Xinan Hao

Positive solutions for singular second order differential equations with integral boundary conditions

Abstract 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 ) = ∫ 0 1 g ( s ) u ( s ) d s , u ( 1 ) = ∫ 0 1 h ( s ) u ( s ) d s , where c ( t ) is allowed to be singular at t = 0 , 1 and f ( u ) may be singular at u = 0 . The existence of positive solutions for the above problem is established by applying the fixed point index theorems under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities

Abstract 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.

Multiple positive solutions for singular nth-order nonlocal boundary value problems in Banach spaces

In this paper, we consider a class of singular nth-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.

Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

This paper investigates the second-order multipoint boundary value problem on the half-line ,, , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

Multiple monotone positive solutions for higher order differential equations with integral boundary conditions

AbstractThis paper investigates the higher order differential equations with nonlocal boundary conditions {u(n)(t)+f(t,u(t),u′(t),…,u(n−2)(t))=0,t∈(0,1),u(0)=u′(0)=⋯=u(n−3)(0)=0,u(n−2)(0)=∫01u(n−2)(s)dA(s),u(n−2)(1)=∫01u(n−2)(s)dB(s). 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.

Existence and uniqueness of positive solutions for fourth-order

This paper deals with the existence and uniqueness of positive solutions to fourth-orderm-point boundary value problems with two parameters. The arguments are based upon a 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 complement the main results of [7, 10].

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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)),0<t<1,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 continuous function and may be unbounded from below with respect to x and y. By applying the topological degree of a completely continuous field and eigenvalue, some new results on the existence of nontrivial solutions for the above boundary value problem are obtained.