Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

The existence and uniqueness theorem for the generalized boundary value problem of the Thomas-Fermi equation:

$$\begin{array}{r}\{\begin{array}{l}{y}^{″}+f(x,y)=0,0<x<\mathrm{\infty },\\ y(0)=1,y(\mathrm{\infty })=0,\end{array}\end{array}$$

where

$$f(x,y)=-y{\left(\frac{y}{x}\right)}^{\frac{p}{p+1}},p>0,0<x<\mathrm{\infty },$$

is proved. Also, highly accurate approximate solutions are obtained explicitly for this new boundary value problem which arises in particular studies of many-electron systems (atoms, ions, molecules, metals, crystals). To the best of our knowledge, the results obtained here are new and provide the lower and upper bounds approximate solutions for the generalized Thomas-Fermi problem.