Abstract
In this paper, we discuss the existence of positive solutions for nonlocal q-integral boundary value problems of fractional q-difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem, some existence results of positive solutions are obtained. In addition, some examples to illustrate our results are given.
MSC:39A13, 34B18, 34A08.
Similar content being viewed by others
1 Introduction
Studies on q-difference equations appeared already at the beginning of the twentieth century in intensive works especially by Jackson [1], Carmichael [2] and other authors such as Poincare, Picard, Ramanujan. Up to date, q-difference equations have evolved into a multidisciplinary subject; for example, see [3–6] and the references therein. For some recent work on q-difference equations, we refer the reader to the papers [7–20], and basic definitions and properties of q-difference calculus can be found in the book [21]. On the other hand, fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. For details, see [22, 23]. Many researchers studied the existence of solutions to fractional boundary value problems; see, for example, [24–33] and the references therein.
The fractional q-difference calculus had its origin in the works by Al-Salam [34] and Agarwal [35]. More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q-difference calculus were made, specifically, q-analogues of the integral and differential fractional operators properties such as the Mittag-Leffler function, the q-Laplace transform, and q-Taylor’s formula [12, 21, 36, 37], just to mention some.
However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stage and many aspects of this theory need to be explored. Recently, there have been some paper considering the existence of solutions to boundary value problems of fractional q-difference equations, for example, [10, 16–20, 38] and the references therein.
In [16], Ferreira considered the Dirichlet type nonlinear q-difference boundary value problem
By applying a fixed point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
In [20], Graef and Kong investigated the boundary value problem with fractional q-derivatives
where is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of λ.
Furthermore, Ahmad, Ntouyas, and Purnaras [10] studied the following nonlinear fractional q-difference equation with nonlocal boundary conditions:
where is the fractional q-derivative of the Caputo type and . The existence of solutions for the problem is shown by applying some well-known tools of fixed point theory such as Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and the Leray-Schauder nonlinear alternative.
In this paper, we deal with the following nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation:
where , , , , and is a parameter, is the q-derivative of Riemann-Liouville type of order α, is continuous, in which . To the authors’ knowledge, no one has studied the existence of positive solutions for the fractional q-difference boundary value problem (1.1). In the present work, we gave the corresponding Green’s function of the boundary value problem (1.1) and its properties. By using the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem, some existence results of positive solutions to the above boundary value problems are enunciated.
2 Preliminaries on q-calculus and lemmas
For the convenience of the reader, below we recall some known facts on fractional q-calculus. The presentation here can be found in, for example, [1, 21, 36, 37].
Let and define
The q-analogue of the power function with is
More generally, if , then
Clearly, if , then . The q-gamma function is defined by
and satisfies .
The q-derivative of a function f is defined by
and the q-derivatives of higher order by
The q-integral of a function f defined in the interval is given by
provided the sum converges absolutely.
If and f is defined in the interval , then its integral from a to b is defined by
Obviously, if on , then .
Similar as done for derivatives, an operator is given by
The fundamental theorem of calculus applies to these operators and , i.e.,
and if f is continuous at , then
The following formulas will be used later, namely, the integration by parts formula
and
where denotes the derivative with respect to the variable t.
Definition 2.1 Let and f be a function defined on . The fractional q-integral of Riemann-Liouville type is and
Definition 2.2 The fractional q-derivative of the Riemann-Liouville type of order is defined by and
where is the smallest integer greater than or equal to α.
Lemma 2.3 Assume that and , then .
Lemma 2.4 Let and f be a function defined on . Then the following formulas hold:
-
(1)
,
-
(2)
.
Lemma 2.5 ([16])
Let and n be a positive integer. Then the following equality holds:
Lemma 2.6 ([36])
Let , , the following is valid:
Particularly, for , , using q-integration by parts, we have
Obviously, we have .
In order to define the solution for the problem (1.1), we need the following lemmas.
Lemma 2.7 Let . Then, for a given , the unique solution of the boundary value problem
subject to the boundary condition
is given by
where
and
Proof Since , we take . In view of Definition 2.1 and Lemma 2.4, we have
Then it follows from Lemma 2.5 that the solution of (2.7) and (2.8) is given by
for some constants . Since , we have .
Using the Riemann-Liouville integral of order β for (2.13), we have
where we have used Lemma 2.4 and Lemma 2.6. Using the boundary condition , we get
Hence, we have
This completes the proof of the lemma. □
Remark 2.8 For the special case where , Lemma 2.7 has been obtained by Ferreira [16].
Lemma 2.9 ([16])
The function defined by (2.11) satisfies the following properties:
Lemma 2.10 The function defined by (2.10) satisfies the following properties:
-
(i)
G is a continuous function and for .
-
(ii)
There exists a positive function such that
where
Proof It is easy to prove that the statement (i) holds. On the other hand, we note that defined by (2.11) is decreasing with respect to t for and increasing with respect to t for . Hence, we have
The proof is completed. □
3 The main results
Let be a Banach space endowed with the norm . Define the cone by .
Define the operator as follows:
It follows from the nonnegativeness and continuity of G and f that the operator satisfies and is completely continuous.
Theorem 3.1 Suppose that is continuous and there exists a function such that
Then the BVP (1.1) has a unique positive solution provided
Proof We will prove that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.
By (2.10), (2.11), and (2.12), for , we obtain the estimate
where .
Consequently,
where .
By introduction, we have
According to (3.3), we can choose m sufficiently large such that
which implies
Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution. □
Remark 3.2 When is a constant, the condition (3.2) reduces to a Lipschitz condition.
For the sake of convenience, we set
where , with , .
Theorem 3.3 Suppose that there exists such that
() is continuous and nondecreasing relative to u, and
Then the BVP (1.1) has one positive solution satisfying
Proof We denote . In what follows, we first show that .
Let ; then . By assumption (), we have
Hence, for any ,
and
Thus, we get .
Let , ; then . Let ; then . We denote , . According to , we have , . Since T is completely continuous, we assert that has a convergent subsequence and there exists such that . Since , then . According to the definition of T and (), we have
which implies
By introduction, we have for , . Thus, there exists such that . From the continuity of T and , we have . The proof is completed. □
Our next existence result is based on Krasnoselskii’s fixed point theorem [39].
Lemma 3.4 (Krasnoselskii’s)
Let E be a Banach space, and let be a cone. Assume , are open subsets of E with , and let be a completely continuous operator such that
Then T has at least one fixed point in .
Theorem 3.5 Let be a nonnegative continuous function on . In addition, we assume that
(H1) There exists a positive constant such that
where , with , , and
(H2) There exists a positive constant with such that
Then the BVP (1.1) has at least one positive solution satisfying .
Proof By Lemma 2.9, we obtain that . Let . For any , according to (H1) and the definitions of and , we obtain
Let . For any , by (H2) and Lemma 2.10, we have
Now, an application of Lemma 3.4 concludes the proof. □
Theorem 3.6 Assume that there exist positive numbers such that
(H3) for and for , ; or
(H4) for and for , ,
where , , κ are given in (H1).
Then the BVP (1.1) has at least n positive solutions with , .
Proof Suppose that the condition (H3) holds. According to the continuity of f, for every pair , there exists with such that
It follows from Theorem 3.5 that every pair presents a positive solution of the BVP (1.1) such that , .
When the condition (H4) holds, the proofs are similar to those in the case (H3). The proof is completed. □
4 Examples
Example 4.1
The fractional boundary value problem
has a unique positive solution.
Proof In this case, , , , . Let
and . It is easy to prove that
By simple calculation, we get
and
which implies that
Obviously, for any , we have
Thus Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution. □
Example 4.2
Consider the following fractional boundary value problem:
where , , , . Choosing , , then , .
A simple computation showed . By Lemma 2.6 and with the aid of a computer, we obtain that
and
Let . Take , , then satisfies
-
(i)
is continuous and nondecreasing relative to u;
-
(ii)
;
-
(iii)
.
So, by Theorem 3.3, the problem (4.2) has one positive solution satisfying
Example 4.3
Consider the following fractional boundary value problem:
where , , . Choosing , , then , .
By calculation, we get . By Lemma 2.6 and with the aid of a computer, we obtain that
and
Let . Take and , . Then satisfies
-
(i)
is continuous;
-
(ii)
, ;
-
(iii)
, .
So, by Theorem 3.5, the problem (4.3) has at least one positive solution with .
References
Jackson FH: q -difference equations. Am. J. Math. 1910, 32: 305-314. 10.2307/2370183
Carmichael RD: The general theory of linear q -difference equations. Am. J. Math. 1912, 34: 147-168. 10.2307/2369887
Finkelstein R, Marcus E: Transformation theory of the q -oscillator. J. Math. Phys. 1995, 36: 2652-2672. 10.1063/1.531057
Freund PGO, Zabrodin AV: The spectral problem for the q -Knizhnik-Zamolodchikov equation and continuous q -Jacobi polynomials. Commun. Math. Phys. 1995, 173: 17-42. 10.1007/BF02100180
Han G, Zeng J: On a q -sequence that generalizes the median Genocchi numbers. Ann. Sci. Math. Qué. 1999, 23: 63-72.
Floreanini R, Vinet L: Quantum symmetries of q -difference equations. J. Math. Phys. 1995, 36: 3134-3156. 10.1063/1.531017
Bangerezako G: Variational q -calculus. J. Math. Anal. Appl. 2004, 289: 650-665. 10.1016/j.jmaa.2003.09.004
Dobrogowska A, Odzijewicz A: Second order q -difference equations solvable by factorization method. J. Comput. Appl. Math. 2006, 193: 319-346. 10.1016/j.cam.2005.06.009
Ismail MEH, Simeonov P: q -difference operators for orthogonal polynomials. J. Comput. Appl. Math. 2009, 233: 749-761. 10.1016/j.cam.2009.02.044
Ahmad B, Ntouyas SK, Purnaras I: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140
El-Shahed M, Hassan HA: Positive solutions of q -difference equation. Proc. Am. Math. Soc. 2010, 138: 1733-1738.
Annaby MH, Mansour ZS: q -Taylor and interpolation series for Jackson q -difference operators. J. Math. Anal. Appl. 2008, 344: 472-483. 10.1016/j.jmaa.2008.02.033
Ahmad B, Ntouyas SK: Boundary value problems for q -difference inclusions. Abstr. Appl. Anal. 2011., 2011: Article ID 292860
Gasper G, Rahman M: Some systems of multivariable orthogonal q -Racah polynomials. Ramanujan J. 2007, 13: 389-405. 10.1007/s11139-006-0259-8
Gauchman H: Integral inequalities in q -calculus. Comput. Math. Appl. 2004, 47: 281-300. 10.1016/S0898-1221(04)90025-9
Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 70
Ferreira RAC: Positive solutions for a class of boundary value problems with fractional q -differences. Comput. Math. Appl. 2011, 61: 367-373. 10.1016/j.camwa.2010.11.012
El-Shahed M, Al-Askar F: Positive solutions for boundary value problem of nonlinear fractional q -difference equation. ISRN Math. Anal. 2011., 2011: Article ID 385459
Liang S, Zhang J: Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q -differences. J. Appl. Math. Comput. 2012, 40: 277-288. 10.1007/s12190-012-0551-2
Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006
Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Kibas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Li C, Luo X, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029
Wang G, Ahmad B, Zhang L: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 2011, 62: 1389-1397. 10.1016/j.camwa.2011.04.004
Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 2010, 217: 480-487. 10.1016/j.amc.2010.05.080
Staněk S: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048
Zhao Y, Huang L, Wang X, Zhu X: Existence of solutions for fractional integro-differential equation with multipoint boundary value problem in Banach spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 172963
Zhao Y, Chen H, Huang L: Existence of positive solutions for nonlinear fractional functional differential equation. Comput. Math. Appl. 2012, 64: 3456-3467. 10.1016/j.camwa.2012.01.081
Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052
Goodrich CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022
Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1300-1309. 10.1016/j.camwa.2009.06.034
Zhang X, Liu L, Wu Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 2012, 55: 1263-1274. 10.1016/j.mcm.2011.10.006
Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 15: 135-140. 10.1017/S0013091500011469
Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060
Rajković PM, Marinković SD, Stanković MS: Fractional integrals and derivatives in q -calculus. Appl. Anal. Discrete Math. 2007, 1: 311-323. 10.2298/AADM0701311R
Annaby MH, Mansour IS: q-Fractional Calculus and Equations. Springer, Berlin; 2012.
Zhao Y, Ye G, Chen H: Multiple positive solutions of a singular semipositone integral boundary value problem for fractional q -derivatives equation. Abstr. Appl. Anal. 2013., 2013: Article ID 643571
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors are highly grateful for the referee’s careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (12JJ2004, 13JJ3106), and the Scientific Research Fund of Hunan Provincial Education Department (12B034).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, YZ, HC, and QZ, contributed to each part of this study equally and read and approved the final version of the manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhao, Y., Chen, H. & Zhang, Q. Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions. Adv Differ Equ 2013, 48 (2013). https://doi.org/10.1186/1687-1847-2013-48
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-48