- Open Access
Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivatives
© Zhao et al.; licensee Springer. 2013
- Received: 17 January 2013
- Accepted: 12 April 2013
- Published: 25 April 2013
In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given.
MSC:34A08, 34B18, 39A13.
- fractional q-difference equation
- nonhomogeneous boundary value problem
- positive solution
Fractional differential equations have attracted considerable interest because of its demonstrated applications in various fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability [1, 2]. Many researchers have studied the existence of solutions (or positive solutions) to fractional boundary value problems; for example, see [3–10] and the references therein.
The early work on q-difference calculus or quantum calculus dates back to Jackson’s papers , basic definitions and properties of quantum calculus can be found in the book . For some recent existence results on q-difference equations, we refer to [13–15] and the references therein.
The fractional q-difference calculus had its origin in the works by Al-Salam  and Agarwal . More recently, there seems to be new interest in the study of this subject and many new developments were made in this theory of fractional q-difference calculus [18–22]. Specifically, fractional q-difference equations have attracted the attentions of several researchers. Some recent work on the existence theory of fractional q-difference equations can be found in [20, 23–31]. However, the study of boundary value problems for nonlinear fractional q-difference equations is still in the initial stage and many aspects of this topic need to be explored.
where and is the fractional q-derivatives of the Caputo type.
where is a parameter, and the uniqueness, existence and nonexistence of positive solutions are considered in terms of different ranges of λ.
where is the fractional q-derivative of the Caputo type, and .
where , is the q-derivative of Riemann-Liouville type of order α, is continuous and semipositone, and may be singular at .
where , , , , and λ is a parameter, is the q-derivative of Riemann-Liouville type of order α, is continuous. 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 and Krasnoselskii’s fixed-point theorem, the uniqueness, existence, and multiplicity of positive solution to the BVP (1.1) are obtained in term of the explicit intervals for the nonhomogeneous term. Our results are different from those of [25, 27].
For the convenience of the reader, below we cite some definitions and fundamental results on q-calculus as well as the fractional q-calculus. The presentation here can be found in, for example, [12, 18, 20, 22].
and satisfies .
where denotes the derivative with respect to the variable t.
where is the smallest integer greater than or equal to α.
Lemma 2.3 ()
Assume that and , then .
Lemma 2.5 ()
Lemma 2.6 ()
In order to define the solution for the problem (1.1), we need the following lemmas.
for some constants . From , we have .
This completes the proof of the lemma. □
, and for all .
for all .
which implies that is an increasing function with respect to t. It is clear that is increasing in t. Therefore, is an increasing function of t for all , and so .
which implies that part (ii) holds. This completes the proof of the lemma. □
According to , we may take , .
Let be a Banach space endowed with the norm . Define the cone by .
If, in addition, on , then the conclusion is true for .
Proof We will show that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.
Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution for any . If , then the condition on and Lemma 2.8 imply that in . This completes the proof of the theorem. □
Remark 3.2 When is a constant, the condition (3.2) reduces to a Lipschitz condition.
Our next existence results is based on Krasnoselskii’s fixed-point theorem .
Lemma 3.3 (Krasnoselskii’s)
, and , , or
, and , .
Then T has at least one fixed point in .
Obviously, K is a cone of nonnegative functions in X.
Lemma 3.4 The operator is completely continuous.
Hence, we have .
which implies that is bounded.
By means of Arzela-Ascoli theorem, is completely continuous.
Theorem 3.5 Suppose that there exists two positive numbers such that one of the following conditions is satisfied
() , ;
() , .
Then the BVP (1.1) has at least one positive solution , such that for . If, in addition, on , then the conclusion is true for .
By Lemma 3.3, the operator T has at least one fixed point , and . Since , , then, the solution is positive for . As in the proof of Theorem 3.1, is a positive solution for . This completes the proof of the theorem. □
Theorem 3.6 Suppose that there exists three positive numbers such that one of the following conditions is satisfied
() , , ;
() , , .
Then the BVP (1.1) has at least two positive solutions such that for . If, in addition, on , then the conclusion is true for .
From Theorem 3.5, the operator T has two fixed point , with . Therefore, the BVP (1.1) has at least two positive solutions for . As in the proof of Theorem 3.1, , are two positive solutions for . This completes the proof of the theorem. □
Denote the integer part of m by . Generally, we have the following theorem.
Theorem 3.7 Suppose that there exists positive numbers such that one of the following conditions is satisfied:
() , , ;
() , , .
Then the BVP (1.1) has at least m positive solutions , , such that for . If, in addition, on , then the conclusion is true for .
has a unique positive solution for any .
Thus, Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution for any . □
where , , , . Choosing , then .
So, by Theorem 3.5, the problem (4.2) has one positive solution such that for .
Dedicated to Professor Hari M Srivastava.
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (Grant No. 11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ3106, 12JJ2004), and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12B034).
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Kibas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Al-Refai M, Ali Hajji M: Monotone iterative sequences for nonlinear boundary value problems of fractional order. Nonlinear Anal. 2011, 74: 3531-3539. 10.1016/j.na.2011.03.006MathSciNetView ArticleGoogle Scholar
- 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.2010.05.080MathSciNetView ArticleGoogle Scholar
- Stanék S: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048MathSciNetView ArticleGoogle Scholar
- 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.2012.01.081MathSciNetView ArticleGoogle Scholar
- Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MathSciNetView ArticleGoogle Scholar
- Goodrich CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022MathSciNetView ArticleGoogle Scholar
- 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.2009.06.034MathSciNetView ArticleGoogle Scholar
- 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.2011.10.006MathSciNetView ArticleGoogle Scholar
- Jackson FH: q -Difference equations. Am. J. Math. 1970, 32: 305-314.View ArticleGoogle Scholar
- Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.View ArticleGoogle Scholar
- Ernst, T: The history of q-calculus and a new method. UUDM Report 2000:16, Department of Mathematics, Uppsala University, 2000, ISSN:1101-3591Google Scholar
- Ahmad B: Boundary-value problems for nonlinear third-order q -difference equations. Electron. J. Differ. Equ. 2011., 2011: Article ID 94Google Scholar
- Goodrich CS: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217: 4740-4753. 10.1016/j.amc.2010.11.029MathSciNetView ArticleGoogle Scholar
- Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 15(2):135-140. 10.1017/S0013091500011469MathSciNetView ArticleGoogle Scholar
- Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060View ArticleGoogle Scholar
- Annaby MH, Mansour IS: q-Fractional Calculus and Equations. Springer, Berlin; 2012.View ArticleGoogle Scholar
- Atici FM, Eloe PW: Fractional q -calculus on a time scale. J. Nonlinear Math. Phys. 2007, 14: 333-344.MathSciNetView ArticleGoogle Scholar
- Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.View ArticleGoogle Scholar
- Goodrich CS: Continuity of solutions to discrete fractional initial value problem. Comput. Math. Appl. 2010, 59: 3489-3499. 10.1016/j.camwa.2010.03.040MathSciNetView ArticleGoogle Scholar
- Rajković PM, Marinković SD, Stanković MS: Fractional integrals and derivatives in q -calculus. Appl. Anal. Discrete Math. 2007, 1: 311-323. 10.2298/AADM0701311RMathSciNetView ArticleGoogle Scholar
- Atici FM, Eloe PW: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241MathSciNetView ArticleGoogle Scholar
- 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.2010.11.012MathSciNetView ArticleGoogle Scholar
- 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 385459Google Scholar
- 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/s12190-012-0551-2MathSciNetView ArticleGoogle Scholar
- 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.2012.03.006MathSciNetView ArticleGoogle Scholar
- Ma J, Yang J: Existence of solutions for multi-point boundary value problem of fractional q -difference equation. Electron. J. Qual. Theory Differ. Equ. 2011, 92: 1-10.MathSciNetView ArticleGoogle Scholar
- Ahmad B, Ntouyas S, Purnaras I: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140Google Scholar
- El-Shahed M, Gaber M: Two-dimensional q -differential transformation and its application. Appl. Math. Comput. 2011, 217: 9165-9172. 10.1016/j.amc.2011.03.152MathSciNetView ArticleGoogle Scholar
- Zhao Y, Chen H, Zhang Q: Existence results for fractional q -difference equations with nonlocal q -integral boundary conditions. Adv. Differ. Equ. 2013., 2013: Article ID 48Google Scholar
- 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 643571Google Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.Google Scholar