Positive solution for singular third-order BVPs on the half line with first-order derivative dependence

In this paper, we investigate the existence of a positive solution to the third-order boundary value problem { -u‴(t)+k2u′(t)=φ(t)f(t,u(t),u′(t)), t>0u(0)=u′(0)=u′(+∞)=0, \left\{ \matrix{- u'''\left( t \right) + {k^2}u'\left( t \right) = \phi \left( t \right)f\left( {t,u\left( t \right),u'\left( t \right)} \right),\,\,\,t > 0 \hfill \cr u\left( 0 \right) = u'\left( 0 \right) = u'\left( { + \infty } \right) = 0, \hfill \cr} \right. where k is a positive constant, ϕ ∈ L¹ (0;+ ∞) is nonnegative and does vanish identically on (0;+ ∞) and the function f : ℝ⁺ × (0;+ ∞) × (0;+ ∞) → ℝ⁺ is continuous and may be singular at the space variable and at its derivative.