Walter G. Kelley

Some existence theorems for nth-order boundary value problems

A number of authors have employed variations of the Waiewski retract method (see [12]) to study second-order boundary value problems. In the area of ordinary differential equations, these include Hukuhara [6, 71, Jackson and Klaasen [8], and Kaplan, Lasota, and Yorke [9]. Bebernes and Kelley [l] have extended the results of Jackson and Klaasen to contingent equations. In this paper, we use a variation of the Waiewski method similar to those developed by Kluczny [ll] and Hukuhara [6] to prove existence theorems for some nth-order boundary value problems, where IZ > 3. Section 2 contains the basic topological results. In Section 3, we show that an existence theorem of Klaasen [lo] extends to nth-order equations under slightly weaker hypotheses. The results of Sections 2 and 3 are brought together in Section 4 to produce existence theorems for a class of boundary value problems. We conclude the paper with an example of a third-order equation from boundary-layer theory. Coppel [2] has also used a topological argument to prove existence for a particular boundary value problem associated with this equation.

Boundary and Interior Layer Phenomena for Singularly Perturbed Systems

Sufficient conditions are given for the existence of a solution to a singularly perturbed boundary value problem for a system of nonlinear equations which exhibits boundary or interior layer behavior for small positive values of the parameter. Examples are included to illustrate the results.

Singularly perturbed difference equations

Comstock and Hsiao have given a method for constructing asymptotic approximations for singularly perturbed linear difference equations with two point boundary conditions and for verifying the corre...

Analytic approximations of travelling wave solutions

A method based on differential inequalities and the maximum principle is developed to construct analytic approdimations of travelling wave solutions of certain reaction–diffusion equations arising in biology. The approximations are asymptotic in the sense that they converge to the unique travelling wave on the real line as the wave speed goes to infinity. The cases of density-independent and density-dependent diffusion coefficients are considered.

BVPs for Nonlinear Second-Order DEs

Our discussion of differential equations up to this point has focused on solutions of initial value problems. We have made good use of the fact that these problems have unique solutions under mild conditions. We have also encountered boundary value problems in a couple of chapters. These problems require the solution or its derivative to have prescribed values at two or more points. Such problems arise in a natural way when differential equations are used to model physical problems. An interesting complication arises in that BVPs may have many solutions or no solution at all. Consequently, we will present some results on the existence and uniqueness of solutions of BVPs. A useful device will be the contraction mapping theorem (Theorem 7.5), which is introduced by the definitions below. One of the goals of this chapter is to show how the contraction mapping theorem is commonly used in differential equations.

The Self-Adjoint Second-Order Differential Equation

In this chapter we are concerned with the second-order (formally) selfadjoint linear differential equation

Existence and Uniqueness Theorems

(p(t)x^\prime)^\prime + q(t)x = h(t).