Positive solutions for second order boundary value problems with derivative terms

This paper weals with the existence of positive solutions of the fully second‐order boundary value problem where is continuous. Under the conditions that the nonlinearity may be superlinear or sublinear growth on x and y, the existence results of positive solutions are obtained. For the superlinear case, a Nagumo‐type condition is presented to restrict the growth of f on y. The superlinear and sublinear growth of the nonlinearity f are described by inequality conditions instead of the usual upper and lower limits conditions. Our discussion is based on the fixed point index theory in cones.     

Existence of solutions for three-point boundary value problems for second order equations

Shooting methods are employed to obtain solutions of the three-point boundary value problem for the second order equation, where is continuous, and and conditions are imposed implying that solutions of such problems are unique, when they exist.

     

Extremal solutions of second order nonlinear periodic boundary value problems

We study a periodic boundary value problem for a nonlinear ordinary differential equation of second order when the nonlinearity is given by a Carathéodory function. We generalize the monotone iterative method to cover the fully nonlinear case.     

Existence of solutions of boundary value problems for second order functional differential equations

In this paper, we deal with the boundary value problem

     

Solving a nonlinear system of second order boundary value problems

In this paper, a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems. The analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Monotone method for a system of second order periodic boundary value problems

The existence of coupled maximal and minimal quasi solutions of a system of second order periodic boundary value problems is discussed when the right hand side possesses a mixed monotone property.     

Positive solutions of even order system periodic boundary value problems

We study a class of even order system boundary value problems with periodic boundary conditions. A series of criteria are obtained for the existence of one, two, any arbitrary number, and even a countably infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also derived. As for the second order case, our results extend, improve, and supplement those in the literature for scalar and system boundary value problems. Several examples are given to demonstrate the applications. Moreover, we obtain conditions for system periodic boundary value problems of a different form to have nontrivial solutions by transforming our main results to such problems.     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales

This paper investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solution is established by using monotone iterative technique     

Iterative procedures for nonlinear second order boundary value problems

Summary Questions of the existence of positive solutions of second order nonlinear boundary value problems with separated boundary conditions are investigated. The nonlinearities are such that the linearization about the trivial solution does not exist or is trivial. The methods are thus applicable when shooting methods or ordinary bifurcation techniques cannot be applied. The conditions on the nonlinearity are quite modest and both super and sublinear problems can be included.This research was performed while the author was visiting at Emory University.Research supported by AFOSR 87-0140.     

A monotone method for constructing extremal solutions of second order scalar boundary value problems

A monotone iterative method is given for the construction of extremal solutions to second order scalar boundary value problems, starting from a pair of upper and lower solutions. The fully nonlinear case is considered.     

Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Positive solutions for boundary value problems of second order difference equations and their computation

We consider the following two classes of second order boundary value problems for difference equation: