Xinan Hao

• PhD
• Professor at Qufu Normal University

In this paper, we study the positive solutions for nonlocal differential equations with concave and convex coefficients: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$...

In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed [Formula: see text] and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds [Formula: see text] and a...

In this paper, we consider the following sublinear biharmonic equations\begin{equation*} \Delta^2 u + V\left( x \right)u =K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*}where $N\geq5,~0<p<1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end...

In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refin...

Abstract In this paper, we consider the following sublinear fractional Schrödinger equation: ( − Δ ) s u + V ( x ) u = K ( x ) | u | p − 1 u , x ∈ R N , $$ (-\Delta)^{s}u + V(x)u= K(x) \vert u \vert ^{p-1}u,\quad x\in \mathbb{R}^{N}, $$ where s , p ∈ ( 0 , 1 ) $s, p\in(0,1)$ , N > 2 s $N>2s$ , ( − Δ ) s $(-\Delta)^{s}$ is a fractional Laplacian ope...

Abstract In this paper we use the fixed point index to study the existence of positive solutions for a system of 2nth-order boundary value problems involving semipositone nonlinearities.

In this paper, by using the cosine family theory, measure of non-compactness and the generalization of Darbo-fixed point theorem, we establish the existence of mild solutions of nonlinear second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces. The result obtained herein generalizes and improves some known...

Abstract In this paper, we study the following Kirchhoff–Schrödinger–Poisson systems: {−(a+b∫R3|∇u|2dx)Δu+V(x)u+ϕu=f(u),x∈R3,−Δϕ=u2,x∈R3, $$\textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=f(u), &x \in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}, &x\in \mathbb{R}^{3}, \end{cases} $$ where a, b are p...

Abstract This paper is concerned with the existence of a sign-changing solution to a class of quasilinear Schrödinger–Poisson systems. There are some technical difficulties in applying variational methods directly to the problem because the quasilinear term makes it impossible to find a suitable space in which the corresponding functional possesses...

In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refin...

We investigate a class of semipositone fractional integral boundary value problem on the half-line. Using the fixed point index theorems in a cone, we obtain the existence result of positive solutions.

Abstract In this paper, by employing fixed point theory, we investigate the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space.

In this paper, we study a higher order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo-Krasnoselskii fixed point theorem on...

We study the existence, multiplicity, and uniqueness results of positive solutions for a fractional thermostat model. Our approach depends on the fixed point index theory, iterative method, and nonsymmetry property of the Green function. The properties of positive solutions depending on a parameter are also discussed.

We investigate a singular fractional differential equation with an infinite-point fractional boundary condition, where the nonlinearity f(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69...

In this paper, we investigate the impulsive fractional q -difference equation with antiperiodic conditions. The existence and uniqueness results of solutions are established via the theorem of nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle. Two examples are given to illustrate our results.

In this paper, we study a integral boundary value problem of fractional differential equation with the nonlinearity depending on fractional derivatives of lower order on an infinite interval. We establish a proper compactness criterion in a special function space. By using the Schauder fixed point theorem and Banach contraction mapping principle, w...

In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces are established under more general conditions by constructing a special Banach space and using cone theory and the Banach contraction mapping principle. The results obtained herein improve and genera...

In this paper, the existence of positive solutions for systems of semipositone singular fractional differential equations with a parameter and integral boundary conditions is investigated. By using fixed point theorem in cone, sufficient conditions which guarantee the existence of positive solutions are obtained. An example is given to illustrate t...

In this paper, we consider a fractional singular three-point boundary value problem with p-Laplacian operator. The nonlinearity f(t,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg...

In this paper we investigate a system of impulsive integral boundary value problems with sign-changing nonlinearities. Using the fixed point theorem in double cones, we prove the existence of multiple positive solutions.

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived i...

In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given t...

Through solving the problem step by step and by applying the method of a C0 semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of...

This paper is concerned with the existence of mild solutions for impulsive semilinear neutral functional integro-differential equations in Banach spaces. The existence result is obtained by using fractional power of operators, Mönch fixed point theorem, the piecewise estimation method and semigroup theory. Applications to partial differential syste...

Through solving equations step by step and by using the generalized Banach fixed point theorem, under simple conditions, the authors present the existence and uniqueness theorem of the iterative solution for nonlinear advection-reaction-diffusion equations with impulsive effects. An explicit iterative scheme for the solution is also derived. The re...

In this paper, we study the existence of positive solutions to the nonlinear fractional order singular andsemipositone nonlocal boundary value problem (Formula Presented.) by using the Leray-Schauder nonlinear alternative and a fixed-point theorem on cones, where (Formula Presented.) is the standard Riemann-Liouville derivative, and f(t, u) is semi...

By using the fixed point theorem for the mixed monotone operator, the existence of unique positive solutions for singular nonlocal boundary value problems of fractional differential equations is established. An example is provided to illustrate the main results.

By using the cone theory and the Banach contraction mapping principle, we study the existence and uniqueness of an iterative solution to the singular nth-order nonlocal boundary value problems.

This paper investigates the higher order differential equations with nonlocal boundary conditions The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone. MSC: 34B10, 34B18.

In this paper, we consider the singular (k,n−k)(k,n−k) conjugate boundary value problems with dependence on the derivatives. The existence and uniqueness results of iterative solutions are obtained by using the cone theory and the Banach contraction mapping principle.

We are concerned with a class of second order impulsive differential equations with integral boundary conditions. Under different combinations of superlineary and sublinearity of nonlinear term and the impulses, various existence, multiplicity, and nonexistence results for positive solutions are derived in dependence of the parameters. The results...

This paper deals with the existence and uniqueness of positive solutions to fourth-order m-point boundary value problems with two parameters. The arguments are based upona specially constructed cone and a fixed point theorem in a cone for a completely continuous operator, due to Krasnoselskii and Zabreiko. The results obtained herein generalize and...

In this paper, we study the existence of positive solutions for the singular second order integral boundary value problem {u″(t)+a(t)u′(t)+b(t)u(t)+c(t)f(u)=0,t∈(0,1),u(0)=∫01g(s)u(s)ds,u(1)=∫01h(s)u(s)ds, where c(t)c(t) is allowed to be singular at t=0,1t=0,1 and f(u)f(u) may be singular at u=0u=0. The existence of positive solutions for the above...

By using the upper-lower solutions method and the fixed-point theorem on cone in a special space, we study the singular boundary value problem for systems of nonlinear second-order differential equations involving two parameters on the half-line. Some results for the existence, nonexistence and multiplicity of positive solutions for the problem are...

The class of second-order differential systems with boundary conditions u '' (t)+f 1 (t,u(t),v(t))=0,t∈(0,1),v '' (t)+f 2 (t,u(t),v(t))=0,t∈(0,1), u ' (0)=v ' (0)=0,u(1)=αu(η),v(1)=αv(η) is considered under sonic conditions concerning the first eigenvelue of the relevant linear problem. By constructing a cone K 1 ×K 2 which is the Cartesian product...

We study the existence of monotone positive solutions for the semipositone right focal boundary value problems [Formula is presented], where λ > 0 is a parameter, n ≥ 3, 1 < k ≤ n-1 is fixed, f may change sign for 0 < t < 1 and we allow f is both semipositone and lower unbounded. Without making any monotone type assumption, the existence results of...

In this paper, we consider a class of singular nnth-order nonlocal boundary value problems in Banach spaces. The existence of multiple positive solutions for the problem is obtained by using the fixed point index theory of strict set contraction operators. To demonstrate the applications of our results, two examples are also given in the paper.

This paper concerns the existence of nontrivial solutions for the following singular boundary value problem with a sign-changing nonlinear term:u(4)(t)=h(t)f(t,u(t),u″(t)),0t1,α1u(0)-β1u′(0)=δ1u(1)+γ1u′(1)=0,α2u″(0)-β2u‴(0)=δ2u″(1)+γ2u‴(1)=0,where h(t) is allowed to be singular at t=0 and/or t=1. Moreover, f(t,x,y):[0,1]×R2→R is a sign-changing con...

This paper is concerned with the existence, nonexistence and multiplicity of positive solutions for the following second order m-point nonhomogeneous singular boundary value problem where may be singular at t=0 and/or t=1. We show that, under suitable conditions, there exists a positive number b∗ such that the above problem has at least two positiv...

The nonlinear nth-order singular nonlocal boundary value problem {u((n))(t) + lambda a(t)f(t, u(t)) = 0, t is an element of (0, 1), u(0) = u'(0) = ... = u((n-2))(0) = 0, u(1) = integral(1)(0) u(s)dA(s) is considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, where integral(1)(0) u(s)dA(s) i...

This paper studies the existence of positive solutions for periodic boundary value problems. The criteria for the existence, nonexistence and multiplicity of positive solutions are established by using the Global continuation theorem, fixed point index theory and approximate method. The results obtained herein generalize and complement some previou...

The authors investigate the second-order multipoint boundary value problem on the half-line u '' (t)+f(t,u(t),u ' (t))=0,t∈ℝ + , αu(0)-βu ' (0)-∑ i=1 n k i u(ξ i )=a≥0, lim t→+∞ u ' (t)=b>0, where α>0, β>0, k i ≥0, 0≤ξ i <∞ (i=1,2,⋯,n), and f:ℝ + ×ℝ×ℝ→ℝ is continuous. We establish sufficient conditions to guarantee the existence of an unbounded sol...

We study the existence and multiplicity of positive solutions for a class of nth-order singular nonlocal boundary value problemsu(n)(t)+a(t)f(t,u)=0, t∈(0,1), u(0)=0, u'(0)=0, …,u(n−2)(0)=0, αu(η)=u(1), where 0<η<1,  0<αηn−1 <1. The singularity may appear at t=0 and/or t=1. The Krasnosel'skii-Guo th...