Kunquan Lan

MULTIPLE POSITIVE SOLUTIONS OF SEMILINEAR DIFFERENTIAL EQUATIONS WITH SINGULARITIES

The existence of positive solutions of a second order differential equation of the form (formula here) with the separated boundary conditions: α z (0) − β z ′(0) = 0 and γ z (1)+δ z ′(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z 3/2 and g = t −1/2 (see [ 12 , 13 , 24 ]), so g has a singularity at 0, was developed in studies of atomic structures (see for example, [ 24 ]) and atomic calculations [ 6 ]. The separated boundary conditions are obtained from the usual Thomas–Fermi boundary conditions by a change of variable and a normalization (see [ 22 , 24 ]). The generalized Emden–Fowler equation, where f = z p , p > 0 and g is continuous (see [ 24 , 28 ]) arises in the fields of gas dynamics, nuclear physics, chemically reacting systems [ 28 ] and in the study of multipole toroidal plasmas [ 4 ]. In most of these applications, the physical interest lies in the existence and uniqueness of positive solutions.

Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations

Positive solutions of systems of Hammerstein integral equations are studied by using the theory of the fixed-point index for compact maps defined on cones in Banach spaces. Criteria for the fixed-point index of the Hammerstein integral operators being 1 or 0 are given. These criteria are generalizations of previous results on a single Hammerstein integral operator. Some of criteria are new and involve the first eigenvalues of the corresponding systems of linear Hammerstein operators. The existence and estimates of the first eigenvalues are given. Applications are given to systems of fractional differential equations with two-point boundary conditions. The Green’s functions of the boundary value problems are derived and their useful properties are provided. As illustrations, the existence of nonzero positive solutions of two specific such boundary value problems is studied.

New fixed point theorems for a family of mappings and applications to problems on sets with convex sections

Some new fixed point theorems for a family of mappings are obtained and applied to problems on sets with convex sections that were first studied by Ky Fan.

A FIXED POINT INDEX FOR GENERALIZED INWARD MAPPINGS OF CONDENSING TYPE

A fixed point index is defined for mappings defined on a cone K which do not necessarily take their values in K but satisfy a weak type of boundary condition called generalized inward. This class strictly includes the well-known weakly inward class. New results for existence of multiple fixed points are established.

Multiple positive solutions of conjugate boundary value problems with singularities

The existence of one or several positive solutions for some nth order differential equations with conjugate boundary conditions is obtained. The approach is to employ the well-known results on the existence of positive solutions for Hammerstein integral equations obtained recently by the author. This avoids utilizing the theory of fixed point index for compact maps defined on cones directly. New properties on the kernels corresponding to the boundary value problems are provided and then employed to prove new properties of nonzero positive solutions for these boundary value problems and new inequalities on functions satisfying the conjugate boundary conditions. Our results improve and generalize many recent results.

Positive Solutions of the Falkner-Skan Equation Arising in the Boundary Layer Theory

The well-known Falkner-Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to lambda pi/2, where lambda \in {mathbb R} is a parameter involved in the equation. It is known that there exists lambda* < 0 such that the equation with suitable boundary conditions has at least one positive solution for each lambda \ge lambda* and has no positive solutions for lambda < lambda*. The known numerical result shows lambda* = -0.1988. In this paper, lambda* \in [-0.4,-0.12] is proved analytically by establishing a singular integral equation which is equivalent to the Falkner-Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner-Skan equation.

Travelling wavefronts of scalar reaction–diffusion equations with and without delays ☆

Abstract This paper deals with the existence of travelling wavefronts for scalar nonlinear reaction–diffusion equations with and without delays in one-dimensional space. New iterative techniques for a class of integral operators of Hammerstein type are established and applied to tackle the existence of travelling wavefronts in a unified way. Our results without delays only require the functions involved to be continuous and satisfy a suitable monotonicity condition. Our results with multiple delays employ the usual C 1 -assumption but generalize the well-known results.

Properties of kernels and multiple positive solutions for three-point boundary value problems

Properties of kernels for a three-point boundary value problem are studied and employed to obtain some results on the existence of multiple positive solutions for the boundary value problem. These results generalize some known results.

A fixed-point theorem and applications to problems on sets with convex sections and to Nash equilibria

A new fixed-point theorem for a family of maps defined on product spaces is obtained. The new result requires the functions involved to satisfy the local intersection properties. Previous results required the functions to have the open lower sections which are more restrictive conditions. New properties of multivalued maps are provided and applied to prove the new fixed-point theorem. Applications to problems on sets with convex sections and to the existence of Nash equilibria for a family of continuous functions are given.

Nonzero positive solutions of systems of elliptic boundary value problems

A new result on existence of nonzero positive solutions of systems of second order elliptic boundary value problems is obtained under some sublinear conditions involving the principle eigenvalues of the corresponding linear systems. Results on eigenvalue problems of such elliptic systems are derived and generalize some previous results on the eigenvalue problems of systems of Laplacian elliptic equations. Applications of our results are given to two such systems with specific nonlinearities.