Existence of positive solutions for a singular third-order two-point boundary value problem on the half-line

Abstract

Abstract In this paper, we consider the following singular third-order two-point boundary value problem on the half-line of the form { x ‴ + ϕ ( t ) f ( t , x , x ′ , x ″ ) = 0 , 0 < t < + ∞ , x ( 0 ) = 0 , x ′ ( 0 ) = a 1 , x ′ ( + ∞ ) = b 1 , $$ \textstyle\begin{cases} x'''+\phi (t)f(t,x,x',x'')=0, \quad 0< t< +\infty , \\ x(0)=0, \qquad x'(0)=a_{1},\qquad x'(+\infty )=b_{1}, \end{cases} $$ where ϕ ∈ C [ 0 , + ∞ ) $\phi \in C[0,+\infty )$ , f ∈ C ( [ 0 , + ∞ ) × ( 0 , + ∞ ) × R 2 , R ) $f\in C([0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}},\mathbb{R})$ may be singular at x = 0 $x=0$ , and a 1 $a_{1}$ , b 1 $b_{1}$ are positive constants. Using the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique, we obtain the existence and qualitative properties of positive solutions for the problem. As applications, an example is given to illustrate our result.

Keywords

• Singular
• Leray–Schauder nonlinear alternative
• Diagonalization method
• Half-line