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

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)=a1,x′(+∞)=b1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} x'''+\phi (t)f(t,x,x',x'')=0, \quad 0< t< +\infty , \\ x(0)=0, \qquad x'(0)=a_{1},\qquad x'(+\infty )=b_{1}, \end{cases} $$\end{document} where ϕ∈C[0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi \in C[0,+\infty )$\end{document}, f∈C([0,+∞)×(0,+∞)×R2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in C([0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}},\mathbb{R})$\end{document} may be singular at x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x=0$\end{document}, and a1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{1}$\end{document}, b1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{1}$\end{document} 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.


Introduction
In this paper, we study the existence and qualitative properties of positive solutions to the singular third-order two-point boundary value problem on the half-line of the form x, x , x ) = 0, 0 < t < +∞, where φ ∈ C[0, +∞) with φ(t) > 0 for t ∈ (0, +∞), f ∈ C([0, +∞) × (0, +∞) × R 2 , R) may be singular at x = 0, and a 1 , b 1 are positive constants with a 1 < b 1 . Third-order differential equations on an infinite interval arise from many physical phenomena, such as free convection problems in boundary layer theory, and the draining or coating fluid flow problems [7,17,19,20]. Hence the third-order boundary value problems on the infinite interval have been extensively studied. For more details on nonsingular third-order boundary value problems on the infinite interval, see, for instance, [1,3,4,8,9,13,15,18,21,22] and the references therein. For singular third-order boundary value problems on the infinite interval, we refer the reader to [2, 5-7, 10, 12, 14, 16, 19, 20] and the references therein.
In recent years, Benbaziz and Djebali [5] considered the following singular third-order multi-point boundary value problem on the half-line: ) satisfies upper and lower-homogeneity conditions in the space variables x, y and may be singular at time variable t = 0. The authors presented sufficient conditions which guarantee the existence of positive solutions to problem (1.2) by using the Krasnosel'skii fixed point theorem on cone compression and expansion of norm type. In [6], Benmezaï and Sedkaoui considered the following singular third-order two-point boundary value problem on the half-line: where κ is a positive constant, φ ∈ L 1 (0, +∞) is nonnegative and does not vanish identically on (0, +∞), the function f : R + × (0, +∞) × (0, +∞) → R + is continuous and may be singular at the space variable and its derivative. They provided sufficient conditions for the existence of a positive solution to problem (1.3) by employing the Krasnosel'skii fixed point theorem on cone compression and expansion of norm type. It is worthy to note that none of the nonlinearity f in works concerned with the singular third-order boundary value problems on the half-line we mentioned above involves the variables x . Up to now, we have not found the works that studied the fully nonlinear case of which f contains explicitly t and every derivative of x up to order two. Motivated and inspired by the above works and [2], in this paper we present sufficient conditions for the existence of positive solutions to problem (1.1) and study the qualitative properties of positive solutions. Our main tool is the Leray-Schauder nonlinear alternative and the diagonalization method together with the truncation function technique.
The rest of this paper is organized as follows. In Sect. 2, we first discuss the existence of positive solutions for singular third-order boundary value problems on the finite interval by the Leray-Schauder nonlinear alternative, and then we investigate the existence of positive solutions to problem (1.1) by using the diagonalization method together with the truncation function technique. In Sect. 3, as application, we give an example to illustrate our result.

Main results
At first, we present some lemmas, which will be useful in the proof of our main results.

Lemma 2.1 ([11]) Assume that is a relatively open subset of a convex set C in a Banach space E. Let T : → C be a compact map and p ∈ . Then either
(1) T has a fixed point in ; or (2) there are x ∈ ∂ and λ ∈ (0, 1) such that x = (1λ)p + λTx.
Take the convex subset C = E and the open set = {x ∈ C : x < M}. Let us define the operator T : → E by By the Arzelà-Ascoli theorem, we can easily prove that T is a compact operator, and Hence ω ∈ . Noticing that the solvability of problem (2.1) λ is equivalent to the solvability of the operator equation x = (1λ)ω + λTx, it follows from the assumption that Hence from Lemma 2.1, T has a fixed point on , and thus problem (2.1) 1 has at least one solution x ∈ . This completes the proof of the lemma.
We now discuss the solvability of problem (1.1) by using the diagonalization method together with the truncation function technique.
Proof We shall complete the proof in two steps.