Abstract and Applied Analysis

Volume 2012, Article ID 391062, 17 pages

http://dx.doi.org/10.1155/2012/391062

## On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 27 December 2011; Revised 10 April 2012; Accepted 14 April 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 M. D. Qassim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

#### 1. Introduction

Fractional derivatives have proved to be very efficient and adequate to describe many phenomena with memory and hereditary processes. These phenomena are abundant in science, engineering (viscoelasticity, control, porous media, mechanics, electrical engineering, electromagnetism, etc.) as well as in geology, rheology, finance, and biology. Unlike the classical derivatives, fractional derivatives have the ability to characterize adequately processes involving a past history. We are witnessing a huge development of fractional calculus and methods in the theory of differential equations. Indeed, after the appearance of the papers by Bagley and Torvik [1–3], researchers started to deal directly with differential equations containing fractional derivatives instead of ignoring them as it used to be the case. For analytical treatments, we may refer the reader to [4–36], and for some applications, one can consult [1–3, 8, 25, 26, 26, 27, 27–31, 33, 34, 37–49] to cite but a few.

We will consider the problem: where is a new type of fractional derivative we will define below and is a given constant. This new fractional derivative interpolates the Hadamard fractional derivative and its Caputo counterpart [26, 34], in the same way the Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. That is why we are naming it after Hilfer and Hadamard.

A nonexistence result for global solutions of the problem (1.1) will be proved when for some and . That is we consider the Cauchy problem: where and show that no solutions can exist for all time for certain values of and . Clearly, sufficient conditions for nonexistence provide necessary conditions for existence of solutions. In addition, we construct an example for which there exist solutions for some powers and in some appropriate space.

The existence and uniqueness of solutions for problem (1.1) has been discussed in [50] in the space defined by where for and .

We also point out here that the case where is the usual Riemann-Liouville fractional derivative has been studied in [26] (see also references therein). There are very few papers [26, 29] dealing with the pure Hadamard case, that is, when .

The rest of the paper is divided into three sections. In Section 2, we present some definitions, notations, and lemmas which will be needed later in our proof. Section 3 is devoted to the nonexistence result and Section 4 contains an example of existence of solutions.

#### 2. Preliminaries

In this section, we present some background material for the forthcoming analysis. For more details, see [25, 26, 33, 42, 51, 52].

*Definition 2.1. *The space consists of those real-valued Lebesgue measurable functions on for which , where
In particular, when , we see that *. *

*Definition 2.2. *Let be a finite interval and , we introduce the weighted space of continuous functions on :
In the space , we define the norm:

*Definition 2.3. *Let be the -derivative, for , we denote by () the Banach space of functions which have continuous -derivatives on up to order and have the derivative of order on such that :
with the norm:
When , we set

*Definition 2.4. *Let be a finite or infinite interval of the half-axis and let . The Hadamard left-sided fractional integral of order is defined by
provided that the integral exists. When , we set

*Definition 2.5. *Let be a finite or infinite interval of the half-axis and let *. *The Hadamard right-sided fractional integral of order is defined by
provided that the integral exists. When *, *we set

*Definition 2.6. *The left-sided Hadamard fractional derivative of order on is defined by
that is,
When , we set

*Definition 2.7. *The right-sided Hadamard fractional derivative of order on is defined by
that is,
When , we set

Lemma 2.8. *If and , then
**
In particular, if , then the Hadamard fractional derivative of a constant is not equal to zero:
**
when .*

Lemma 2.9. *Let , and .** If , then is bounded from into . In particular, is bounded in .**If , then is bounded from into . In particular, is bounded in .*

This lemma justifies the following one

Lemma 2.10 (the semigroup property of the fractional integration operator ). *Let , and . If , then, for ,
**
holds at any point . When , this relation is valid at any point .*

Lemma 2.11. *Let and . If , then the fractional derivatives and exist on and , respectively, ) and can be represented in the forms:
**
respectively.*

Lemma 2.12 (fractional integration by Parts). *Let and . If and , then
**
where .*

*Definition 2.13. *The fractional derivative of order on defined by
where , is called the Hadamard-Caputo fractional derivative of order *. *

Now, motivated by the Hilfer fractional derivative introduced in [41, 42], we introduce the new fractional derivative which we call Hilfer-Hadamard fractional derivative of order and type : The Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative. This new one interpolates the Hadamard fractional derivative and its caputo counterpart. Indeed, for , we find the Hadamard fractional derivative as defined in Definition 2.6 and, for , we find its Caputo type counterpart (Definition 2.13).

Theorem 2.14 (Young’s inequality). *If and are nonnegative real numbers and and are positive real numbers such that , then one has
**
Equality holds if and only if .*

#### 3. Nonexistence Result

Before we state and prove our main result, we will start with the following lemma.

Lemma 3.1. *If and , then
*

*Proof. *Since , then on we have for some positive constant . Therefore,
As , we see that
In a similar manner, we prove the second part of the lemma.

The proof of the next result is based on the test function method developed by Mitidieri and Pokhozhaev in [52].

Theorem 3.2. *Assume that and . Then, Problem (1.2) does not admit global nontrivial solutions in , where
**
when .*

*Proof. *Assume that a nontrivial solution exists for all time . Let be a test function satisfying is non-increasing and such that
for some and some () such that . Multiplying the inequality in (1.2) by and integrating over , we get
Observe that the integral in left-hand side exists and the one in the right-hand side exists for when . Moreover, from the definition of , we can rewrite (3.6) as
By virtue of Lemma 2.12 (after extending by zero outside ), we may deduce from (3.7) that
Notice that Lemma 2.12 is valid in our case since ( implies that on for some positive constant )
Let , then by the definition of the Gamma function,
Hence, (and ) for some .

An integration by parts in (3.8) yields
or
because (see Lemma 3.1) and
Multiplying by inside the integral in the left hand side of (3.12), we see that
It appears from Definition 2.7 that
and from Lemma 2.11, we see that
Since and
the last equality becomes
Note that and by the same argument as the one used at the beginning of the proof we may show that since .

Therefore, Lemma 2.12 again allows us to write
and by the semigroup property Lemma 2.10
On the other hand,
As is nonincreasing, we have for all and , . Also, it is clear that
Therefore,
Definition 2.4 allows us to write
By the same argument as the one used at the beginning of the proof, we may show that . Moreover, it is easy to see that (for, otherwise, we consider with some sufficiently large ). Thus, we can apply Lemma 2.12 to get
Next, we multiply by inside the integral in the right-hand side of (3.25):
For , we have (because and ). It follows that
By using the Young inequality (see Theorem 2.14), with and such that , in the right-hand side of (3.27), we find
Clearly, from (3.14) and (3.28), we see that
or
Therefore, by Definition 2.5, we have
The change of variable yields
Another change of variable gives
We may assume that the integral term in the right-hand side of (3.33) is convergent, that is,
for some positive constant , for otherwise we consider with some sufficiently large . Therefore
If , then
as . Finally, from (3.35), we obtain
We reach a contradiction since the solution is not supposed to be trivial.

In the case we have and the relation (3.35) ensures that
Moreover, it is clear that
This relation, together with (3.27) (relations (3.28) and (3.31) also are used without ), implies that
for some positive constant , with
due to the convergence of the integral in (3.38). This is again a contradiction.

If , we have (because and ). Then, the change of variables and in (3.27) yields
or
The expression may be assumed bounded (or else we use with a large value of ). Hence,
for some positive constant .

Although we are concerend here about nonexistence of solutions, using standard techniques, one may show the existence of local solutions of Problem (1.1) with . However, according to Theorem 3.2, such a solution cannot be continued for all time in . This is a phenomenon which occurs often in parabolic and hyperbolic problems with sources of polynomial type. In the absence of strong dissipations, these sources are the cause of blowup in finite time (of local solutions). For this reason, they are called blowup terms.

#### 4. Example

For our example, we need the following lemma.

Lemma 4.1. *The following result holds for the fractional derivative operator :
**
where and .*

*Proof. *We observe from Lemma 2.8 that
Therefore,
which, in light of the definition of , yields
From Lemma 2.8 again, we have
The proof is complete.

*Example 4.2. *Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order and type :
with real , (). Suppose that the solution has the following form:
Our aim next is to find the values of and . By using Lemma 4.1 we have
Therefore,
It can be directly shown that and . If , that is, , then (4.6) has the exact solution:
This solution satisfies the initial condition when . Note that there is an overlap of the interval of existence in this example and the interval of nonexistence in the previous theorem. This may be explained by the fact that this solution is in but not in .

#### Acknowledgments

The authors wish to express their thanks to the referees for their suggestions. The authors are also very grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through the Project no. In101003.

#### References

- R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,”
*Journal of Rheology*, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar - R. L. Bagley and P. J. Torvik, “A different approach to the analysis of viscoelastically damped structures,”
*AIAA Journal*, vol. 21, pp. 741–748, 1983. View at Google Scholar - R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of real material,”
*Journal of Applied Mechanics*, vol. 51, no. 2, pp. 294–298, 1984. View at Google Scholar - R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 179750, 25 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,”
*Advances in Difference Equations*, vol. 2009, Article ID 981728, 47 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,”
*Boundary Value Problems*, vol. 2011, article 36, 9 pages, 2011. View at Publisher · View at Google Scholar - B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,”
*Computers and Mathematics with Applications*. In press. View at Publisher · View at Google Scholar - V. V. Anh and R. Mcvinish, “Fractional differential equations driven by Lévy noise,”
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 16, no. 2, pp. 97–119, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. de Andrade, C. Cuevas, and J. P. C. dos Santos, “Existence results for a fractional equation with state-dependent delay,”
*Advances in Difference Equations*, vol. 2011, Article ID 642013, 15 pages, 2011. View at Google Scholar · View at Zentralblatt MATH - M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order,”
*Surveys in Mathematics and its Applications*, vol. 3, pp. 1–12, 2008. View at Google Scholar · View at Zentralblatt MATH - S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,”
*Computers & Mathematics with Applications*, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. A. El-Sayed, “Fractional differential equations,”
*Kyungpook Mathematical Journal*, vol. 28, no. 2, pp. 119–122, 1988. View at Google Scholar · View at Zentralblatt MATH - A. M. A. El-Sayed, “On the fractional differential equations,”
*Applied Mathematics and Computation*, vol. 49, no. 2-3, pp. 205–213, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. A. El-Sayed, “Fractional order evolution equations,”
*Journal of Fractional Calculus*, vol. 7, pp. 89–100, 1995. View at Google Scholar · View at Zentralblatt MATH - A. M. A. El-Sayed and S. A. Abd El-Salam, “Weighted Cauchy-type problem of a functional differ-integral equation,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 30, pp. 1–9, 2007. View at Google Scholar · View at Zentralblatt MATH - A. M. A. El-Sayed and S. A. Abd El-Salam, “${L}_{p}$-solution of weighted Cauchy-type problem of a diffre-integral functional equation,”
*International Journal of Nonlinear Science*, vol. 5, no. 3, pp. 281–288, 2008. View at Google Scholar - K. M. Furati and N.-E. Tatar, “Power-type estimates for a nonlinear fractional differential equation,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 62, no. 6, pp. 1025–1036, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. M. Furati and N.-E. Tatar, “An existence result for a nonlocal fractional differential problem,”
*Journal of Fractional Calculus*, vol. 26, pp. 43–51, 2004. View at Google Scholar · View at Zentralblatt MATH - K. M. Furati and N.-E. Tatar, “Behavior of solutions for a weighted Cauchy-type fractional differential problem,”
*Journal of Fractional Calculus*, vol. 28, pp. 23–42, 2005. View at Google Scholar · View at Zentralblatt MATH - K. M. Furati and N.-E. Tatar, “Long time behavior for a nonlinear fractional model,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 1, pp. 441–454, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. A. Kilbas, B. Bonilla, and J. J. Trujillo, “Existence and uniqueness theorems for nonlinear fractional differential equations,”
*Demonstratio Mathematica*, vol. 33, no. 3, pp. 583–602, 2000. View at Google Scholar · View at Zentralblatt MATH - A. A. Kilbas, B. Bonilla, J. J. Trujillo et al., “Fractional integrals and derivatives and differential equations of fractional order in weighted spaces of continuous functions,”
*Doklady Natsionalnoy Akademii Nauk Belarusi*, vol. 44, no. 6, 2000 (Russian). View at Google Scholar - A. A. Kilbas and S. A. Marzan, “Cauchy problem for differential equation with Caputo derivative,”
*Fractional Calculus & Applied Analysis*, vol. 7, no. 3, pp. 297–321, 2004. View at Google Scholar · View at Zentralblatt MATH - A. A. Kilbas and S. A. Marzan, “Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions,”
*Differential Equations*, vol. 41, no. 1, pp. 84–89, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006, Edited by Jan van Mill. - A. A. Kilbas and A. A. Titioura, “Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions,”
*Mathematical Modelling and Analysis*, vol. 12, no. 3, pp. 343–356, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Kosmatov, “Integral equations and initial value problems for nonlinear differential equations of fractional order,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 7, pp. 2521–2529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Kou, J. Liu, and Y. Ye, “Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 142175, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lakshmikantham, “Theory of fractional functional differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Lin, “Global existence theory and chaos control of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 1, pp. 709–726, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science, 1987, (Translation from Russian 1993). - C. Yu and G. Gao, “Existence of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 310, no. 1, pp. 26–29, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Zhou, “Existence and uniqueness of solutions for a system of fractional differential equations,”
*Fractional Calculus & Applied Analysis*, vol. 12, no. 2, pp. 195–204, 2009. View at Google Scholar - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. - L. Gaul, P. Klein, and S. Kempe, “Damping description involving fractional operators,”
*Mechanical Systems and Signal Processing*, vol. 5, pp. 81–88, 1991. View at Google Scholar - W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach of selfsimilar protein dynamics,”
*Biophysical Journal*, vol. 68, pp. 46–53, 1995. View at Google Scholar - T. T. Hartley, C. F. Lorenzo, H. K. Qammar et al., “Chaos in a fractional order Chua system,”
*IEEE Transactions on Circuits & Systems I*, vol. 42, no. 8, pp. 485–490, 1995. View at Google Scholar - R. Hilfer,
*Fractional Time Evolution, Applications of Fractional Calculus in Physics*, World Scientific, New-Jersey, NJ, USA, 2000. View at Zentralblatt MATH - R. Hilfer, “Experimental evidence for fractional time evolution in glass materials,”
*Chemical Physics*, vol. 284, pp. 399–408, 2002. View at Google Scholar - R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,”
*American Society of Mechanical Engineers*, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Laskri and N.-E. Tatar, “The critical exponent for an ordinary fractional differential problem,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1266–1270, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanis,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Vienna, Austria, 1997. View at Google Scholar - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity*, Imperial College Pres, World Scientific, London, UK, 2010. View at Publisher · View at Google Scholar - B. Mandelbrot, “Some noises with 1/f spectrum, a bridge between direct current and white noise,”
*IEEE Transactions on Information Theory*, vol. 13, no. 2, pp. 289–298, 1967. View at Google Scholar - F. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,”
*Journal of Chemical Physics*, vol. 103, pp. 7180–7186, 1995. View at Google Scholar - I. Podlubny, I. Petráš, B. M. Vinagre, P. O'Leary, and L'. Dorčák, “Analogue realizations of fractional-order controllers,”
*Nonlinear Dynamics*, vol. 29, no. 1–4, pp. 281–296, 2002. View at Publisher · View at Google Scholar - K. M. Furati, M. D. Kassim, and N.-E. Tatar, “Existence and stability for a differential problem with Hilfer-Hadamard fractional derivative,” submitted.
- P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 1–15, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - È. Mitidieri and S. I. Pokhozhaev, “A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,”
*Proceedings of the Steklov Institute of Mathematics*, vol. 234, pp. 1–383, 2001. View at Google Scholar