Existence of positive solutions for a generalized and fractional ordered Thomas-Fermi theory of neutral atoms

The singular boundary value problem we discuss is as follows: (C)D(0+)(alpha)u(t) = lambda q(t)f (t, u(t)), 0 < t < 1, alpha(1)u(0) + alpha(2)u'(0) = a, beta(1)u(1) + beta(2)u'(1) = b, where 1 < alpha <= 2, lambda > 0 is a parameter, D-C(0+)alpha is the Caputo fractional derivative. We present the existence of positive solutions for a fractional boundary value problem modeled from the Thomas-Fermi equation subjected to Sturm-Liouville boundary conditions.