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.

中文翻译：

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建立涉及广义 Emden 方程的边值问题解的存在性和唯一性，采用类似 Thomas-Fermi 的理论

本文的目的是为与包含类托马斯费米理论的广义 Emden 方程相关的边值问题构建坚实的数学基础。相对论和非相对论 Thomas-Fermi 方程的边界值问题作为特殊情况包括在内。这些边值问题的解的存在性和唯一性问题构成了关于这些物理现象的数学模型实际上是否适定的基本且重要的研究领域。然而，这些问题对于广义 Emden 问题和相对论 Thomas-Fermi 问题仍然是开放的。在这里，我们通过证明这些边值问题都承认唯一的解来推进当前对解的存在性和唯一性的理解。我们的方法涉及通过应用微分不等式和不动点理论的论证来分析问题。新结果保证了唯一解的存在，确保包含 Thomas-Fermi 理论的广义 Emden 方程建立在坚实的数学基础上。