Positive solutions of semi-positone nonlinear operator equations in Banach spaces with lattice structure

By using global bifurcation theories we obtain an existence result for positive solutions of some nonlinear operator equations. The main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions.     

Global structure of positive solution sets of nonlinear operator equations

In this paper we investigate the existence of unbounded maximal subcontinua of positive solution sets of some nonlinear operator equations that bifurcates from infinity. The main feature of the paper is that the nonlinearity may not be of asymptotically linear type and may not be positone. The methods used to show our main results are different from that of Rabinowitz’ well-known global bifurcation theorems and that of some other related papers. We first obtain a sequence of unbounded subcontinua located in a pipe that bifurcates from nontrivial solutions by using a topological degree argument. Then using a result on subcontinua of superior limit in metric spaces we obtain the main results concerning unbounded maximal subcontinua of positive solution sets of the nonlinear operator equation. The main results can be applied to the semipositone problems to obtain the existence of positive solutions.     

Structure of positive solution sets of sub-linear semi-positone operator equations

In this paper, first we obtain some results on the structure of the positive solution set of a nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of nonlinear operator equations. We use global bifurcation theories to show our main results.     

Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition

The main purpose of this paper is to present the existence results of solutions and positive solutions of nonlinear high-order fractional boundary value problems with integral boundary condition. By using the Banach fixed point theorem and the Krasnosel’skii fixed point theorem, we obtain the existence and uniqueness of real solution. By the Guo–Krasnosel’skii fixed point theorem on the cone, we obtain a desired result for guaranteeing the existence of positive solution. Several interesting examples relevant to the main results are also considered.     

非正非线性算子方程正解的存在性及其应用

首先使用全局分歧理论得到了含参数非线性算子方程解集无界连通分支存在的结果,然后根据算子的正连通性得到了一类非正非线性算子方程正解的存在结果.使用本文的主要结果在无需假设非线性项为正的条件下可以得到某些微分边值问题正解的存在结果.     

Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u ₀-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.     

Positive solutions for multipoint boundary value problem of fractional differential equations

In this paper, we study the existence of positive solutions for a multi-point boundary value problem of nonlinear fractional differential equations. By applying a monotone iterative method, some existence results of positive solutions are obtained. In addition, an example is included to demonstrate the main result.     

2m阶Dirichlet边值问题的多解(英文)

本文讨论了非线性2m阶Dirichlet边值问题多解的存在性.在非线性项满足一定条件时,通过有效地利用锥中的不动点指数理论和解的反对称延拓法,得到关于变号解的一些新的存在结果.确切地说,得到该2m阶边值问题至少存在两个正解,两个负解和多个变号解.     

Existence of positive almost automorphic solutions to nonlinear delay integral equations

This paper is concerned with the existence of positive almost automorphic solutions to some nonlinear delay integral equations. We first establish a new fixed point theorem for mixed monotone operator in a cone, and then, with its help, we obtain existence theorems of positive almost automorphic solutions. Some examples are given to illustrate our results. As one will see, even in the case of almost periodicity, our theorems extend some earlier results, and moreover, the approach dealing with the integral equation arising in an epidemic problem in this paper is also new.     

Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with Dirichlet boundary conditions. We obtain some new existence theorems of solutions for the nonlinear impulsive problem by using critical point theory. We extend and improve some recent results.     

一类非线性Schr(o)dinger方程的整体小解

本文研究带有非线性项|u|~pu的高阶非线性Schr(?)dinger方程的Cauchy问题．对于p的某一取值范围。我们证明了此问题整体解的存在唯一性,并得到了解关于初值的连续依赖性及解具有较强的衰减估计。     

分数阶撖分方程边值问题正解的存在性及唯一性

根据非线性项的不同,用两个不动点定理研究一类分数阶微分方程正解的存在性及唯一性,且其解可找到迭代序列逼近.最后列举两个例子说明其结果的应用.     

Existence of solutions for a damped nonlinear impulsive problem with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for second‐order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple Solutions for the Eigenvalue Problem of Nonlinear Fractional Differential Equations

In this article, multiple solutions for the eigenvalue problem of nonlinear fractional differential equation is considered. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Multiple Solutions of Nonlinear Impulsive Differential Equations with Dirichlet Boundary Conditions via Variational Method

In this paper, we consider the existence of multiple solutions for second-order nonlinear impulsive differential equations with Dirichlet boundary condition. We obtain some existence theorems of solutions for the nonlinear problem when the impulsive functions satisfies the superlinear growth conditions by critical point theory. We extend and improve some recent results.     

Existence and nonexistence results for a class of fractional boundary value problems

In this paper, we consider a fractional boundary value problem involving Riemann-Liouville fractional derivative and depending on a parameter. we obtain the existence and nonexistence results of positive solutions when the nonlinear term satisfies different requirements of superlinearity, sublinearity and the parameter lies in some intervals.     

On sign-changing solution for a fourth-order asymptotically linear elliptic problem

In this paper we deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.     

The existence of nonoscillatory solutions of higher order nonlinear neutral equations

In this work, we consider the existence of nonoscillatory solutions of variable coefficient higher order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations of first, second and higher order. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.     

On the Existence of Weak Solutions of Nonlinear Integral Equations in Banach Spaces

In this paper, we use the techniques of measure of weak noncompactness and Henstock–Kurzweil–Pettis integrals to discuss the existence theorem of weak solutions for a class of the nonlinear integral equations and obtain a new result, which improves and extends some relevant results for differential and integral equations in Banach spaces.