Journal of Function Spaces
Volume 2018 (2018), Article ID 6974046, 8 pages
https://doi.org/10.1155/2018/6974046

Research Article

Separated Boundary Value Problems of Sequential Caputo and Hadamard Fractional Differential Equations

Jessada Tariboon,¹ Asawathep Cuntavepanit,² Sotiris K. Ntouyas,3,4 and Woraphak Nithiarayaphaks¹

¹Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
²Division of Sciences and Liberal Arts, Mahidol University Kanchanaburi Campus, Kanchanaburi 71150, Thailand
³Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
⁴Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Jessada Tariboon; jessada.t@sci.kmutnb.ac.th

Received 24 July 2018; Accepted 31 October 2018; Published 15 November 2018

Guest Editor: Lishan Liu

Copyright © 2018 Jessada Tariboon 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.

In this paper, we discuss the existence and uniqueness of solutions for new classes of separated boundary value problems of Caputo-Hadamard and Hadamard-Caputo sequential fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples.

1. Introduction

Fractional differential equations have been of increasing importance for the past decades due to their diverse applications in science and engineering such as biophysics, bioengineering, virology, control theory, signal and image processing, blood flow phenomena, etc.; see [1–6]. Many interesting results of the existence of solutions of various classes of fractional differential equations have been obtained; see [7–15] and the references therein.

Sequential fractional differential equations are also found to be of much interest [16, 17]. In fact, the concept of sequential fractional derivative is closely related to the nonsequential Riemann-Liouville derivatives, for details, see [3]. For some recent results on boundary value problems for sequential fractional differential equations; see [18–22] and references cited therein.

In this paper, we discuss existence and uniqueness of solutions for two sequential Caputo-Hadamard and Hadamard-Caputo fractional differential equations subject to separated boundary conditions as

and

where and are the Caputo and Hadamard fractional derivatives of orders and , respectively, , is a continuous function, and , .

It can be observed that the sequential Caputo-Hadamard and Hadamard-Caputo fractional differential equations in (1) and (2) are different type when and , since

and

for , respectively.

The rest of the paper is arranged as follows. In Section 2, we establish basic results that lay the foundation for defining a fixed point problem equivalent to the given problems (1) and (2). The main results, based on Banach’s contraction mapping principle, Krasnoselskii’s fixed point theorem, and nonlinear alternative of Leray-Schauder type, are obtained in Section 3. Illustrating examples are discussed in Section 4.

2. Preliminaries

In this section, we introduce some notations and definitions of fractional calculus [4, 5] and present preliminary results needed in our proofs later.

Definition 1 (see [5]). For an at least -times differentiable function , the Caputo derivative of fractional order is defined as

where denotes the integer part of the real number

Definition 2 (see [5]). The Riemann-Liouville fractional integral of order of a function is defined as

provided the integral exists.

Definition 3 (see [5]). For an at least -times differentiable function , the Caputo-type Hadamard derivative of fractional order is defined as

where , .

Definition 4 (see [5]). The Hadamard fractional integral of order is defined as

provided the integral exists.

Lemma 5 (see [5]). For , the general solution of the fractional differential equation is given by

where , .

In view of Lemma 5, it follows that

for some ,   ().

Lemma 6 (see [23]). Let or   and , where . Then, we have

where , .

In order to define the solution of the boundary value problem (1), we consider the linear variant

where

Lemma 7. Let

Then, the unique solution of the separated boundary value problem of sequential Caputo and Hadamard fractional differential equation (12) is given by the integral equation

Proof. Taking the Riemann-Liouville fractional integral of order to the first equation of (12), we get

Again taking the Hadamard fractional integral of order to the above equation, we obtain

Substituting in (15)-(16) and applying the first boundary condition of (12), it follows that

For in equations (15)-(16) and using the second boundary condition of (12), it yields

Solving the linear system of (17) and (18) for finding two constants , we get

and

Substituting constants and in (16), we get the integral equation (14). The converse follows by direct computation. The proof is completed.

In the same way, we can prove the following lemma, which concerns a linear variant of problem (2):

where

Lemma 8. Let

Then, the unique solution of the separated boundary value problem of sequential Caputo and Hadamard fractional differential equation (21) is given by the integral equation

3. Main Results

We set some abbreviate notations for sequential Riemann-Liouville and Hadamard fractional integrals of a function with two variables as

and

where . Also we use this one for a single Riemann-Liouville and Hadamard fractional integrals of orders and , respectively.

In this section, we will use fixed point theorems to prove the existence and uniqueness of solution for problems (1) and (2). To accomplish our purpose, we define the Banach space , of all continuous functions on to endowed with the norm In addition, we define the operator by

where is defined by (13) and Note that the separated boundary value problem (1) has solutions if and only if has fixed points.

For computational convenience we put

To prove the existence theorems of problem (2), we define the operator by

Now, we prove the existence and uniqueness result for problem (1). For problem (2) the proof is similar and omitted.

Theorem 9. Suppose that
there exists a function , such that

If , where , then the separated boundary value problem (1) has a unique solution on .

Proof. Firstly, we define a ball as , where the constant satisfies

where . Next, we will show that For any and using the triangle inequality , we have

which implies that Let , then

which yields that Since , we deduce that the operator is a contraction. By Banach contraction mapping principle the operator has a unique fixed point, which leads that problem (1) has a unique solution on .

Theorem 10. Let in Theorem 9 holds. If , where

then the separated boundary value problem (2) has a unique solution on .

Our second existence result is based on Krasnoselskii’s fixed point theorem.

Theorem 11 ((Krasnoselskii’s fixed point theorem) [24]). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let be operators such that

(a) where ;(b) is compact and continuous;(c) is a contraction mapping.

Then there exists such that .

Theorem 12. Let be a continuous function satisfying in Theorem 9. In addition, assume that
() , and .
If

then the separated boundary value problem (1) has at least one solution on .

Proof. Let , where a constant satisfying and We decompose the operator into two operators and on with

Note that the ball is a closed, bounded, and convex subset of the Banach space .
Now, we will show that for satisfying condition (a) of Theorem 11. Setting , then we have

This means that . To prove that is a contraction mapping, for , we have

by condition , which is a contraction, by (34). Therefore, the condition (c) of Theorem 11 is satisfied. Next we will show that the operator is compact and continuous. By using the continuity of the function on , we can conclude that the operator is continuous. For , it follows that

where

which implies that the set is uniformly bounded. Now we are going to prove that is equicontinuous. For , such that and for , we have

which is independent of and also tends to zero as . Hence the set is equicontinuous. Therefore the set is relatively compact. By applying the Arzelá-Ascoli theorem, the operator is compact on . Therefore the operators and satisfy the assumptions of Theorem 11. By the conclusion of Theorem 11, we get that the separated boundary value problem (1) has at least one solution on . This completes the proof.

Theorem 13. Assume that and are fulfilled. If , then the separated boundary value problem (2) has at least one solution on .

The above theorem can be proved by applying Krasnoselskii’s fixed point theorem to the operator defined in (28).

Remark 14. If the operators and are interchanged, then we have the existence results as follows:

(i)If , then problem (1) has at least one solution on .(ii)If , then problem (2) has at least one solution on , where

However, in application to existence theory, the computation of values and is easier than and , respectively.

The third existence result will be proved by applying Leray-Schauder nonlinear alternative.

Theorem 15 ((nonlinear alternative for single valued maps) [25]). Let be a Banach space, a closed, convex subset of an open subset of , and Suppose that is a continuous; compact (that is, is a relatively compact subset of ) map. Then either

(i) has a fixed point in or(ii)there is a (the boundary of in ) and with

Let us state and prove the existence theorem.

Theorem 16. Suppose that
() there exist a continuous nondecreasing function and a function such that

() there exists a constant such that

Then the separated boundary value problem (1) has at least one solution on .

Proof. Let the operator be defined in (26). Let us prove that the operator maps bounded sets (balls) into bounded sets in . For a constant , we define a bounded ball . Then for , one has

which implies that

After that we will show that the operator maps bounded sets into equicontinuous sets of . Let be any two points in such that . Then for , we have

As , the right-hand side of the above inequality tends to zero independently of . Hence, by applying the Arzelá-Ascoli theorem, the operator is completely continuous.
The result will be followed from the Leray-Schauder nonlinear alternative if we prove the boundedness of the set of the solutions to equation for Let be a solution of the operator equation . Then, for , by directly computation, we have

which leads to

From the assumption , there exists a positive constant such that Let us set

It is easy to see that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Therefore, by the nonlinear alternative of Leray-Schauder type (Theorem 15), we deduce that the operator has a fixed point which is a solution of problem (1). The proof is completed.

Theorem 17. Assume that the condition in Theorem 16 is satisfied. If a positive constant satisfying

then the separated boundary value problem (2) has at least one solution on .

The next two special cases can be obtained by setting , and , with two constants , .

Corollary 18. Let be a continuous function satisfying , for all . Then

(i)if , then the separated boundary value problem (1) has at least one solution on ;(ii)if , then the separated boundary value problem (2) has at least one solution on .

4. Examples

In this section, we present some examples to illustrate our results.

Example 1. Consider the following sequential Caputo-Hadamard fractional differential equations with separated boundary conditions

Here , , , , , , , and . From given information, we find that , , and which yield .

(i) Let with

It follows that

Then condition is satisfied with . Thus . Hence, by Theorem 9, problem (51) with (52) has a unique solution on .

(ii) Given by

Observe that the function defined in (54) satisfies with . But the Theorem 9 can not be applied to this case because the value of . However, by the benefit of Theorem 12, we have . By the conclusion of Theorem 12, problem (51) with (54) has at least one solution on .

Example 2. Consider the following sequential Hadamard-Caputo fractional differential equations with separated boundary conditions

Here , , , , , , , and . From above information, we can find that , , and which can be computed the value of .

(i) The function is defined by

Setting and , we see that the condition of Theorem 16 is satisfied with the above function . In addition, we can find that . Then there exists a constant such that satisfying inequality (50). Therefore, applying Theorem 17, problem (55) with (56) has at least one solution on .

(ii) Let by

It is easy to see that the function defined in (57) can be expressed as . Then . Using (ii) of the Corollary 18, the problem (55) with (57) has at least one solution on .

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.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok, Contract no. KMUTNB-60-ART-105.

References

K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, New York, NY, USA, Elsevier, 2006.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.

J. R. Graef, L. Kong, and Q. Kong, “Application of the mixed monotone operator method to fractional boundary value problems,” Fractional Differential Calculus, vol. 2, no. 1, pp. 554–567, 2011.

Z. Bai and W. Sun, “Existence and multiplicity of positive solutions for singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 63, no. 9, pp. 1369–1381, 2012.

S. K. Ntouyas, J. Tariboon, and W. Sudsutad, “Boundary value problems for Riemann-Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions,” Mediterranean Journal of Mathematics, vol. 13, no. 3, pp. 939–954, 2016.

B. Ahmad, A. Alsaedi, S. K. Ntouyas, and J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Springer, Cham, 2017.

T. A. Maraaba, F. Jarad, and D. Baleanu, “On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives,” Science China Mathematics, vol. 51, no. 10, pp. 1775–1786, 2008.

T. Abdeljawad, F. Jarad, and D. Baleanu, “Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo derivatives,” J. Math. Phys, vol. 49, Article ID 083507, p. 10, 2008.

Y. Adjabi, F. Jarad, D. Baleanu, and T. Abdeljawad, “On Cauchy problems with CAPuto Hadamard fractional derivatives,” Journal of Computational Analysis and Applications, vol. 21, no. 4, pp. 661–681, 2016.

Y. Y. Gambo, R. Ameen, F. Jarad, and T. Abdeljawad, “Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives,” Advances in Difference Equations, Paper No. 134, 13 pages, 2018.

R. P. Agarwal, B. Ahmad, and A. Alsaedi, “Fractional-order differential equations with anti-periodic boundary conditions: a survey,” Boundary Value Problems, Paper No. 173, 27 pages, 2017.

M. Klimek, “Sequential fractional differential equations with Hadamard derivative,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4689–4697, 2011.

C. Bai, “Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 211–231, 2011.

B. Ahmad, S. K. Ntouyas, R. P. Agarwal, and A. Alsaedi, “Existence results for sequential fractional integro-differential equations with nonlocal multi-point and strip conditions,” Boundary Value Problems, vol. 2016, no. article 205, 2016.

A. Alsaedi, S. K. Ntouyas, R. P. Agarwal, and B. Ahmad, “On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions,” Advances in Difference Equations, vol. 2015, article 33, 2015.

B. Ahmad and S. K. Ntouyas, “Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions,” Applied Mathematics and Computation, vol. 266, pp. 615–622, 2015.

B. Ahmad and J. J. Nieto, “Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order,” journal of function spaces and applications, vol. 2013, Article ID 149659, 8 pages, 2013.

B. Ahmad and S. K. Ntouyas, “On Higher-Order Sequential Fractional Differential Inclusions with Nonlocal Three-Point Boundary Conditions,” Abstract and Applied Analysis, vol. 2014, Article ID 659405, 10 pages, 2014.

F. Jarad, T. Abdeljawad, and D. Baleanu, “Caputo-type modification of the Hadamard fractional derivatives,” Advances in Difference Equations, vol. 2012, article 142, 2012.

M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1, pp. 123–127, 1955.

A. Granas and J. Dugundji, Fixed Point Theory and Applications, Springer, New York, NY, USA, 2003.