Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

In [14], Hadamard introduced a new definition of fractional derivatives which differs from the Riemann-Liouville and Caputo fractional derivatives in the sense that its kernel integral contains the logarithmic function of an arbitrary exponent. Hadamard fractional derivatives are viewed as a generalization of the operator δ = xðd/dxÞ: For further details, properties, and generalizations of this type of derivative, we refer the reader to [5,[15][16][17][18][19][20][21] and the references therein.
In [23], Ahmad and Ntouyas studied the following problem: where H D α is the Hadamard fractional derivative order α ∈ ð1, 2 and g : ½1, e × ℝ ⟶ ℝ is a continuous function satisfying where L > 0 denotes a convenient Lipschitz constant. The authors used the classical Banach fixed point theorem to obtain the existence and uniqueness of a solution for the abovementioned problem.
In [24], the authors studied the existence of solutions for a fractional boundary value problem involving Hadamardtype fractional differential inclusions and integral boundary conditions. Their approach was based on standard fixed point theorems for multivalued maps. In [25], the authors used some classical ideas of fixed point theory to investigate the existence and uniqueness of solutions of a boundary value problem comprising nonlinear Hadamard fractional differential equations and nonlocal nonconserved boundary conditions in terms of the Hadamard integral. In [15], the authors studied a Cauchy problem for a differential equation with a left Caputo-Hadamard fractional derivative. By using Banach's fixed point theorem, they proved the existence and uniqueness of the solution in the space of continuously differentiable functions.
The primary objective of this paper is to address the existence and qualitative properties of a solution for the following problem: where H D α is the Hadamard fractional derivative of order α ∈ ð1, 2, λ ≥ 0, and f satisfies (H 1 ) f : ð1, eÞ × ½0, ∞Þ ⟶ ½0, ∞ÞÞ is continuous, such that for each each fixed r ∈ ð1, eÞ, s ⟶ f ðr, sÞ is nondecreasing on ½0, ∞Þ.
The remainder of this paper is organized as follows. In Section 2, some relevant properties of Hadamard fractional calculus are presented. Additionally, we construct Green's function and establish certain interesting inequalities. Theorem 1 is proven in Section 3. To illustrate our existence results, some examples are provided at the end of Section 3.

Preliminaries
We recall some relevant properties concerning Hadamard fractional derivative. For more details, the reader can see Section 2.7 of [19].
For γ = 0, we define H I 0 h = h: Definition 4. Let γ > 0 and ½γ its integer part. The Hadamard fractional derivative of order γ of the function h is defined as where n = ½γ + 1 and δ = rðd/drÞ.
In Figure 1, we give the representation of the Green function G 3/2 ðt, sÞ with the contours and the projections on some coordinate planes. In particular, one can see that G 3/2 ðt, sÞ is nonnegative.

Journal of Function Spaces
(iv) For each r, ξ, s ∈ ð1, eÞ, the following holds: Proof. It is easy to check that (i) holds.

From (18), we have
By symmetry, one can verify that Hence, the required results follow from (24) and (25).
(i) The property follows from Lemma 8 (ii).