Uniqueness and Existence of Solution for a System of Fractional q -Difference Equations

Abstract

We prove the existence and uniqueness of solution for a system of fractional differential equations. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach's fixed-point theorem.

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

Wen-Xue Zhou 1,2 and Hai-zhong Liu 1

Academic Editor:Felix Sadyrbaev

1, College of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
2, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 28 March 2014; Revised 27 May 2014; Accepted 13 June 2014; 7 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

This paper is mainly concerned with the uniqueness and existence of solution for a system of fractional q -difference equations given by [figure omitted; refer to PDF] where ^{D C q α} , ^{D C q β} is the fractional q -derivatives of the Caputo type, 1 < α , β ...4; 2 , _{α i} ( i = 1,2 , 3,4 ) , _{β i} ( i = 1,2 , 3,4 ) , _{γ i} ( i = 1,2 , 3,4 ) , and _{η i} ( i = 1,2 , 3,4 ) are arbitrary real constants, and f , g : [ 0,1 ] × R [arrow right] R are given continuous functions.

In the last few years, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. [1], Miller and Ross [2], Oldham and Spanier [3], Podlubny [4], and Samko et al. [5].

Some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala ([6-8]), Babakhani and Daftardar-Gejji ([9-11]), Bai [12], and so on. Also, there are some papers which deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, topological degree theory, etc.)--see ([13-18]) and the references therein. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications. The reader is referred to the papers ([19-22]) and the references cited therein.

The pioneer work on q -difference calculus or quantum calculus dates back to Jackson's papers ([23, 24]), while a systematic treatment of the subject can be found in [25, 26]. For some recent existence results on q -difference equations, see [27-29] and the references therein.

There has also been a growing interest on the subject of discrete fractional equations on the time scale Z . Some interesting results on the topic can be found in a series of papers [30-38]. Fractional q -difference equations have recently attracted the attention of several researchers. For some earlier work on the topic, we refer to [39, 40], whereas some recent work on the existence theory of fractional q -difference equations can be found in [41-45]. However, the study of boundary value problems of fractional q -difference equations is at its infancy and much of the work on the topic is yet to be done.

From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors. To the best of our knowledge, there have been few papers which deal with problem (1) for nonlinear fractional differential equation. Motivated by all the works above, in this paper we discuss problem (1). Using nonlinear alternative of Leray-Schauder type, we will give the existence and uniqueness of solution for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order (1).

The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the uniqueness and existence of a solution for the nonlinear fractional differential equation boundary value problem (1). In last section, we give two examples to illustrate our work.

2. Preliminaries and Lemmas

In this section, we cite some definitions and fundamental results of the q -calculus as well as of the fractional q -calculus ([46, 47]). We also give a lemma that will be used in obtaining the main results of the paper.

Let q ∈ ( 0,1 ) and define [47] [figure omitted; refer to PDF] The q -analogue of the power ^{( a - b ) n} is [figure omitted; refer to PDF] If α is not a positive integer, then [figure omitted; refer to PDF] Note that if b = 0 , then ^{a ( α )} = ^{a α} . The q -gamma function is defined by [figure omitted; refer to PDF] and satisfies _{Γ q} ( x + 1 ) = _{[ x ] q Γ q} ( x ) (see [47]).

The q -derivative of a function f is here defined by [figure omitted; refer to PDF] and q -derivatives of higher order by [figure omitted; refer to PDF] The q -integral of a function f defined in the interval [ 0 , b ] is given by [figure omitted; refer to PDF] If a ∈ [ 0 , b ] and f is defined in the interval [ 0 , b ] , its integral from a to b is defined by [figure omitted; refer to PDF] Similarly, as done for derivatives, an operator ^{I q n} can be defined, namely, by [figure omitted; refer to PDF] The fundamental theorem of calculus applies to these operators _{I q} and _{D q} ; that is, [figure omitted; refer to PDF] and if f is continuous at x = 0 , then [figure omitted; refer to PDF] Basic properties of the two operators can be found in the book that is mentioned in [8]. We now point out three formulas that will be used later ( _{D i q} denotes the derivative with respect to variable i ) [43]: [figure omitted; refer to PDF]

Remark 1.

We note that if α > 0 and a ...4; b ...4; t , then ^{( t - a ) ( α )} ...5; ^{( t - b ) ( α )} [43].

Definition 2 (see [40]).

Let α ...5; 0 and let f be a function defined on [ 0,1 ] . The fractional q -integral of the Riemann-Liouville type is ( ^{I RL q 0} f ) ( x ) = f ( x ) and [figure omitted; refer to PDF]

Definition 3 (see [48]).

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

Definition 4 (see [48]).

The fractional q -derivative of the Caputo type of order α ...5; 0 is defined by [figure omitted; refer to PDF] where [ α ] is the smallest integer greater than or equal to α .

Lemma 5.

Let α , β ...5; 0 and let f be a function defined on [ 0,1 ] . Then the next 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 6 (see [42]).

Let α ...5; 0 and n ∈ N . Then the following equality holds: [figure omitted; refer to PDF]

Lemma 7 (see [48]).

Let α > 0 and n ∈ ^{R +} \ N . Then the following equality holds: [figure omitted; refer to PDF]

For convenience, one introduces the following notations: [figure omitted; refer to PDF]

From Lemmas 5 and 7, we can obtain the following lemma.

Lemma 8.

Let h ∈ C [ 0,1 ] and Δ ...0; 0 ; then the unique solution of the linear fractional boundary value problem [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

The following lemma is fundamental in the proofs of our main result.

Lemma 9 (see [49]; nonlinear alternative of Leray-Schauder type).

Let E be a Banach space with M ⊆ E closed and convex. Assume that U is a relatively open subset of C with 0 ∈ U and F : U ¯ [arrow right] C is 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 exist u ∈ ∂ U and λ ∈ ( 0,1 ) with u = λ F u .

3. Main Results

In this section, we will discuss the uniqueness and existence of solutions for boundary value problem (1).

First of all, we define the Banach space X = { u |" u ∈ C [ 0,1 ] } endowed with the norm || u || = _{max ... t ∈ [ 0,1 ]} | u ( t ) | . For ( u , v ) ∈ X × X , let || ( u , v ) || = max ... { || u || , || v || } ; then ( X × X , || ( · , · ) || ) is a Banach space.

For convenience, we set [figure omitted; refer to PDF] and let [black triangle up] ...0; 0 . Note [figure omitted; refer to PDF]

Employing Lemma 8, system (1) can be expressed as [figure omitted; refer to PDF] where _{b 1} , _{b 2} , _{b 3} , _{b 4} are given by (21), and _{b 1} , _{b 2} , _{b 3} , _{b 4} are given by (22)

From Lemma 8 in Section 2, we can obtain the following lemma.

Lemma 10.

Suppose that f ( t , v ) and g ( t , u ) are continuous; then ( u , v ) ∈ X × X is a solution of BVP (1) if and only if ( u , v ) ∈ X × X is a solution of the integral equations (24).

Let ( u , v ) ∈ X × X ; define an operator T : X × X [arrow right] X × X as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] then, by Lemma 10, the fixed point of operator T coincides with the solution of system (1).

In the first result, we prove uniqueness of solution of the boundary value problem (1) via Banach's contraction principle.

Theorem 11.

Assume that f , g : [ 0,1 ] × R [arrow right] R are continuous functions and the following conditions hold:

(H1) there exist two q -integrable functions _{L 1} , _{L 2} : [ 0,1 ] [arrow right] R that satisfy [figure omitted; refer to PDF]

In addition, assume that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where Δ and [black triangle up] are given by (19) and (22), respectively. Then system (1) has a unique solution.

Proof.

Let us set _{sup ... t ∈ [ 0,1 ]} | f ( t , 0 ) | = _{M 1} < ∞ , _{sup ... t ∈ [ 0,1 ]} | g ( t , 0 ) | = _{M 2} < ∞ , [figure omitted; refer to PDF]

Define [figure omitted; refer to PDF]

For v ∈ _{B r 1} , we obtain [figure omitted; refer to PDF] Then, for v ∈ _{B r 1} , t ∈ [ 0,1 ] , we have [figure omitted; refer to PDF] In view of (31), we obtain [figure omitted; refer to PDF] From the last estimate we deduce that _{r 1} = _{M 1 A 1} / ( 1 - _{κ 1} ) .

By a similar way as done above we have [figure omitted; refer to PDF] and _{r 2} = _{M 2 A 2} / ( 1 - _{κ 2} ) .

Therefore, we obtain [figure omitted; refer to PDF] From the last estimate we can choose r = max ... { _{r 1} , _{r 2} } ; then, for every ( u , v ) ∈ U , we have T U ⊂ U .

In order to show that T is a contraction, let u , v , _{u 1} , _{v 1} ∈ X , and, for any t ∈ [ 0,1 ] , we get [figure omitted; refer to PDF] which, in view of _{κ 1} < 1 and (31), implies that [figure omitted; refer to PDF]

Similarly, we have || _{T 2} u - _{T 2 u 1} || ...4; _{κ 2} || u - _{u 1} || .

Thus, we have [figure omitted; refer to PDF]

Since _{κ 1} < 1 , _{κ 2} < 1 , therefore, the operator T is a contraction. Hence, by Banach's contraction principle, the operator T has a unique fixed point, which is the unique solution of the system (1). This completes the proof.

The second result is based on the nonlinear alternative of Leray-Schauder type (Lemma 9).

Theorem 12.

Assume that f , g : [ 0,1 ] × R [arrow right] R are continuous functions and the following conditions hold:

(H2) there exist four functions _{p i} ( t ) , _{q i} ( t ) ∈ ^{L 1} ( [ 0,1 ] , ^{R +} ) , i = 1,2 , and two nondecreasing functions [straight phi] , ψ : ^{R +} [arrow right] ^{R +} , such that [figure omitted; refer to PDF]

: where ( t , x ) , ( t , y ) ∈ [ 0,1 ] × R .

(H3) There exists a constant M > 0 such that [figure omitted; refer to PDF]

: where [figure omitted; refer to PDF]

Then system (1) has at least one solution on [ 0,1 ] .

Proof.

Consider the operator T : X × X [arrow right] X × X defined by (25). The proof consists of several steps. As a first step, it will be shown that T maps bounded sets into bounded sets in X × X . For a positive number r , let U = { ( u , v ) ∈ X × X : || ( u , v ) || ...4; r } be bounded set in X × X ; then, for ( u , v ) ∈ U , we have [figure omitted; refer to PDF]

As before, it can be shown that [figure omitted; refer to PDF] Similarly, we have [figure omitted; refer to PDF] Thus, T maps bounded sets into bounded sets in X × X .

Next, we show that T maps bounded sets into equicontinuous sets of X × X . Let _{t 1} , _{t 2} ∈ [ 0,1 ] with _{t 1} < _{t 2} and ( u , v ) ∈ U , where U is a bounded set of X × X . Then taking into consideration the inequality ^{( _{t 2} - q s ) ( α - 1 )} - ^{( _{t 1} - q s ) ( α - 1 )} ...4; ( _{t 2} - _{t 1} ) , for 0 < _{t 1} < _{t 2} , we obtain [figure omitted; refer to PDF] Clearly, the right-hand side of the above inequality tends to zero independently of v ∈ U as _{t 2} [arrow right] _{t 1} . Thus, it follows by the Arzelá-Ascoli theorem that _{T 1} is completely continuous. Similarly, _{T 2} is completely continuous. Therefore, T : X × X [arrow right] X × X is completely continuous.

Let us set Ω = { ( u , v ) ∈ U : || ( u , v ) || < M } . Note that the operator T : Ω ¯ [arrow right] X × X is continuous and completely continuous. From the choice of Ω , assume that there is ( u , v ) ∈ ∂ Ω such that ( u , v ) = λ T ( u , v ) , for some λ ∈ ( 0,1 ) . By (H3), we obtain [figure omitted; refer to PDF] which is a contradiction. In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 9), we deduce that T has a fixed point ( u , v ) ∈ Ω ¯ which is a solution of the system (1). The proof is complete.

In the sequel we present two examples which illustrate Theorems 11 and 12.

4. Examples

Example 1.

Consider the following fractional q -difference nonlocal boundary value problem: [figure omitted; refer to PDF] In this case, α = 3 / 2 , β = 7 / 4 , _{α 1} = 1 , _{β 1} = 1 / 2 , _{α 2} = 1 / 4 , _{β 2} = 3 / 4 , _{α 3} = 1 , _{β 3} = 1 / 2 , _{α 4} = 1 / 4 , _{β 4} = 3 / 4 , _{η 1} = 1 / 3 , _{η 2} = 2 / 3 , _{η 3} = 1 / 3 , _{η 4} = 2 / 3 , _{γ 1} = 1 = _{γ 2} , _{γ 3} = 1 = _{γ 4} , and _{L 1} , _{L 2} are constants to be fixed later on. Moreover, Δ = 5 / 8 , [black triangle up] = 5 / 8 , _{δ 1} = 26 / 15 , _{δ 2} = 4 / 3 , _{δ 3} = 26 / 15 , and _{δ 4} = 4 / 3 . Consider [figure omitted; refer to PDF]

Clearly, we have [figure omitted; refer to PDF]

Choose [figure omitted; refer to PDF]

Hence all the assumptions of Theorem 11 are satisfied. Therefore, by Theorem 11, the problem (50) has a unique solution.

Example 2.

Consider the following fractional boundary value problem: [figure omitted; refer to PDF] In this case, α = 3 / 2 , β = 3 / 2 , _{α 1} = 1 , _{β 1} = 1 / 2 , _{α 2} = 1 / 4 , _{β 2} = 3 / 4 , _{α 3} = 1 , _{β 3} = 1 / 2 , _{α 4} = 1 / 4 , _{β 4} = 3 / 4 , _{η 1} = 1 / 3 , _{η 2} = 2 / 3 , _{η 3} = 1 / 3 , _{η 4} = 2 / 3 , _{γ 1} = 1 = _{γ 2} , _{γ 3} = 1 = _{γ 4} , and _{L 1} , _{L 2} are constants to be fixed later on. Moreover, Δ = 5 / 8 , [black triangle up] = 5 / 8 , _{δ 1} = 26 / 15 , _{δ 2} = 4 / 3 , _{δ 3} = 26 / 15 , and _{δ 4} = 4 / 3 . Clearly [figure omitted; refer to PDF] Clearly _{p 1} = 1 / 8 , _{q 1} = 1 , [straight phi] ( M ) = M , _{p 2} = 1 / 8 , _{q 2} = 1 , and ψ ( M ) = M . Consequently, _{ω 1} [approximate] 0.567129414 , _{ω 2} [approximate] 4.536963312 , _{[varpi] 1} [approximate] 0.567129414 , _{[varpi] 2} [approximate] 4.536963312 , and conditions (43) imply that M > 10.48055997 . Thus, all the assumptions of Theorem 12 are satisfied. Therefore, the conclusion of Theorem 12 applies to problems (54).

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11161027, 11262009, and 11226132), the Scientific Research Projects in Colleges and Universities of Gansu Province of China (2013A-043), the Fundamental Research Funds for the Gansu Universities (213061), the Fundamental Research Funds for the Gansu Universities (212084), the Youth Science Foundation of Lanzhou Jiaotong University (2012019), and the National Natural Science Foundation of China (11226132). The authors are thankful to the referees for their careful reading of the paper and insightful comments.

Conflict of Interests

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

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