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

Abstract

Listen

Translate article

Abstract In this paper, we investigate the existence of a positive solution to the third-order boundary value problem { - u ‴ ( t ) + k 2 u ′ ( t ) = φ ( t ) f ( t , u ( t ) , u ′ ( t ) ) , t > 0 u ( 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.