Nonlinear Boundary Value Problems on Semi-Infinite Intervals using Weighted Spaces: An Upper and Lower Solution Approach

This paper presents a lower and upper solution approach for singular second order boundary value problems on the half line and establishes the existence of positive, unbounded and monotone solutions. The project is supported by the fund of National Natural Science(10571111) and the fund of Natural Science of Shandong Province(Y2005A07).     

On The Existence of a Countable Set of Positive Solutions for a Nonlocal Boundary-Value Problem with Vector-Valued Response

ABSTRACT

The existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated by some variational methods based on the idea of the Fenchel conjugate. As a consequence of a duality developed here, we obtain the existence of a countable set of solutions for our problem that are minimizers to a certain integral functional. We derive (also in the superlinear case) a measure of a duality gap between primal and dual functional for approximate solutions.     

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.     

Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.     

Nonexistence and Existence of Multiple Positive Solutions for Superlinear Three-point Boundary Value Problems via Index Theory

This paper discusses both the nonexistence of positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x, y） is superlinear in y at y = 0 and the existence of multiple positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x,y） is superlinear in x at ＋∞.     

Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

Abstract

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results.     

Inclusion problems via subsolution–supersolution method with applications to hemivariational inequalities

Abstract

Existence and location of solutions to a Dirichlet problem driven by (p, q)-Laplacian and containing a (convection) multivalued term fully depending on the solution and its gradient are established through the method of subsolution–supersolution. This result extends preceding works, in particular improving the growth condition for the lower order terms and allowing multivalued nonlinearities. A criterion for the existence of positive solutions with a priori estimates is obtained. Finally, an application to hemivariational inequalities is given.     

The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

On the substantiation of a projection method for the stationary Fokker-Planck equation

A projection-type condition is discussed that is sufficient for the stationary Fokker-Planck equation Δu - div(u f) = 0 to be solvable within a class of probability density functions. Based on existence theorems and estimates of positive solutions obtained by the author, a fairly large class of vector fields f satisfying this condition is proposed.     

Double Solutions of Impulsive Dynamic Boundary Value Problems on a Time Scale

A double fixed-point theorem is applied to obtain the existence of at least two positive solutions of a right focal boundary value problem for a second order impulsive dynamic equation on a time scale.     

Periodicity in a “Food-limited” Population Model with Toxicants and Time Delays

Abstract   With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, we study the global existence of positive periodic solutions of a “food-limited population model with toxicants and time delays. Some new results are obtained. Supported by the National Natural Sciences Foundation of China (No.10171010), the Key Project on Sciences and Technology of the Ministry of Education of China (No. Key 01061) and the Natural Sciences Foundation for Young Scholar of Northeast Normal University.     

带导数项的非齐次边值问题正解的存在性

运用Schauder不动点定理讨论了带导数项的非齐次边值问题:u"+a(t)f(t,u,u')=0,00.正解的存在性.其中:f关于u是超线性增长的.     

Existence of periodic solutions to a system with functional response on time scales

This paper investigates the existence of periodic solutions of a three-species food-chain diffusive system with Beddington-DeAngelis functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the system. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the method is unified to provide the existence of the desired solutions for continuous differential equations and discrete difference equations.     

Existence of Positive Solutions for Delay Differential Equations with Impulses

A powerful comparison result and some effective sufficient conditions for existence of positive solutions of the delay differential equation with impulses[formula]are obtained by establishing the equivalence of existence of positive solutions of (∗) and solutions of an integral equation.     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

Stochastic dynamics of an HIV/AIDS epidemic model with treatment

Abstract

We investigate a stochastic HIV/AIDS epidemic model with treatment. The model allows for two stages of infection namely the asymptomatic phase and the symptomatic phase. We prove existence of global positive solutions. We show that the solutions are stochastically ultimately bounded and stochastically permanent. We also study asymptotic behaviour of the solution to the stochastic model around the disease-free equilibrium of the underlying deterministic model. Our theoretical results are illustrated by way of numerical simulations.     

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

Abstract

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.     

Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals

The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR INFINITE DELAY FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, the existence of positive periodic solutions for infinite delay functional differential equations(FDEs for short) is discussed by utilizing a fixed point theorem on a cone in Banach space. Some results on the existence of positive periodic solutions are derived.     

次线性奇异三点边值问题的正解

该文主要研究二阶次线性奇异三点边值问题的正解的存在性,利用上下解方法和比较定理给出了C[0,1] 和 C¹[0,1]$ 正解存在的充分必要条件,其中的非线性项 f(t,x) 可以在 x=0, t=0 和 t=1 处奇异.