Existence and uniqueness of solutions of boundary value problems by matching solutions.

Abstract

In this dissertation, we investigate the existence and uniqueness of boundary value problems for the third and nth order differential equations by matching solutions. Essentially, we consider the interval [a, c] of a BVP as the union of the two intervals [a, b] and [b, c], analyze the solutions of the BVP on each, and then match the proper ones to be the unique solution on the whole domain. In the process of matching solutions, boundary value problems with different boundaries, especially at the matching point b, would be quite different for the requirements of conditions on the nonlinear term. We denote the missing derivatives in the boundary conditions at the matching point b by k₁ and k₂. We show how y(ᵏ²)(b) varies with respect to y(ᵏ¹)(b), where y is a solution of the BVP on [a, b] or [b, c]. Under certain conditions on the nonlinear term, we can get a monotone relation between y(ᵏ²)(b) and y(ᵏ¹)(b), on [a, b] and [b, c], respectively. If the monotone relations are different on [a, b] and [b, c], then we can finally get a unique value for y(ᵏ¹)(b) where the k₂nd derivative of two solutions on [a, b] and [b, c] are equal and we can join the two solutions together to obtain the unique solution of our original BVP. If the relations are the same, then we will arrive at the situation that the k₂nd order derivatives of two solutions at b on [a, b] and [b, c] are decreasing with respect to the k₁st derivatives at b at different rates, and by analyzing the relations more in detail, we can finally get a unique value for the k₁st derivative of solutions of BVP's on [a, b] and [b, c], which are matched to be a unique solution of the BVP on [a, c]. In our arguments, we use the Mean Value Theorem and the Rolle's Theorem many times. As the simplest models, third order BVP's are considered first. Then, in the following chapters, nth order problems are studied. Lastly, we provide an example and some ideas for our future work.