Existence of Solutions for Fractional q-Integrodifference Equations with Nonlocal Fractional q-Integral Conditions

Abstract

We study a class of fractional q-integrodifference equations with nonlocal fractional q-integral boundary conditions which have different quantum numbers. By applying the Banach contraction principle, Krasnoselskii's fixed point theorem, and Leray-Schauder nonlinear alternative, the existence and uniqueness of solutions are obtained. In addition, some examples to illustrate our results are given.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Suphawat Asawasamrit 1 and Jessada Tariboon 1 and Sotiris K. Ntouyas 2

Academic Editor:H. M. Srivastava

1, Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
2, Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 May 2014; Revised 22 June 2014; Accepted 22 June 2014; 9 July 2014

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.

1. Introduction

In this paper, we deal with the following nonlocal fractional q -integral boundary value problem of nonlinear fractional q -integrodifference equation: [figure omitted; refer to PDF] where 0 < p , q , r , z < 1 , 1 < α ...4; 2 , β , γ , δ > 0 , λ ∈ R are given constants, ^{D q α} is the fractional q -derivative of Riemann-Liouville type of order α , ^{I [varphi] ψ} is the fractional [varphi] -integral of order ψ with [varphi] = p , r , z , and ψ = β , γ , δ , f : [ 0 , T ] × R × R [arrow right] R is a continuous function.

The early work on q -difference calculus or quantum calculus dates back to Jackson's paper [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]. Motivated by recent interest in the study of fractional-order differential equations, the topic of q -fractional equations has attracted the attention of many researchers. The details of some recent development of the subject can be found in ([5-17]) and the references cited therein, whereas the background material on q -fractional calculus can be found in a recent book [18].

In this paper, we will study the existence and uniqueness of solutions of a class of boundary value problems for fractional q -integrodifference equations with nonlocal fractional q -integral conditions which have different quantum numbers. So, the novelty of this paper lies in the fact that there are four different quantum numbers . In addition, the boundary condition of (1) does not contain the value of unknown function x at the right side of boundary point t = T . One may interpret the q -integral boundary condition in (1) as the q -integrals with different quantum numbers are related through a real number λ .

The paper is organized as follows. In Section 2, for the convenience of the reader, we cite some definitions and fundamental results on q -calculus as well as the fractional q -calculus. Some auxiliary lemmas, needed in the proofs of our main results, are presented in Section 3. Section 4 contains the existence and uniqueness results for problem (1) which are shown by applying Banach's contraction principle, Krasnoselskii's fixed point theorem, and Leray-Schauder's nonlinear alternative. Finally, some examples illustrating the applicability of our results are presented in Section 5.

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, [6, 18, 19].

For q ∈ ( 0,1 ) , define [figure omitted; refer to PDF]

The q -analogue of the power function ( 1 - b ^{) k} with k ∈ _{N 0} : = { 0,1 , 2 , ... } is [figure omitted; refer to PDF] More generally, if γ ∈ R , then [figure omitted; refer to PDF]

We use the notation ^{0 ( γ )} = 0 for γ > 0 . The q -gamma function is defined by [figure omitted; refer to PDF] Obviously, _{Γ q} ( x + 1 ) = [ x _{] q Γ q} ( x ) .

The q -derivative of a function h is defined by [figure omitted; refer to PDF] and q -derivatives of higher order are given by [figure omitted; refer to PDF] The q -integral of a function h defined on the interval [ 0 , b ] is given by [figure omitted; refer to PDF] If a ∈ [ 0 , b ] and h is defined in the interval [ 0 , b ] , then its integral from a to b is defined by [figure omitted; refer to PDF] Similar to derivatives, an operator ^{I q k} is given by [figure omitted; refer to PDF] The fundamental theorem of calculus applies to operators _{D q} and _{I q} ; that is, [figure omitted; refer to PDF] and if h is continuous at x = 0 . Then [figure omitted; refer to PDF]

Definition 1.

Let ν ...5; 0 and h be a function defined on [ 0 , T ] . The fractional q -integral of Riemann-Liouville type is given by ( ^{I q 0} h ) ( x ) = h ( x ) and [figure omitted; refer to PDF]

Definition 2.

The fractional q -derivative of Riemann-Liouville type of order ν ...5; 0 is defined by ( ^{D q 0} h ) ( x ) = h ( x ) and [figure omitted; refer to PDF] where l is the smallest integer greater than or equal to ν .

Definition 3.

For any m , n > 0 , [figure omitted; refer to PDF] is called the q -beta function.

The expression of q -beta function in terms of the q -gamma function can be written as [figure omitted; refer to PDF]

Lemma 4 (see [4]).

Let α , β ...5; 0 , and f be a function defined in [ 0 , T ] . Then, the following formulas hold:

(1) ( ^{I q β I q α} f ) ( x ) = ( ^{I q α + β} f ) ( x ) ;

(2) ( ^{D q α I q α} f ) ( x ) = f ( x ) .

Lemma 5 (see [6]).

Let α > 0 and ν be a positive integer. Then, the following equality holds: [figure omitted; refer to PDF]

3. Some Auxiliary Lemmas

Lemma 6.

Let α , β > 0 , and 0 < q < 1 . Then one has [figure omitted; refer to PDF]

Proof.

Using the definitions of q -analogue of power function and q -beta function, we have [figure omitted; refer to PDF] The proof is complete.

Lemma 7.

Let α , β , γ > 0 , and 0 < p , q , r < 1 . Then one has [figure omitted; refer to PDF]

Proof.

Taking into account Lemma 6, we have [figure omitted; refer to PDF] This completes the proof.

Lemma 8.

Let β , γ > 0 , λ ∈ R , and 0 < p , q , r < 1 . Then, for y ∈ C ( [ 0 , T ] , R ) , the unique solution of boundary value problem, [figure omitted; refer to PDF] subject to the nonlocal fractional condition, [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

From 1 < α ...4; 2 , we let n = 2 . Using the Definition 2 and Lemma 4, (22) can be expressed as [figure omitted; refer to PDF] From Lemma 5, we have [figure omitted; refer to PDF] for some constants _{k 1} , _{k 2} ∈ R . It follows from the first condition of (23) that _{k 2} = 0 . Applying the Riemann-Liouville fractional p -integral of order β > 0 for (27) with _{k 2} = 0 and taking into account of Lemma 6, we have [figure omitted; refer to PDF] In particular, we have [figure omitted; refer to PDF] Using the Riemann-Liouville fractional r -integral of order γ > 0 and repeating the above process, we get [figure omitted; refer to PDF] The second nonlocal condition of (23) implies [figure omitted; refer to PDF] Substituting the values of _{k 1} and _{k 2} in (27), we get the desired result in (24).

4. Main Results

In this section, we denote C = C ( [ 0 , T ] , R ) as the Banach space of all continuous functions from [ 0 , T ] to R endowed with the norm defined by || x || = _{sup ... t ∈ [ 0 , T ]} | x ( t ) | . In view of Lemma 8, we define an operator Q : C [arrow right] C by [figure omitted; refer to PDF] where Ω ...0; 0 is defined by (25). It should be noticed that problem (1) has solutions if and only if the operator Q has fixed points.

For the sake of convenience of proving the results, we set [figure omitted; refer to PDF]

The first result on the existence and uniqueness of solutions is based on the Banach contraction mapping principle.

Theorem 9.

Let f : [ 0 , T ] × R × R [arrow right] R be a continuous function satisfying the assumption:

( _{H 1} ) there exist constants _{L 1} , _{L 2} > 0 such that [figure omitted; refer to PDF] for each t ∈ [ 0 , T ] and _{w 1} , _{w 2} , _{w - 1} , _{w - 2} ∈ R .

If [figure omitted; refer to PDF] where Λ is given by (33), then the boundary value problem (1) has a unique solution on [ 0 , T ] .

Proof.

We transform problem (1) into a fixed point problem, x = Q x , where the operator Q is defined by (32). By applying the Banach contraction mapping principle, we will show that Q has a fixed point which is the unique solution of problem (1).

Setting _{sup ... t ∈ [ 0 , T ]} | f ( t , 0,0 ) | = M < ∞ and choosing [figure omitted; refer to PDF] where θ ...4; [straight epsilon] < 1 , and the constant Ψ defined by (34), we will show that Q _{B r} ⊂ _{B r} , where _{B r} = { x ∈ C : || x || ...4; r } . For any x ∈ _{B r} , we have [figure omitted; refer to PDF] The assumption ( _{H 1} ) implies that [figure omitted; refer to PDF] for all t ∈ [ 0 , T ] and _{w 1} , _{w 2} ∈ R .

Then, by using Lemmas 6 and 7, we have [figure omitted; refer to PDF] Then, we have || Q x || ...4; r which yields Q _{B r} ⊂ _{B r} .

Next, for any x , y ∈ C and for each t ∈ [ 0 , T ] , we have [figure omitted; refer to PDF] The above result implies that || Q x - Q y || ...4; Λ || x - y || . As Λ < 1 , Q is a contraction. Hence, by the Banach contraction mapping principle, we deduce that Q has a fixed point which is the unique solution of problem (1).

The second existence result is based on Krasnoselskii's fixed point theorem.

Lemma 10 (Krasnoselskii's fixed point theorem [20]).

Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) A x + B y ∈ M whenever x , y ∈ M ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists z ∈ M such that z = A z + B z .

Theorem 11.

Assume that f : [ 0 , T ] × R × R [arrow right] R is a continuous function satisfying the assumption ( _{H 1} ) . In addition one supposes that

( _{H 2} ) | f ( t , _{w 1} , _{w 2} ) | ...4; κ ( t ) , for all ( t , _{w 1} , _{w 2} ) ∈ [ 0 , T ] × R × R and κ ∈ C ( [ 0 , T ] , ^{R +} ) .

If [figure omitted; refer to PDF] then the boundary value problem (1) has at least one solution on [ 0 , T ] .

Proof.

Let us set _{sup ... t ∈ [ 0 , T ]} | κ ( t ) | = || κ || and choose a suitable constant ρ as [figure omitted; refer to PDF] where Ψ is defined by (34). Now, we define the operators _{Q 1} and _{Q 2} on the set _{B ρ} = { x ∈ C : || x || ...4; ρ } as [figure omitted; refer to PDF] Firstly, we will show that the operators _{Q 1} and _{Q 2} satisfy condition (a) of Lemma 10. For x , y ∈ _{B ρ} , we have [figure omitted; refer to PDF] Therefore ( _{Q 1} x ) + ( _{Q 2} y ) ∈ _{B ρ} . Further, condition ( _{H 1} ) coupled with (42) implies that _{Q 2} is contraction mapping. Therefore, condition (c) of Lemma 10 is satisfied.

Finally, we will show that _{Q 1} is compact and continuous. Using the continuity of f and ( _{H 2} ) , we deduce that the operator _{Q 1} is continuous and uniformly bounded on _{B ρ} . We define _{sup ... ( t , w 1 , w 2 ) ∈ [ 0 , T ] × ^{B ρ 2}} | f ( t , _{w 1} , _{w 2} ) | = N < ∞ . For _{t 1} , _{t 2} ∈ [ 0 , T ] with _{t 1} < _{t 2} and x ∈ _{B ρ} , we have [figure omitted; refer to PDF] Actually, as _{t 1} - _{t 2} [arrow right] 0 the right-hand side of the above inequality tends to zero independently of x ∈ _{B ρ} . Therefore, _{Q 1} is relatively compact on _{B ρ} . Applying the Arzelá-Ascoli theorem, we get that _{Q 1} is compact on _{B ρ} . Thus all assumptions of Lemma 10 are satisfied. Therefore, the boundary value problem (1) has at least one solution on [ 0 , T ] . The proof is complete.

Using the Leray-Schauder nonlinear alternative, we give the third result.

Lemma 12 (nonlinear alternative for single-valued maps [21]).

Let E be a Banach space, let C be a closed, convex subset of E , let U be an open subset of C , and let 0 ∈ U . Suppose that F : U ¯ [arrow right] C is a continuous, compact (i.e., F ( U ¯ ) is a relatively compact subset of C ) map. Then either

(i) F has a fixed point in U ¯ , or

(ii) there is a u ∈ ∂ U (the boundary of U in C ) and λ ∈ ( 0,1 ) with u = λ F ( u ) .

For the sake of convenience of proving the last result, we set [figure omitted; refer to PDF]

Theorem 13.

Assume that f : [ 0 , T ] × R × R [arrow right] R is a continuous function. In addition one supposes that

( _{H 3} ) there exist a continuous nondecreasing function ψ : [ 0 , ∞ ) [arrow right] ( 0 , ∞ ) and a function p ∈ C ( [ 0 , T ] , ^{R +} ) such that [figure omitted; refer to PDF]

( _{H 4} ) there exists a constant K > 0 such that [figure omitted; refer to PDF] where _{Φ 1} and _{Φ 2} are defined by (47) and (48), respectively, and [figure omitted; refer to PDF] Then the boundary value problem (1) has at least one solution on [ 0 , T ] .

Proof.

Firstly, we will show that the operator Q , defined by (32), maps bounded sets (balls) into bounded sets in C . For a positive number R , we set a bounded ball in C as _{B R} = { x ∈ C : || x || ...4; R } . Then, for t ∈ [ 0 , T ] , we have [figure omitted; refer to PDF] Therefore, we conclude that || Q x || ...4; G .

Secondly, we will show that Q maps bounded sets into equicontinuous sets of C . Let _{t 1} , _{t 2} ∈ [ 0 , T ] with _{t 1} < _{t 2} and _{B R} be a bounded set of C ( [ 0 , T ] , R ) as in the previous step, and let x ∈ _{B R} . Then we have [figure omitted; refer to PDF] Obviously the right-hand side of the above inequality tends to zero independently of x ∈ _{B R} as _{t 1} [arrow right] _{t 2} . Therefore, by applying the Arzelá-Ascoli theorem, we deduce that Q : C [arrow right] C is completely continuous.

The result will follow from the Leray-Schauder nonlinear alternative once we have proved the boundedness of the set of all solutions to the equation x ( t ) = ω ( Q x ) ( t ) for some 0 < ω < 1 . Let x be a solution. Then, for t ∈ [ 0 , T ] , we have [figure omitted; refer to PDF]

As before, one can easily find that [figure omitted; refer to PDF] which can alternatively be written as [figure omitted; refer to PDF] In view of ( _{H 4} ) , there exists K such that || x || ...0; K . Let us set [figure omitted; refer to PDF] Note that the operator Q : U ¯ [arrow right] C ( 0 , T , R ) is continuous and completely continuous. From the choice of U , there is no x ∈ ∂ U such that x = ω Q x for some ω ∈ ( 0,1 ) . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 12), we deduce that Q has a fixed point x ∈ U ¯ which is a solution of problem (1). This completes the proof.

5. Examples

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

Example 1.

Consider the following nonlocal fractional q -integral boundary value problem: [figure omitted; refer to PDF]

Here α = 3 / 2 , q = 1 / 2 , δ = 7 / 5 , z = 3 / 4 , λ = 1 / 5 , β = 1 / 2 , p = 3 / 5 , η = 5 / 2 , γ = 5 / 2 , r = 2 / 3 , ξ = 3 / 2 , T = 3 , and f ( t , x , ^{I z δ} x ) = ( 2 sin π t / ( ^{e t} + 4 ^{) 2} ) ( | x | / ( 2 + | x | ) ) + ( ^{e - t 2} / ( ( 6 + t ^{) 2} ) ) ^{I 3 / 4 7 / 5} x + 1 / 2 .

Since | f ( t , _{w 1} , _{w 2} ) - f ( t , _{w - 1} , _{w - 2} ) | ...4; ( 1 / 25 ) | _{w 1} - _{w - 1} | + ( 1 / 36 ) | _{w 2} - _{w - 2} | , then ( _{H 1} ) is satisfied with _{L 1} = 1 / 25 and _{L 2} = 1 / 36 . By using the Maple program, we find that [figure omitted; refer to PDF] Hence, by Theorem 9, the nonlocal boundary value problem (58) has a unique solution on [ 0,3 ] .

Example 2.

Consider the following nonlocal fractional q -integral boundary value problem: [figure omitted; refer to PDF]

Here α = 9 / 5 , q = 2 / 3 , δ = 3 / 5 , z = 1 / 10 , λ = 1 / 50 , β = 1 / 10 , p = 1 / 5 , η = 2 / 3 , γ = 2 / 9 , r = 1 / 8 , ξ = 1 / 2 , T = 1 , and f ( t , x , ^{I z δ} x ) = ( ^{tan - 1} ( π x / 2 ) ) / ( 4 ^{π 2} + ^{t 2} ) + ( 1 + sin ( π t ) ) / ( 30 π ) + ^{I 1 / 10 3 / 5} x .

By using the Maple program, we find that [figure omitted; refer to PDF] Clearly, [figure omitted; refer to PDF] Choosing p ( t ) = 1 + sin π t and ψ ( | _{w 1} | ) = ( 1 / 120 π ) ( 15 | _{w 1} | + 4 ) , we can show that [figure omitted; refer to PDF] which implies that K > 0.0645811 . Hence, by Theorem 13, the nonlocal boundary value problem (60) has at least one solution on [ 0,1 ] .

Acknowledgments

This research was funded by King Mongkut's University of Technology North Bangkok, Contract no. KMUTNB-GOV-57-09. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM), Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] F. H. Jackson, " q -Difference Equations," The American Journal of Mathematics , vol. 32, no. 4, pp. 305-314, 1910.

[2] V. Kac, P. Cheung Quantum Calculus , Springer, New York, NY, USA, 2002.

[3] W. A. Al-Salam, "Some fractional {$q$}-integrals and {$q$}-derivatives," Proceedings of the Edinburgh Mathematical Society , vol. 15, pp. 135-140, 1967.

[4] R. P. Agarwal, "Certain fractional {$q$}-integrals and {$q$}-derivatives," Proceedings of the Cambridge Philosophical Society , vol. 66, pp. 365-370, 1969.

[5] T. Ernst, "The history of q -calculus and a new method," UUDM Report , no. 2000:16, Department of Mathematics, Uppsala University, 2000.

[6] R. A. C. Ferreira, "Nontrivial solutions for fractional q -difference boundary value problems," Electronic Journal of Qualitative Theory of Differential Equations , no. 70, pp. 1-10, 2010.

[7] C. S. Goodrich, "Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions," Computers and Mathematics with Applications , vol. 61, no. 2, pp. 191-202, 2011.

[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo Theory and applications of fractional differential equations , vol. 204, of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, Netherlands, 2006.

[9] H. M. Srivastava, "Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials," Applied Mathematics & Information Sciences , vol. 5, no. 3, pp. 390-444, 2011.

[10] J. Ma, J. Yang, "Existence of solutions for multi-point boundary value problem of fractional q -difference equation," Electronic Journal of Qualitative Theory of Differential Equations , no. 92, pp. 1-10, 2011.

[11] J. R. Graef, L. Kong, "Positive solutions for a class of higher order boundary value problems with fractional q -derivatives," Applied Mathematics and Computation , vol. 218, no. 19, pp. 9682-9689, 2012.

[12] B. Ahmad, S. K. Ntouyas, I. K. Purnaras, "Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations," Advances in Difference Equations , vol. 2012, article 140, 2012.

[13] B. Ahmad, S. K. Ntouyas, "Existence of solutions for nonlinear fractional q -difference inclusions with nonlocal Robin (separated) conditions," Mediterranean Journal of Mathematics , vol. 10, no. 3, pp. 1333-1351, 2013.

[14] B. Ahmad, S. K. Ntouyas, A. Alsaedi, H. Al-Hutami, "Nonlinear q-fractional differential equations with nonlocal and sub-strip type boundary conditions," The Electronic Journal of Qualitative Theory of Differential Equations , vol. 26, pp. 1-14, 2014.

[15] B. Ahmad, J. J. Nieto, A. Alsaedi, H. Al-Hutami, "Existence of solutions for nonlinear fractional q -difference integral equations with two fractional orders and nonlocal four-point boundary conditions," Journal of the Franklin Institute: Engineering and Applied Mathematics , vol. 351, no. 5, pp. 2890-2909, 2014.

[16] A. Alsaedi, B. Ahmad, H. Al-Hutami, "A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions," Abstract and Applied Analysis , vol. 2013, 2013.

[17] X. Li, Z. Han, S. Sun, "Existence of positive solutions of nonlinear fractional q -difference equation with parameter,", article 260 Advances in Difference Equations , vol. 2013, 2013.

[18] M. H. Annaby, Z. S. Mansour q-Fractional Calculus and Equations , vol. 2056, of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2012.

[19] H. M. Srivastava, P. W. Karlsson Multiple Gaussian Hypergeometric Series , John Wiley & Sons, 1985.

[20] M. A. Krasnoselskii, "Two remarks on the method of successive approximations," Uspekhi Matematicheskikh Nauk , vol. 10, pp. 123-127, 1955.

[21] A. Granas, J. Dugundji Fixed Point Theory , Springer, New York, NY, USA, 2003.