are then presented using the Mountain Pass Theorem for locally Lipschitz functionals and the Concentration-Compactness Principle. Finally, an inverse translation reveals the presence of positive solutions to the original quasilinear equations.

Interval of the existence of positive solutions for a boundary value problem for system of three second-order differential equations

The aim of this paper is to estimate an interval of the existence of positive solutions for a boundary value problem for the system of three nonlinear second-order ordinary differential equations. Krasnosel'skiĭ–Precup fixed point theorem is used to determine this interval theoretically. For Dirichlet boundary conditions theoretical result is compared with result obtained numerically.

Existence of positive solutions for weakly coupled Schrödinger system with supercritical growth

Existence of positive solutions for weakly coupled Schrödinger system with supercritical growth

Positive Solutions for a Fractional Boundary Value Problem with Lidstone Like Boundary Conditions

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L1-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

EXISTENCE OF POSITIVE SOLUTIONS OF NONLINEAR STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

The authors study the existence of positive solutions of nonlinear Sturm- Liouville boundary value problem: (p(t)y’ (t))’+ λf(t, y(t)) = 0, 0 < t < 1 with mixed boundary conditions ay(0) − bp(0)y'(0) = 0, cy(1) + dp(1)y'(1) = 0, for λ > 0 on a suitable interval. The methods involve the applications of a fixed point theorem for operators on a cone in a Banach space. An attempt has been made to establish the existence of a positive solutions of the above boundary value problem under certain sufficient conditions on f(t, y). Examples are given at the end of the paper to justify our results.

Positive Solutions of the Discrete Fractional Oscillation Equation

This article establishes sufficient conditions for the existence of positive solutions to initial value problems associated with the discrete fractional oscillation equation using the well-known conical shell fixed point theorem. Two numerical examples illustrate the applicability of the established results.

Positive solutions to the planar logarithmic Choquard equation with exponential nonlinearity

In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

Periodic dynamics of predator-prey system with Beddington–DeAngelis functional response and discontinuous harvesting

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

More Effective Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of a Class of Third-Order Functional Differential Equations

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.

Existence Results for Systems of Nonlinear Second-Order and Impulsive Differential Equations with Periodic Boundary

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, in the general case, some Banach space. In this work, sufficient conditions for the existence of solutions for a system with parameters are obtained. The results are obtained by using fixed point theorems for operators on a cone. Our approach is based on Schaefer’s fixed point theorem more precisely. In addition, the existence of positive solutions is also investigated.

Dynamic analysis and optimal control of a stochastic investor sentiment contagion model considering sentiments isolation with random parametric perturbations

Investor sentiment contagion has a profound influence on economic and social development. This paper explores the diverse influences of various investor sentiments in modern society on the economy and society. It also investigates the interference of various uncertain factors on investor sentiments in the modern economy and society. On this basis, the dual-system stochastic SPA2G2R model was constructed, incorporating positive and negative sentiments, as well as a supervision and isolation mechanism. The global existence of positive solutions was established, and sufficient conditions for the disappearance and steady distribution of investor sentiment were calculated. An optimal control strategy for the stochastic model was put forward, with numerical simulation supporting the theoretical analysis results. A comparison with parameter changes in the deterministic model was also conducted. The research reveals a competitive relationship between different investor sentiments. Enhancing societal guidance mechanisms promotes positive investor sentiment contagion. Timely control by the supervisory department effectively curbs the spread of investor sentiment. Additionally, white noise promotes investor sentiment contagion, suggesting effective regulation through control of noise intensity and disturbance parameters.

Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearities

We consider the following quasi-linear Schrödinger equation−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju+V(x)u=f(u),x∈RN,N≥3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on aij and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f(u)=|u|p−2u, the case p∈(2,4) is also considered.