Abstract
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.
1. Introduction
Fractional-order differential equations appear extensively in a variety of applications in science and engineering; see, for instance, [1–13] and the references therein.
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 For further details, properties, and generalizations of this type of derivative, we refer the reader to [5, 15–21] and the references therein.
The study of existence, uniqueness, and global asymptotic behavior of a continuous solution of fractional differential equations involving Hadamard fractional derivatives has been investigated by several researchers; see, for example, [22–32].
In [23], Ahmad and Ntouyas studied the following problem: where is the Hadamard fractional derivative order and is a continuous function satisfying where 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 Hadamard-type 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 is the Hadamard fractional derivative of order , and satisfies
(H1) is continuous, such that for each each fixed is nondecreasing on .
(H2) For all
The following are some examples of functions that satisfy hypotheses (H1) and (H2). (i) which is not a Lipschitz function on (ii) where and It is to be noted that is singular at (iii) where is any nonnegative continuous function on and
Before stating our main result, we explain some notations.
1.1. Notations
(i) If then(ii)(iii) is Green’s function of the operator on with and (iv) is the unique solution of the problem (v)Assuming and we define
It will be proven that
The main result of this paper can be stated as follows.
Theorem 1. Let and assume that hypotheses (H2) and (H2) are satisfied. Then, for , problem (3) has a solution satisfying for all ,
Remark 2. (i)For we have (ii)If then satisfies (6).
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.
2. Preliminaries
We recall some relevant properties concerning Hadamard fractional derivative. For more details, the reader can see Section 2.7 of [19].
Definition 3. The Hadamard fractional integral of order of the function is defined as
For we define
Definition 4. Let and its integer part. The Hadamard fractional derivative of order of the function is defined as where and .
Example 5. (Property 2.24 of [19]).
If then
In particular, if and then
Lemma 6. (see [5]).
Let and . Then,
(i) and (ii)The equality is valid on if, and only if,
where and is the smallest integer greater than or equal to (iii)If then
where and is the smallest integer greater than or equal to
Lemma 7. Let and
The unique solution of the problem is given by where
Proof. By Lemma 6, the solution of problem (12) can be written as
Since and we obtain and
Therefore, where is given by (14).
In Figure 1, we give the representation of the Green function with the contours and the projections on some coordinate planes. In particular, one can see that is nonnegative.
(a) and contours
(b) Projection on
(c) Projection on
Lemma 8. Let Then, (i)(ii)On one has where In particular, (iii)On one has (iv)For each the following holds:
Proof. It is easy to check that (i) holds.
To prove (ii), for we have where
Therefore, inequalities in (18) follow from the fact that
By using (18) and the fact that and we obtain (19).
Next, we aim at proving (iv) Let and put
From (18), we have
By symmetry, one can verify that
Hence, the required results follow from (24) and (25).
3. Proof of Theorem 1
We aim at proving Theorem 1. First, we need to establish some preliminary results. For we denote by (i)(ii)For (iii)For (iv)For we recall that is the unique solution of problem (4).
Note that for ,
We recall that .
Proposition 9. Let and then (i)(ii)Let be such that the function then and it is the unique solution of the problem
Proof. (i)The property follows from Lemma 8 (ii).(ii)From (i), and by using again Lemma 8 (ii), we have which implies by Example 5 (i) that is bounded on
Therefore, we have where
We claim that where
Indeed, from (32) and (14), we have
Now using the fact that we deduce that
On the other hand, (i)if then by using (35), we get (ii)if then obviously
Hence,
So, (33) follows from (34), (36), and (39).
Next, we claim that
Indeed, from (33), we have
From the hypothesis, the function is continuous and integrable near while the function becomes continuous and integrable near 1. So and are differentiable on
On the other hand by observing that we deduce that is differentiable on
Therefore,
Applying for the second time the -derivative, we obtain
By Lemma 8 (ii) and (iii), for each we have
This implies by the dominated convergence theorem that
Similarly, we have .
Finally, the uniqueness follows from Lemma 6 (ii).
Remark 10. The property of the above proposition remains true for
Lemma 11. Let and then (i)(ii)For all , (iii)The family is relatively compact in
Proof. (i)As consequence of Lemma 8 (iv) and definition of we obtain (ii)Observe that for each we have
Using this fact, Fatou’s lemma, and (26), we deduce that
That is,
Similarly, since , we obtain
Hence, (47) follows by combining (51) and (52). (iii)It follows from (ii) and (i) that the family is uniformly bounded
Since the function and we deduce by (53) that is equicontinuous in and becomes relatively compact in by Ascoli’s theorem
Proof of Theorem 1. We let
By hypotheses (H1) and (H2) and (28), we have
Define
Using (54) and (47), we obtain
Therefore,
Let and
For define by
By using , and Lemma 11 (iii) we prove that is relatively compact in
From (58), , and (55), we deduce that
Next, by simple arguments, one can prove that is a compact operator.
Therefore, it has a fixed point satisfying
Let Then, and satisfies
Since it follows from (28) that
By using (61), and , we deduce that the function
Hence, from (60), Proposition 9 (ii), and (4), we conclude that is a solution of problem (3).
Example 12. Let . Then, for some and each , problem has a solution in satisfying
Observe that the nonlinearity considered in this example is singular at
Example 13. Let and .
Then, there exists a constant such that for problem admit a solution in satisfying
In particular, for , and we have from (4) and (55),
Therefore, by choosing some continuous functions in (66), we obtain the following graph for with and a numerical value of the constant In Figure 2, we collect the graph of functions , and in Table 1, we summarize the numerical value of .
Example 14. Let and Then, Theorem 1 can be applied for where
4. Conclusion
In this paper, we have considered singular nonlinear Hadamard fractional boundary value problems. By using estimates on Green’s function and the Schauder fixed point theorem, we have proven the existence of a positive solution which blows up.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research group NO (RG-1435-043).