Abstract
In this article, we study the existence and uniqueness of solutions for multi-strip fractional q-integral boundary value problems of nonlinear fractional q-difference equations. By using the Banach contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory some interesting results are obtained. Some examples are presented to illustrate the results.
MSC:34A08, 34B18, 39A13.
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1 Introduction
In this article, we investigate the following nonlinear fractional q-difference equation for multi-strip fractional q-integral boundary condition:
where , , , , for all are given constants, is the fractional q-derivative of Riemann-Liouville type of order α, is the fractional -integral of order and is a continuous function.
q-Difference calculus or quantum calculus was initiated by Jackson [1]. Basic definitions and properties of quantum calculus can be found in the book [2]. The fractional q-difference calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. For some recent work on the subject, we refer to [5–12] and the references cited therein.
Strip conditions appear in the mathematical modeling of certain real world problems. For motivation, discussion on multi-strip boundary conditions, examples and a consistent bibliography on these problems, we refer to the papers [13–20] and the references therein. As it is pointed out in [20], the boundary condition in (1.1) can be interpreted in the sense that a controller at the right-end of the considered interval is influenced by a discrete distribution of finite many nonintersecting strips of arbitrary length expressed in terms of fractional integral boundary conditions.
The significance of investigating problem (1.1) is that the multi-strip fractional q-integral boundary condition is very general and includes many conditions as special cases. In particular, if for , then the condition of (1.1) is reduced to the multi-strip q-integral condition as follows:
Moreover, we emphasize that we have different quantum numbers and as far as we know this is new in the literature.
The rest of the paper is organized as follows. In Section 2 we briefly give some basic notations, definitions and lemmas. In Section 3 we collect some auxiliary results needed in the proofs of our main results. Section 4 contains the main results concerning existence and uniqueness results for problem (1.1), which are shown by applying the Banach contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory. Some examples are presented in Section 5 to illustrate the results.
2 Preliminaries
To make this paper self-contained, below we recall some known facts on fractional q-calculus. The presentation here can be found in, for example, [21, 22].
For , define
The q-analogue of the power function with is
More generally, if , then
Note if , then . We also use the notation for . The q-gamma function is defined by
Obviously, .
The q-derivative of a function h is defined by
and q-derivatives of higher order are given by
The q-integral of a function h defined on the interval is given by
If and h is defined in the interval , then its integral from a to b is defined by
Similar to derivatives, an operator is given by
The fundamental theorem of calculus applies to these operators and , i.e.,
and if h is continuous at , then
Definition 2.1 Let and h be a function defined on . The fractional q-integral of Riemann-Liouville type is given by and
Definition 2.2 The fractional q-derivative of Riemann-Liouville type of order is defined by and
where l is the smallest integer greater than or equal to ν.
Definition 2.3 For any ,
is called the q-beta function.
From [2], the expression of q-beta function in terms of the q-gamma function can be written as
Lemma 2.4 [4]
Let and f be a function defined in . Then the following formulas hold:
-
(1)
,
-
(2)
.
Lemma 2.5 [22]
Let and ν be a positive integer. Then the following equality holds:
3 Some auxiliary lemmas
Lemma 3.1 Let and . Then we have
Proof Using the definitions of q-analogue of power function and q-beta function, we have
The proof is complete. □
Lemma 3.2 Let and . Then we have
Proof Taking into account Lemma 3.1, we have
This completes the proof. □
For convenience, we set a nonzero constant
Lemma 3.3 Let , , , and for all . Then, for a given , the unique solution of the linear q-difference equation
subject to the multi-strip fractional q-integral condition
is given by
where Λ is defined by (3.3).
Proof Since , we take . In view of Definition 2.2 and Lemma 2.4, the linear q-difference equation (3.4) can be written as
Using Lemma 2.5, we obtain
for some constants . Since , we get .
Applying the Riemann-Liouville fractional -integral of order with for (3.7) and taking into account Lemma 3.1, we have
Repeating the above process with and using the second condition of (3.5), we get a constant as follows:
Substituting the values of constants and in the linear solution (3.7), the desired result in (3.6) is obtained. □
4 Main results
Let denote the Banach space of all continuous functions from to ℝ endowed with the supremum norm defined by . In view of Lemma 3.3, we define an operator by
with . It should be noticed that problem (1.1) has solutions if and only if the operator has fixed points.
For the sake of convenience, we put
The first existence and uniqueness result is based on the Banach contraction mapping principle.
Theorem 4.1 Let be a continuous function satisfying the assumption
(H1) there exists a constant such that for each and .
If
where a constant Φ is given by (4.2), then the multi-strip boundary value problem (1.1) has a unique solution on .
Proof We transform problem (1.1) into a fixed point problem, , where the operator is defined by (4.1). Applying the Banach contraction mapping principle, we will show that the operator has a fixed point which is a unique solution of problem (1.1).
Setting and choosing
with L Φ satisfying (4.3), we will show that , where the set . For any , and taking into account Lemma 3.2, we have
It follows that .
For and for each , we have
The above result leads to . As , by (4.3), therefore is a contraction. Hence, by the Banach contraction mapping principle, we deduce that has a fixed point which is the unique solution of problem (1.1). □
Next, we prove the existence of at least one solution by using Krasnoselskii’s fixed point theorem.
Lemma 4.2 (Krasnoselskii’s fixed point theorem [23])
Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that .
Theorem 4.3 Assume that is a continuous function satisfying assumption (H1). In addition, we suppose that
(H2) , and .
If the following condition holds
then the multi-strip boundary value problem (1.1) has at least one solution on .
Proof We define and choose a suitable constant R such that
where Φ is defined by (4.2). Furthermore, we define the operators and on by
and
It should be noticed that .
For any , we have
Therefore . Obviously, condition (4.4) implies that is a contraction mapping.
Finally, we will show that is compact and continuous. The continuity of f coupled with assumption (H2) implies that the operator is continuous and uniformly bounded on . We define . For , and , we have
Actually, as the right-hand side of the above inequality tends to zero independently of u. So is relatively compact on . Therefore, by the Arzelá-Ascoli theorem, is compact on . Thus all the assumptions of Lemma 4.2 are satisfied. Thus, the boundary value problem (1.1) has at least one solution on . The proof is complete. □
Remark 4.4 In the above theorem we can interchange the roles of the operators and to obtain the second result replacing (4.4) by the following condition:
Now, our third existence result is based on Leray-Schauder’s nonlinear alternative.
Lemma 4.5 (Nonlinear alternative for single-valued maps [24])
Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
-
(i)
F has a fixed point in , or
-
(ii)
there is (the boundary of U in C) and with .
Theorem 4.6 Assume that is a continuous function. In addition we suppose that:
(H3) there exist a continuous nondecreasing function and a function such that
(H4) there exists a constant such that
where Φ is defined by (4.2).
Then the multi-strip boundary value problem (1.1) has at least one solution on .
Proof Firstly, we will show that the operator defined by (4.1) maps bounded sets (balls) into bounded sets in . For a positive number ρ, let be a bounded ball in . Then, for , we have
Therefore, we deduce that .
Secondly, we will show that maps bounded sets into equicontinuous sets of . Let , with and . Then we have
Obviously, the right-hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Let u be a solution of problem (1.1). Then, for , and following similar computations as in the first step with (H3), we have
Consequently, we have
In view of (H4), there exists a constant such that . Let us set
Note that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by nonlinear alternative of Leray-Schauder type (Lemma 4.5), we deduce that has a fixed point in , which is a solution of the boundary value problem (1.1). This completes the proof. □
As the forth result, we prove the existence of solutions of (1.1) by using Leray-Schauder degree theory.
Theorem 4.7 Let be a continuous function. Assume that
(H5) there exist constants , where Φ are given by (4.2), and such that
Then the multi-strip boundary value problem (1.1) has at least one solution on .
Proof Let be the operator defined by (4.1). We will prove that there exists at least one solution of the operator equation .
Setting a ball , where a constant radius , by
it is sufficient to show that satisfies
Now, we set
As shown in Theorem 4.6, we have that the operator is continuous, uniformly bounded and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map is completely continuous. If (4.5) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that
where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, we have for at least one . Let us assume that for some . Then, for all , we have
Taking norm and solving for , we get
Choosing , then we deduce that (4.5) holds. This completes the proof. □
5 Examples
In this section, we present some examples to illustrate our results.
Example 5.1 Consider the following multi-strip fractional q-integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , and . Since
then (H1) is satisfied with . Using the Maple program, we find that
Therefore, we get
Hence, by Theorem 4.1, the boundary value problem (5.1) has a unique solution on .
Example 5.2 Consider the following multi-strip fractional q-integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , , , , , and . By using the Maple program, we find that
Clearly,
Choosing and , we can show that
which implies that . Hence, by Theorem 4.6, the boundary value problem (5.2) has at least one solution on .
Example 5.3 Consider the following multi-strip fractional q-integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . By using the Maple program, we find that
We observe that
Therefore, we have and
Hence, by Theorem 4.7, the boundary value problem (5.3) has at least one solution on .
Authors’ information
The fourth author is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
References
Jackson FH: q -Difference equations. Am. J. Math. 1970, 32: 305-314.
Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.
Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 15(2):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
Ernst, T: The history of q-calculus and a new method. UUDM Report 2000:16, Department of Mathematics, Uppsala University (2000)
Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.
Goodrich CS: 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
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.
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
Ahmad B, Ntouyas SK, Purnaras IK: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140
Ahmad B, Ntouyas SK: Existence of solutions for nonlinear fractional q -difference inclusions with nonlocal Robin (separated) conditions. Mediterr. J. Math. 2013, 10: 1333-1351. 10.1007/s00009-013-0258-0
Li X, Han Z, Sun S: Existence of positive solutions of nonlinear fractional q -difference equation with parameter. Adv. Differ. Equ. 2013., 2013: Article ID 260
Ahmad B, Ntouyas SK: Existence of solutions for fractional differential inclusions with nonlocal strip conditions. Arab J. Math. Sci. 2012, 18: 121-134. 10.1016/j.ajmsc.2012.01.005
Ahmad B, Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with strip conditions. Bound. Value Probl. 2012., 2012: Article ID 55
Ahmad B, Ntouyas SK: Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differ. Equ. 2012, 98: 1-22.
Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415
Ahmad B, Ntouyas SK, Alsaedi A, Al-Hutami H: Nonlinear q -fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2014, 26: 1-12.
Ahmad B, Nieto JJ, Alsaedi A, Al-Hutami H: Existence of solutions for nonlinear fractional q -difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 2014, 351: 2890-2909. 10.1016/j.jfranklin.2014.01.020
Alsaedi A, Ahmad B, Al-Hutami H: A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 410505
Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013, 20: 1-19.
Annaby MH, Mansour ZS Lecture Notes in Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.
Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.
Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.
Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.
Acknowledgements
The research of J. Tariboon and S. Asawasamrit is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Pongarm, N., Asawasamrit, S., Tariboon, J. et al. Multi-strip fractional q-integral boundary value problems for nonlinear fractional q-difference equations. Adv Differ Equ 2014, 193 (2014). https://doi.org/10.1186/1687-1847-2014-193
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DOI: https://doi.org/10.1186/1687-1847-2014-193