Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

Patanarapeelert, Nichaphat; Sitthiwirattham, Thanin

doi:https://doi.org/10.1155/2017/7895186

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Preliminaries Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 7895186 | https://doi.org/10.1155/2017/7895186

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

Nichaphat Patanarapeelert¹and Thanin Sitthiwirattham²

Academic Editor: Chris Goodrich

Received25 Sept 2017

Revised15 Nov 2017

Accepted27 Nov 2017

Published17 Dec 2017

Abstract

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

1. Introduction

The quantum calculus is the subject of calculus without limits and deals with a set of nondifferentiable functions. The quantum operators are widely used in mathematic fields such as hypergeometric series, complex analysis, orthogonal polynomials, combinatorics, hypergeometric functions, and the calculus of variations. The quantum calculus is also found in many applications, such as quantum mechanics and particle physics [1–9].

In 1949, the Hahn difference operator was introduced by Hahn [10]. This operator is a combination of two well-known operators, the forward difference operator and the Jackson -difference operator. The Hahn difference operator is defined by We note that The Hahn difference operator has been used in finding families of orthogonal polynomials as well to determine some approximation problems (see [11–13]).

The right inverse of the Hahn difference operator was proposed by Aldwoah in 2009 [14, 15]. This operator is in the terms of the Jackson -integral containing the right inverse of [16] and Nörlund sum containing the right inverse of [16].

In 2010, Malinowska and Torres [17, 18] introduced the Hahn quantum variational calculus. In 2013, Malinowska and Martins [19] presented the generalized transversality conditions for the Hahn quantum variational calculus. Next, Hamza and Ahmed [20, 21] established the theory of linear Hahn difference equations. The authors also study the existence and uniqueness of solution for the initial value problems for Hahn difference equations by employing the method of successive approximations; in addition, they proved Gronwall’s and Bernoulli’s inequalities with respect to the Hahn difference operator and investigated the mean value theorems for this calculus. In 2016, Hamza and Makharesh [22] investigated Leibniz’s rule and Fubini’s theorem associated with Hahn difference operator. In the same year, Sitthiwirattham [23] studied the nonlocal boundary value problem for nonlinear Hahn difference equation as given bywhere , , , , , , is a given function, and is a given functional.

In 2010, Čermák and Nechvátal [24] established the fractional -difference operator and the fractional -integral for . In 2011, Čermák et al. [25] studied linear fractional difference equations with discrete Mittag-Leffler functions for in 2011. Rahmat [26, 27] presented the -Laplace transform and some -analogues of integral inequalities on discrete time scales for . In 2016, Du et al. [28] studied the monotonicity and convexity for nabla fractional -difference for , . Since fractional Hahn operators require the condition , we note that the operators mentioned above are not fractional Hahn operators. Recently, the fractional Hahn operators have been introduced by Brikshavana and Sitthiwirattham [29].

In order to gain further insight into fractional Hahn operators, in this paper, we study the boundary value problem for fractional Hahn difference equation. Particularly, we consider a Riemann-Liouville fractional Hahn difference boundary value problem for a fractional Hahn integrodifference equation of the formand a fractional Hahn integral boundary value problem for a Caputo fractional Hahn difference equation of the formwhere ; , , , , , , and ; is given function; are given functionals; and for , define

In Section 2, we provide some basis definitions, properties, and lemma used as material for this work. In Sections 3 and 4, we prove the existence results of problems (4) and (5), respectively, and we prove the existence and uniqueness of a solution by using the Banach fixed-point theorem and the existence of at least one solution by using the Schauder fixed-point theorem. Finally, some examples are provided to illustrate our results in the last section.

2. Preliminaries

In this section, we suggest some notations, definitions, and lemmas which are used in the main results. Let and and define The -analogue of the power function with is defined by The -analogue of the power function with is defined by In general, for , we define We note that and and use the notation for . The -gamma and -beta functions are defined by

Definition 1. For , , and defined on an interval containing , the Hahn difference of is defined by and . Providing that is differentiable at , we call the -derivative of and say that is -differentiable on .

Remarks 2. We give some properties for the Hahn difference as follows:(1).(2).(3).(4)

Letting with and , we define the -interval by Observe that, for each , the sequence is uniformly convergent to .

We also define the forward jump operator as and the backward jump operator as for .

Definition 3. Let be any closed interval of containing , and . Assuming that is a given function, we define -integral of from to by where Providing that the series converges at and , we consider as -integrable on and the sum to the right-hand side of above equation will be called the Jackson-Nörlund sum.

We note that the actual domain of function defined on

We next introduce the fundamental theorem of Hahn calculus in the following lemma.

Lemma 4 (see [14]). Let be continuous at . Define Then, is continuous at . Furthermore, exists for every and Conversely, one has

Lemma 5 (see [23]). Let , , and be continuous at . Then,

Lemma 6 (see [23]). Let and . Then,

We next introduce fractional Hahn integral and fractional Hahn difference of Riemann-Liouville and Caputo types as follows.

Definition 7. For , and defined on , the fractional Hahn integral is defined byand .

Definition 8. For , , , and defined on , the fractional Hahn difference of the Riemann-Liouville type of order is defined by The fractional Hahn difference of the Caputo type of order is defined by and .

Lemma 9 (see [29]). Let , , , and . Then, for some , and

Lemma 10 (see [29]). Let , , , and . Then, for some , and

Next, we give some auxiliary lemmas for simplifying our calculations.

Lemma 11 (see [29]). Let , , and . Then,

In the following, we give a lemma that deals with the linear variant of problem (4) and gives a representation of the solution.

Lemma 12. Let , , , , , , and and let be a given function. Then the problemhas the unique solutionwhere the functions and are defined by respectively. The constants , , and are defined byrespectively.

Proof. Taking fractional Hahn -integral of order for (27), we obtainTaking fractional Hahn -difference of order for (34), we obtainSubstituting into (34) and using the first condition of (27), we haveBy letting , into (35) and employing the second condition of (27), we obtainTo find and , we solve the system of equations (36) and (37). Then, we obtain where , , , , and are defined as (29)–(33), respectively.
Substituting the constants and into (34) and by Cramer’s rule, it is implied that (28) is the uniqueness solution.

We next prove a lemma that deals with the linear variant of problem (5) and gives a representation of the solution.

Lemma 13. Let , , , , , and and let be a given function and be given functionals. Then, the problemhas the unique solution

Proof. Taking fractional Hahn -integral of order for (39), we obtainTaking fractional Hahn -integral of order for (41), we haveLetting into (41) and employing the first condition of (39), we get Substituting into (42) and employing the second condition of (39), we obtain Substituting the constants and into (41) and by Cramer’s rule, it is implied that (40) is the uniqueness solution.

We next introduce Schauder’s fixed-point theorem used to prove the existence of a solution to (4) and (5).

Lemma 14 ([30] (Arzelá-Ascoli theorem)). A set of functions in with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on .

Lemma 15 (see [30]). If a set is closed and relatively compact, then it is compact.

Lemma 16 ([31] (Schauder’s fixed-point theorem)). Let be a complete metric space, let be a closed convex subset of , and let be the map such that the set is relatively compact in . Then the operator has at least one fixed point

3. Existence Results for Problem (4)

In this section, we prove the existence results for problem (4). Let be a Banach space of all function with the norm defined by where , , , , and . Define an operator bywhere , , and are defined as (31)–(33), respectively, and the functions and are defined byObviously, problem (4) has solutions if and only if the operator has fixed points.

Theorem 17. Assume that is continuous and is continuous with . In addition, suppose that the following conditions hold:
There exist constants such that, for each and ,, whereThen, problem (4) has a unique solution.

Proof. To prove that is contraction, we considerfor each and . We find thatFinally, for each and , where and are defined on (49)-(50), respectively.
implies that is a contraction. Therefore, by using Banach fixed-point theorem, has a fixed point which is a unique solution of problem (4).

We next show the existence of a solution to (4) by the following Schauder’s fixed-point theorem.

Theorem 18. Suppose that holds. Then, problem (4) has at least one solution.

Proof. We organize the proof into three steps.
Step 1. Verify map bounded sets into bounded sets in . We set and choose a constantwhere and are defined on (49)-(50), respectively. Denote that For each and , we obtainSimilarly with Theorem 17, we obtain Therefore . This implies that is uniformly bounded.
Step 2. Show that is continuous on .
Letting , there exists such that, for each and for all , Then, we haveThis means that the operator is continuous on .
Step 3. Examine that is equicontinuous with . For any with , we haveIf , the right-hand side of the above inequality tends to be zero. So is relatively compact on .
This means that the set is an equicontinuous set. As a consequence of Steps I to III together with the Arzelá-Ascoli theorem, we get that is completely continuous. By Schauder’s fixed-point theorem, we can conclude that problem (4) has at least one solution. The proof is done.

4. Existence Results for Problem (5)

In this section, we present the existence results for problem (5). Let be a Banach space of all function with the norm defined bywhere , , , , , and Define an operator byObviously, problem (5) has solutions if and only if the operator has fixed points.

Theorem 19. Assume that is continuous and . In addition, suppose that the following conditions hold:
There exist constants such that, for each and , There exist constants such that, for each , , whereThen, problem (5) has a unique solution.

Proof. To show that is contraction, we consider for each and .
For each and , we find thatHence, from (67), it is implied that
By , it is implied that is a contraction. Therefore, by using Banach fixed-point theorem, has a fixed point which is a unique solution of problem (5).

We also deduce the existence of a solution to (5) by the following Schauder’s fixed-point theorem.

Theorem 20. Suppose that hold. Then, problem (5) has at least one solution.

Proof. We divide the proof into three steps.
Step 1. Verify map bounded sets into bounded sets in . We set , , and and choose a constantwhere are defined on (65). Denote that For each and , we obtainSimilar to Theorem 19, we obtain So, we have . This implies that is uniformly bounded.
Step 2. Show that is continuous on .
Letting , there exist such that, for each and for all ,Then, we have By a similar proof to the above, we obtain Hence, . This means that the operator is continuous on .
Step 3. Examine that is equicontinuous with . For any with ,If , the right-hand side of the above inequalities (75) tends to be zero. So is relatively compact on .
Therefore the set is an equicontinuous set. As a consequence of Steps I to III together with the Arzelá-Ascoli theorem, we get that is completely continuous. By Schauder’s fixed-point theorem, we can conclude that problem (5) has at least one solution. The proof is completed.

5. Some Examples

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

Example 1. Consider the following fractional Hahn boundary value problem:Here , , , , , , , , , , , , , and .
Sincethen is satisfied with and
Also, we haveWe can show that Therefore, we find that Hence, by Theorem 17, problem (76) has a unique solution on .

Example 2. Consider the following fractional Hahn boundary value problem:Here , , , , , , , , , , , , , and .
Since then and are satisfied with and and and .
Therefore, we get Hence, by Theorem 19, problem (81) has a unique solution on .

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-GOV-60-76).

References

M. H. Annaby and Z. S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2012.
View at: Publisher Site | MathSciNet
V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.
View at: Publisher Site | MathSciNet
D. L. Jagerman, Difference Equations with Applications to Queues, Dekker, New York, NY, USA, 2000.
K. A. Aldwoah, A. B. Malinowska, and D. F. M. Torres, “The power quantum calculus and variational problems,” Dynamics of Continuous, Discrete & Impulsive Systems Series B: Applications & Algorithms, vol. 19, no. 1-2, pp. 93–116, 2012.
View at: Google Scholar | MathSciNet
A. M. C. Cruz, N. Martins, and D. F. M. Torres, “Symmetric differentiation on time scales,” Applied Mathematics Letters, vol. 26, no. 2, pp. 264–269, 2013.
View at: Google Scholar
B. Cruz and M. C. Artur, Symmetric Quantum Calculus [Ph.D. thesis], Aveiro University, Aveiro, Portugal, 2012.
G.-C. Wu and D. Baleanu, “New applications of the variational iteration method from differential equations to q-fractional difference equations,” Advances in Difference Equations, vol. 2013, p. 21, 2013.
View at: Publisher Site | Google Scholar
J. Tariboon and S. K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive difference equations,” Advances in Difference Equations, vol. 2013, article 282, 2013.
View at: Publisher Site | Google Scholar
R. Álvarez-Nodarse, “On characterizations of classical polynomials,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 320–337, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
W. Hahn, “Über Orthogonalpolynome, die q-Differenzenlgleichungen genügen,” Mathematische Nachrichten, vol. 2, pp. 4–34, 1949.
View at: Google Scholar
R. S. Costas-Santos and F. Marcellain, “Second structure Relation for q-semiclassical polynomials of the Hahn Tableau,” Journal of Mathematical Analysis and Applications, vol. 329, pp. 206–228, 2007.
View at: Google Scholar
K. H. Kwon, D. W. Lee, S. B. Park, and B. H. Yoo, “Hahn class orthogonal polynomials,” Kyungpook Mathematical Journal, vol. 38, no. 2, pp. 259–281, 1998.
View at: Google Scholar
M. Foupouagnigni, Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre-Freud equations recurrence coefficients [Ph.D. thesis], Université Nationale du Bénin, Bénin, 1998.
K. A. Aldwoah, Generalized time scales and associated difference equations [Ph.D. thesis], Cairo University, cairo, Egypt, 2009.
M. H. Annaby, A. E. Hamza, and K. A. Aldwoah, “Hahn difference operator and associated Jackson-Nörlund integrals,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 133–153, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
F. H. Jackson, “Basic integration,” Quarterly Journal of Mathematics, vol. 2, pp. 1–16, 1951.
View at: Publisher Site | Google Scholar | MathSciNet
A. B. Malinowska and D. F. Torres, “The Hahn quantum variational calculus,” Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 419–442, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
A. B. Malinowska and D. F. Torres, “Quantum variational calculus,” in Spinger Briefs in Electrical and Computer Engineering-Control, Automation and Robotics, Springer, Berlin, Germany, 2014.
View at: Google Scholar
A. B. Malinowska and N. Martins, “Generalized transversality conditions for the Hahn quantum variational calculus,” Optimization. A Journal of Mathematical Programming and Operations Research, vol. 62, no. 3, pp. 323–344, 2013.
View at: Publisher Site | Google Scholar
A. E. Hamza and S. M. Ahmed, “Existence and uniqueness of solutions of Hahn difference equations,” Advances in Difference Equations, vol. 4, no. 2, pp. 441–461, 2013.
View at: Google Scholar
A. E. Hamza and S. M. Ahmed, “Existence and uniqueness of solutions of Hahn difference equations,” Advances in Difference Equations, vol. 2013, article 316, 2013.
View at: Publisher Site | Google Scholar
A. E. Hamza and S. D. Makharesh, “Leibniz’ rule and Fubinis theorem associated with Hahn difference operator,” Journal of Advanced Mathematical, vol. 12, no. 6, pp. 6335–6345, 2016.
View at: Google Scholar
T. Sitthiwirattham, “On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different q, ω -derivatives,” Advances in Difference Equations, vol. 2016, no. 1, article no. 116, 2016.
View at: Publisher Site | Google Scholar
J. Čermák and L. Nechvátal, “On (q,h)-analogue of fractional calculus,” Journal of Nonlinear Mathematical Physics, vol. 17, no. 1, pp. 51–68, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
J. Čermák, T. Kisela, and L. Nechvátal, “Discrete Mittag-Leffler functions in linear fractional difference equations,” Abstract and Applied Analysis, vol. 2011, Article ID 565067, 21 pages, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
M. R. S. Rahmat, “The (q,h)-Laplace transform on discrete time scales,” Computers and Mathematics with Applications, vol. 62, no. 1, pp. 272–281, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
M. R. S. Rahmat, “On some (q,h)-analogues of integral inequalities on discrete time scales,” Computers and Mathematics with Applications, vol. 62, no. 4, pp. 1790–1797, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
F. Du, B. Jia, L. Erbe, and A. Peterson, “Monotonicity and convexity for nabla fractional (q,h)-difference,” Journal of Difference Equations and Applications, vol. 22, no. 9, pp. 1224–1243, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
T. Brikshavana and T. Sitthiwirattham, “On fractional Hahn calculus,” Advances in Difference Equations, vol. 2017, article 354, 2017.
View at: Google Scholar
D. H. Griffel, Applied Functional Analysis, Ellis Horwood, Chichester, UK, 1981.
View at: MathSciNet
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, Florida, Fla, USA, 1988.

Copyright

Copyright © 2017 Nichaphat Patanarapeelert and Thanin Sitthiwirattham. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

901

Downloads

880

Citations