Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line

and Applied Analysis 3 Definition 7. The Riemann-Liouville fractional integral of order β > 0 of a function h : (0,∞) → R is given by I β h (t) = 1 Γ (β) ∫ t 0 (t − s) β−1 h (s) ds, t > 0, (15) provided that the right-hand side is pointwise defined on (0,∞). Definition 8. The Riemann-Liouville fractional derivative of order β > 0 of a function h : (0,∞) → R is given by D β h (t) = 1 Γ (n − β) ( d dt ) n


Introduction
Fractional differential equations arise in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic.They also serve as an excellent tool for the description of hereditary properties of various materials and processes (see [1][2][3]).In consequence, the subject of fractional differential equations has been gaining much importance and attention.Most of the related results focused on developing the global existence and uniqueness of solutions on finite intervals (see [4][5][6][7][8][9][10][11][12]) and the references therein).However, to the best of our knowledge, there exist few articles dealing with the existence of solutions to fractional differential equations on the half-line; see, for instance, [13][14][15][16][17][18][19][20][21].In [17], by using the recent Leggett-Williams norm-type theorem due to O'Regan and Zima, the author established the existence of positive solutions for fractional boundary value problems of resonance on infinite intervals.On the other hand, in [20], Su and Zhang studied the following fractional differential problem on the half-line by using Schauder's fixed point theorem: () =  (, ,  −1 ) ,  ∈ (0, ∞) , 1 <  ≤ 2, where   is the standard Riemann-Liouville fractional derivative (see Definition 8 below).
In [21], by means of the Leray-Schauder alternative theorem, Zhao and Ge proved the existence of solutions to the following boundary value problem: where  ∈ R and 0 <  < ∞.
In this paper, we aim at studying the existence, uniqueness, and the exact asymptotic behavior of a positive solution to the following fractional boundary value problem: where 1 <  < 2,  ∈ (−1, 1), and  is a nonnegative continuous function on (0, ∞) that may be singular at 0. To state our result, we need some notations.We first introduce the following Karamata classes.
Definition 2. The class K ∞ is the set of all Karamata functions  defined on [1, ∞) by where  > 0 and It is easy to verify the following.
Remark 4 (see [22]).Let  be a function in K ∞ , and then there exists  ≥ 0 such that for every  > 0 and  ≥ 1 we have As a typical example of function belonging to the class K, we quote where   are real numbers, log   = log ∘ log ∘ ⋅ ⋅ ⋅ log  ( times), and  is a sufficiently large positive real number such that  is defined and positive on (0, ], for some  > 1. In the sequel, we denote by  + ((0, ∞)) the set of nonnegative Borel measurable functions in (0, ∞) and by  2− ([0, ∞)) the set of all functions  such that  →  2− () is continuous on [0, ∞).
Finally, for  ∈ R, we put  + = max(, 0).Throughout this paper we assume that the function  is nonnegative on (0, ∞) and satisfies the following condition: (H)  ∈ ((0, ∞)) such that where In what follows, we put and we define the function  on (0, ∞) by where, for  ∈ (0, ), and, for  ≥ 1, Our main result is the following.
The content of this paper is organized as follows.In Section 2, we present some properties of the Green function   (, ) of the operator  → −   on (0, ∞) with Dirichlet conditions lim  → 0  2− () = 0 and lim  → ∞  1− () = 0. Next, we give some fundamental properties of the two Karamata classes K and K ∞ and we establish sharp estimates on some potential functions.In Section 3, exploiting the results of the previous section and using the Schauder fixed point theorem, we prove Theorem 5.

Preliminaries
2.1.Fractional Calculus and Green Function.For the convenience of the reader, we recall in this section some basic definitions on fractional calculus (see [2,24,25]) and we give some properties of the Green function   (, ).Definition 7. The Riemann-Liouville fractional integral of order  > 0 of a function ℎ : (0, ∞) → R is given by provided that the right-hand side is pointwise defined on (0, ∞).
Definition 8.The Riemann-Liouville fractional derivative of order  > 0 of a function ℎ : (0, ∞) → R is given by where  = [] + 1 provided that the right-hand side is pointwise defined on (0, ∞).Here [] means the integer part of the number .
So we have the following properties (see [2]).
Proof.From Corollary 14, the function    is well defined in (0, ∞).Using Proposition 13 and Lemma 12 (ii), we get This implies that  2− (  ||) is finite on (0, ∞).So by using Fubini's theorem, we obtain Observe that by considering the substitution  =  + ( − ), we obtain Using this fact and ( 22) we deduce that Now, assume that  ≤ , and then by (36) we have On the other hand, if  ≤  and  ∈ (0, ), we have So combining (38) and (39), we obtain This implies that and Moreover, using Proposition 13 and the dominated convergence theorem, we deduce that lim  → 0 Finally, we need to prove the uniqueness.Let , V ∈  2− ([0, ∞)) be two solutions of (33) and put  =  − V.

Sharp Estimates on the Potential of Some Karamata
Functions.We collect in this paragraph some properties of functions belonging to the Karamata class K (resp., K ∞ ) and we give estimates on some potential functions.

Lemma 19 .
Let  be a function in K ∞ .Then one has lim