Abstract
This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence and uniqueness result is established by using the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main result.
MSC:26A33, 39A05, 39A10, 39A12.
Similar content being viewed by others
1 Introduction
For , such that is a nonnegative integer, we define and throughout this paper. It is also worth noting that, in what follows, for any Banach-valued function u defined on , we appeal to the convention , where with and θ is the zero element of a given Banach space.
In this paper, we will consider the existence of a unique solution to the following discrete fractional mixed type sum-difference equation boundary value problem in the Banach space E:
where , , denotes the discrete Riemann-Liouville fractional difference of order α, is continuous, θ represents the zero element of E, and
where , , .
Discrete fractional calculus is a generalization of ordinary difference and summation on arbitrary order that can be non-integer, and it has gained considerable popularity due mainly to its demonstrated applications in describing some real-world phenomena [1, 2]. Among all the topics, the branch of discrete fractional boundary value problems is currently undergoing active investigation; see, for example, [3–17] and the references therein.
Boundary value problems for differential equations in Banach spaces have been studied by many authors [18–36]. Especially for the study of nonlinear mixed type integro-differential equations which arise from many nonlinear problems in science [36], a series of excellent results have been obtained in recent years [21, 23, 30–36].
On the other hand, it is well known that discrete analogues of differential equations can be very useful in applications [37, 38], in particular for using computer to simulate the behavior of solutions for certain dynamic equations. However, compared to continuous case, significantly less is known about discrete difference calculus in Banach spaces [39–45]. Furthermore, as far as we know, the theory of discrete fractional mixed type sum-difference equations boundary value problems in Banach spaces is still a new research area. So, in this paper, we focus on this gap and provide some sufficient conditions for the existence and uniqueness of solutions to problem (1.1).
The remainder of this paper is organized as follows. Section 2 preliminarily presents some necessary basic knowledge for the theory of discrete fractional calculus in Banach spaces. In Section 3, the existence and uniqueness result for the solution to problem (1.1) will be established with the help of the contraction mapping principle. Finally, in Section 4, two concrete examples are provided to illustrate the possible applications of the established analytical result.
2 Preliminaries
In this section, we firstly present the definitions for the discrete Riemann-Liouville fractional difference and the discrete fractional sum for Banach-valued functions similar to the corresponding definitions for real-valued functions [46–49].
Definition 2.1 ([46])
For any t and ν, the falling factorial function is defined as
provided that the right-hand side is well defined. We appeal to the convention that if is a pole of the gamma function and is not a pole, then .
Definition 2.2 The ν th discrete fractional sum of a function , for , is defined by
Also, we define the trivial sum , .
Definition 2.3 The ν th discrete Riemann-Liouville fractional difference of a function , for , is defined by
where n is the smallest integer greater than or equal to ν and is the n th order forward difference operator. If , then .
Remark 2.1 From Definitions 2.2 and 2.3, it is easy to see that maps functions defined on to functions defined on and maps functions defined on to functions defined on , where n is the smallest integer greater than or equal to ν. Also, it is worth reminding the reader that the t in (or ) represents an input for the function (or ) and not for the function f. For ease of notation, throughout this paper we omit the subscript a in and when it does not lead to domain confusion and general ambiguity.
Now, we present the following two results, which are analogues to the ordinary case for the real-valued function.
Lemma 2.1 Let and . Then
Lemma 2.2 Let , and p be a positive integer. Then
Remark 2.2 Lemma 2.1 and Lemma 2.2 are natural analogues of Theorem 2.2 in [46] and Theorem 2.2 in [47] for real-valued functions. Their proofs are similar to the ordinary case. So, here we omit them. Additionally, by using Lemma 2.1, we can easily obtain the equality , holds, for any Banach-valued function f.
At last, we need to state the following lemmas, which will be important in the sequel.
Lemma 2.3 Let and . Then
where , , and n is the smallest integer greater than or equal to ν.
Proof By Definition 2.3, Lemma 2.1, Lemma 2.2 and Remark 2.2, we have
Setting , ; then we get (2.1). So the proof is complete. □
Lemma 2.4 ([48])
Let and be given. Then
for any t for which both sides are well defined. Furthermore, for , and ,
and
3 Main results
In this section, we establish the existence of a unique solution to problem (1.1). To accomplish this, we firstly list here the following conditions.
(C1) There exist constants and such that
(C2) , and there exist nonnegative numbers a, b, c and a function with such that
for , .
Next, we define
equipped with the norm
Furthermore, by means of the linear functional analysis theory, we can easily prove that is a Banach space.
Next, we state and prove the following lemmas, which will be used to establish the existence result of solutions to problem (1.1).
Lemma 3.1 If (C1) and (C2) hold, then, for any ,
Proof Setting in (C2), we have
So, for any , , using (C2) again produces
Summating both sides of (3.2), we can get (3.1). The proof is completed. □
Lemma 3.2 Let be given and , . The unique solution of
is
where
Proof Suppose that satisfies the equation of problem (3.3), then Lemma 2.3 implies that
for some , , . By , we get .
Furthermore, in view of Lemma 2.4, we have
Substituting in (3.5) gives .
Repeating the above steps with , we can get
Therefore,
By virtue of Lemma 2.4 again, we have
Using the condition in (3.7), we obtain
Now, substitution of into (3.6) gives
where is defined by (3.4). The proof is complete. □
Remark 3.1 From the expression of , we can easily find that and for .
With the above auxiliary results in hand, we now establish the main result as follows.
Theorem 3.1 If (C1), (C2) hold and
then problem (1.1) has a unique solution u in X.
Proof Define an operator by
where , and due to Lemma 3.1, we have
Therefore,
here and σ is defined by (3.8). So, the operator ℱ is well defined. Furthermore, from Lemma 3.2, we can transform problem (1.1) as an operator equation , and it is clear to see that u is a solution of problem (1.1) is equivalent to a fixed point of ℱ.
Next, for any , we denote
In view of (C2), we have
So we get
which, together with the assumption that , implies that ℱ is a contraction mapping. By means of the Banach contraction mapping principle, we get that ℱ has a unique fixed point in E; that is problem (1.1) has a unique solution. This completes the proof. □
4 Examples
In this section, we illustrate the possible application of the above established analytical result with the following two concrete examples.
Example 4.1 Consider the following problem:
Conclusion Problem (4.1) has a unique solution such that as for .
Proof Let . Evidently, is a Banach space with the norm for any . Then the infinite discrete fractional difference system (4.1) can be regarded as a boundary value problem of the form (1.1) in the Banach space E. In this situation, , , ,
and , in which
where and , , . From the expression of , it is easy to see that is continuous. Furthermore, for any , , we have
and therefore,
where , , , which imply that (C2) holds together with the following facts:
and
On the other hand, we can verify that
So (C1) is also satisfied. Finally, by a simple calculation, we can obtain
Thus, all the conditions of Theorem 3.1 are satisfied and our conclusion follows from Theorem 3.1. □
Example 4.2 Consider the following problem:
Here, represents the discrete Riemann-Liouville fractional difference of order for the function with respect to its first variable t.
Conclusion Problem (4.2) has a unique solution such that for each given , is continuous for .
Proof Let ; then is a Banach space equipped with the norm , . Define by ; then the discrete fractional partial difference system (4.2) can be transformed into the form of problem (1.1), where , ,
and
for . It is obvious that f is continuous.
Choosing , , and , ; then we can verify that , , , , and
holds for any , .
Clearly, all the conditions of Theorem 3.1 are fulfilled. Therefore, we can conclude that problem (4.2) has a unique solution. □
References
Wu G, Baleanu D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 2014, 75: 283-287. 10.1007/s11071-013-1065-7
Atıcı F, Şengül S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369: 1-9. 10.1016/j.jmaa.2010.02.009
Atıcı F, Eloe P: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241
Goodrich C: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Differ. Equ. 2010, 5: 195-216.
Goodrich C: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl. 2010, 59: 3489-3499. 10.1016/j.camwa.2010.03.040
Goodrich C: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041
Goodrich C: 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.2010.11.029
Goodrich C: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 2012, 18: 397-415. 10.1080/10236198.2010.503240
Goodrich C: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022
Goodrich C: On semipositone discrete fractional boundary value problems with non-local boundary conditions. J. Differ. Equ. Appl. 2013, 19: 1758-1780. 10.1080/10236198.2013.775259
Dahal R, Duncan D, Goodrich C: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 2014, 20: 473-491. 10.1080/10236198.2013.856073
Holm M:Solutions to a discrete, nonlinear, fractional boundary value problem. Int. J. Dyn. Syst. Differ. Equ. 2011, 3: 267-287.
Pan Y, Han Z, Sun S, Hou C: The existence of solutions to a class of boundary value problems with fractional difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 275
Ferreira R: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 2013, 19: 712-718. 10.1080/10236198.2012.682577
Chen F, Zhou Y: Existence and Ulam stability of solutions for discrete fractional boundary value problem. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 459161
Lv W: Existence of solutions for discrete fractional boundary value problems with a p -Laplacian operator. Adv. Differ. Equ. 2012., 2012: Article ID 163
Lv W: Solvability for discrete fractional boundary value problems with a p -Laplacian operator. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 679290
Liu Y: Boundary value problems for second order differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 2003, 135: 569-583. 10.1016/S0096-3003(02)00070-X
Zhang X, Feng M, Ge W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 2008, 69: 3310-3321. 10.1016/j.na.2007.09.020
Zhang X, Feng M, Ge W: Existence and nonexistence of positive solutions for a class of n th-order three-point boundary value problems in Banach spaces. Nonlinear Anal. 2009, 70: 584-597. 10.1016/j.na.2007.12.028
Zhang X, Feng M, Ge W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 2010, 233: 1915-1926. 10.1016/j.cam.2009.07.060
Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 2008, 222: 351-363. 10.1016/j.cam.2007.11.003
Feng M, Pang H: A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 2009, 70: 64-82. 10.1016/j.na.2007.11.033
Chen H, Li P: Existence of solutions of three-point boundary value problems in Banach spaces. Math. Comput. Model. 2009, 49: 780-788. 10.1016/j.mcm.2008.05.003
Chen H, Zhao Y: Triple positive solutions for nonlinear boundary value problems in Banach space. Comput. Math. Appl. 2009, 58: 1780-1787. 10.1016/j.camwa.2009.07.061
Jiang W, Wang B: Positive solutions for second-order multi-point boundary value problems in Banach spaces. Electron. J. Differ. Equ. 2011., 2011: Article ID 22
Hao X, Liu L, Wu Y, Xu N: Multiple positive solutions for singular n th-order nonlocal boundary value problems in Banach spaces. Comput. Math. Appl. 2011, 61: 1880-1890. 10.1016/j.camwa.2011.02.017
Su X: Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 2011, 74: 2844-2852. 10.1016/j.na.2011.01.006
Guo D, Lakshmikantham V: Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces. J. Math. Anal. Appl. 1988, 129: 211-222. 10.1016/0022-247X(88)90243-0
Guo D: A boundary value problem for n th-order integro-differential equations in a Banach space. Appl. Math. Comput. 2003, 136: 571-592. 10.1016/S0096-3003(02)00082-6
Guo D: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in a Banach space. Nonlinear Anal. 2004, 56: 985-1006. 10.1016/j.na.2003.10.023
Guo D: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 2005, 63: 618-641. 10.1016/j.na.2005.05.023
Xu Y, Zhang H: Multiple positive solutions of a boundary value problem for a class of 2 n th-order m -point singular integro-differential equations in Banach spaces. Appl. Math. Comput. 2009, 214: 607-617. 10.1016/j.amc.2009.04.021
Xu Y, Zhang H: Multiple positive solutions of a m -point boundary value problem for 2 n th-order singular integro-differential equations in Banach spaces. Nonlinear Anal. 2009, 70: 3243-3253. 10.1016/j.na.2008.04.026
Zhang L, Ahmad B, Wang G, Agarwal R: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.
Lakshmikantham V: Some problems in integro-differential equations of Volterra type. J. Integral Equ. 1985, 10: 137-146.
Hilscher R, Zeidan V: Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey. J. Differ. Equ. Appl. 2005, 11: 857-875. 10.1080/10236190500137454
Kelley W, Peterson A: Difference Equations: An Introduction with Applications. Academic Press, New York; 1991.
Agarwal R, O’Regan D: Difference equations in Banach spaces. J. Aust. Math. Soc. A 1998, 64: 277-284. 10.1017/S1446788700001762
Agarwal R, O’Regan D: A fixed-point approach for nonlinear discrete boundary value problems. Comput. Math. Appl. 1998, 36: 115-121.
González C, Jiménez-Melado A: An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions of difference equations in Banach spaces. J. Math. Anal. Appl. 2000, 247: 290-299. 10.1006/jmaa.2000.6877
Tabor J: Oscillation of linear difference equations in Banach spaces. J. Differ. Equ. 2003, 192: 170-187. 10.1016/S0022-0396(03)00040-8
Bay N, Phat V: Stability analysis of nonlinear retarded difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 951-960. 10.1016/S0898-1221(03)00068-3
González C, Jiménez-Melado A: Set-contractive mappings and difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 1235-1243. 10.1016/S0898-1221(03)00094-4
Agarwal R, Thompson H, Tisdell C: Difference equations in Banach spaces. Comput. Math. Appl. 2003, 45: 1437-1444. 10.1016/S0898-1221(03)00100-7
Atıcı F, Eloe P: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2: 165-176.
Atıcı F, Eloe P: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137: 981-989.
Holm M: Sum and difference compositions in discrete fractional calculus. CUBO 2011, 13: 153-184. 10.4067/S0719-06462011000300009
Miller K, Ross B: Fractional difference calculus. In Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications. Nihon University, Koriyama; 1989:139-152.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The research is supported by the National Natural Science Foundation of China 11261032, the Longdong University Grant XYZK-1207 and XYZK-1402.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Lv, W., Feng, J. Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces. Adv Differ Equ 2014, 184 (2014). https://doi.org/10.1186/1687-1847-2014-184
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-184