Existence results for boundary value problems for ordinary differential equations and difference equations

Abstract

This thesis investigates existence results for boundary value problems for systems of second-order ordinary differential equations with impulses, for the vector ordinary p-Laplacian and also for the finite difference equations arising from the vector ordinary p-Laplacian. The thesis begins with systems of second-order ordinary differential equations x''=f(t,x,x') subject to nonlinear boundary conditions and nonlinear impulses. Many papers on boundary value problems involving this differential equation have been published, but some made strong assumptions on f or presented complicated proofs. Additionally, very few authors have explored the theory of impulsive boundary value problems for such equations, although these problems have a large number of applications in the modelling of physical problems such as mechanical systems with impact. In this thesis, we extend the notion of admissible bounding set O which is a subset of [0,1] x {R}d, of the Hartman-Nagumo conditions and of compatibility to show the problem has a solution xnwith (t,x(t)) enO. To establish the existence, we reformulate the problem as an equivalent nonlinear equation and homotope it to a new nonlinear equation. This equation is equivalent to a simple linear system of ordinary differential equations subject to Dirichlet boundary conditions and impulses. We show that there exists a unique solution to this problem. Then, by the Leray Index Theorem and homotopy invariance, we show that the degree of the nonlinear equation is non-zero, and hence establish existence results for the original problem. Our proof is simpler and requires weaker assumptions on f than those of earlier works closely related to this problem. More importantly, a key technique used in our proof offers a fresh starting point for future research to establish the most natural and general existence results for boundary value problems. Next, we study the Dirichlet problem for the vector ordinary p-Laplacian (║x'║p-2x')] = f(t,x,x').We establish a general result for a broader class of fnthan earlier works about this problem by imposing fewer assumptions on f. We extend the concept of admissible bounding set and introduce the notion of a p-admissible bounding set O to guarantee a priori bounds on potential solutions. Moreover, we introduce the p-Mawhin-Urena-Nagumo condition and the p-Hartman-Nagumo condition to obtain a priori bounds on the derivatives of solutions. We apply our first result to establish the existence results. For p, we turn the vector ordinary p-Laplacian into an equivalent system of second-order ordinary differential equations to prove existence. For pnwe approximate the p-Laplacian and turn the approximation into a system of second-order ordinary differential equations. Then using our first result, we show solutions to the approximation problem exist and converge to solutions to the original problem. Finally, we investigate boundary value problems for systems of finite difference equations arising from discretizing the vector ordinary p-Laplacian. Many papers have presented important results for discrete p-Laplacian problems that do not arise from discretization of the vector ordinary p-Laplacian. However, there appears to have been little research done on the discrete problems arising from the discretization of boundary value problems for the vector ordinary p-Laplacian. This thesis fills this gap in the literature. We establish a more general result than previous works do by imposing weaker assumptions on fnand by removing a strong assumption. Existence and convergence theorems follow, once the assumptions of our second result and the compatibility of boundary conditions are introduced.