Abstract
The purpose of this article is to construct a firm mathematical foundation for the boundary value problem associated with a generalized Emden equation that embraces Thomas-Fermi-like theories. Boundary value problems for the relativistic and non-relativistic Thomas-Fermi equations are included as special cases. Questions of existence and uniqueness of solutions to these boundary value problems form a fundamental and important area of investigation regarding whether these mathematical models for physical phenomena are actually well-posed. However, these questions have remained open for the generalized Emden problem and the relativistic Thomas-Fermi problem. Herein we advance current understanding of existence and uniqueness of solutions by proving that these boundary value problems each admit a unique solution. Our methods involve an analysis of the problems through arguments that apply differential inequalities and fixed-point theory. The new results guarantee the existence of a unique solution, ensuring the generalized Emden equation that embraces Thomas-Fermi theory sits on a firm mathematical foundation.