Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions

Abstract

We study the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard-type fractional differential equations and nonlocal fractional integral boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.

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Phollakrit Thiramanus 1 and Sotiris K. Ntouyas 2 and Jessada Tariboon 1

Academic Editor:Bashir Ahmad

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 1 April 2014; Accepted 10 June 2014; 26 June 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 investigate the following Hadamard boundary value problem: [figure omitted; refer to PDF] where ^Dq denotes the Hadamard fractional derivative of order q,f:[1,e]×R[arrow right]R is a continuous function, _ηi ,_ξj ∈(1,e) , _λi ,_μj ∈R , for all i=1,2,...,m , j=1,2,...,n , _η1 <_η2 <...<_ηm , _ξ1 <_ξ2 <...<_ξn , and ^J[varphi] is the Hadamard fractional integral of order [varphi]>0 ([varphi]=_αi ,_βj ,i=1,2,...,m,j=1,2,...,n) .

We mention that integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering, cellular systems, heat transmission, plasma physics, and thermoelasticity.

Condition (2) is a general form of the integral boundary conditions considered in [1] and covers many special cases. For example, if _αi =_βj =1 , for all i=1,2,...,m and j=1,2,...,n , then condition (2) reduces to [figure omitted; refer to PDF]

Fractional differential equations provide appropriate models for describing real world problems, which cannot be described using classical integer order differential equations. The theory of fractional differential equations has received much attention over the past years and has become an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [2-5]. Some recent contributions to the subject can be seen in [1, 6-20] and references cited therein.

It has been noticed that most of the work on this topic is based on Riemann-Liouville and Caputo type fractional differential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [21], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains logarithmic function of arbitrary exponent. Details and properties of Hadamard fractional derivative and integral can be found in [2, 22-26]. For some recent results on Hadamard boundary value problem we refer to [27, 28].

We establish a variety of results for the problem (1)-(2) by using classical fixed point theorems. The first result, Theorem 4, relies on Banach contraction mapping principle and concerns an existence and uniqueness result for the solutions of the problem (1)-(2). A second existence and uniqueness result is proved in Theorem 7, via nonlinear contractions and a fixed point theorem due to Boyd and Wong. Existence results are proved in the third result, Theorem 9, by using Krasnoselskii fixed point theorem, and in the fourth result, Theorem 12, by using nonlinear alternative of Leray-Schauder type.

The paper is organized as follows. In Section 2, we recall some preliminary concepts that we need in the sequel and prove a preliminary lemma. Section 3 contains the main results for the problem (1)-(2). In Section 4, some illustrative examples are discussed.

2. Preliminaries

In this section, we introduce some notations and definitions of fractional calculus [2] and present preliminary results needed in our proofs later.

Definition 1.

The Hadamard derivative of fractional order q for a function f: [1,∞)[arrow right]R is defined as [figure omitted; refer to PDF] where [q] denotes the integer part of the real number q , log...(·)=_log...e (·) , and Γ is the Gamma function.

Definition 2.

The Hadamard fractional integral of order q for a function f: [1,∞)[arrow right]R is defined by [figure omitted; refer to PDF] provided the integral exists.

For convenience, we set [figure omitted; refer to PDF]

Lemma 3.

Let Λ...0;0 , 1<q...4;2 , _αi ,_βj >0 , and _ηi ,_ξj ∈(1,e) for i=1,2,...,m , j=1,2,...,n , and h∈C([1,e],R) . The unique solution of the following fractional differential equation, [figure omitted; refer to PDF] subject to the boundary condition, [figure omitted; refer to PDF] is given by the integral equation [figure omitted; refer to PDF]

Proof.

Applying the Hadamard fractional integral of order q to both sides of (7), we have [figure omitted; refer to PDF] where _z1 ,_z2 ∈R .

The condition of x(1)=0 implies _z2 =0 . Therefore, [figure omitted; refer to PDF] For any p>0 , by Definition 2, it follows that [figure omitted; refer to PDF] The second condition of (8) with (12) leads to [figure omitted; refer to PDF] Substituting the value of a constant _z1 into (11), we obtain (9) as required. The proof is completed.

3. Main Results

Let C=C([1,e],R) denote the Banach space of all continuous functions from [1,e] to R endowed with the norm defined by ||x||=_{sup...t∈[1,e]} |x(t)| . As in Lemma 3, we define an operator F: C[arrow right]C by [figure omitted; refer to PDF] with Λ...0;0 . It should be noticed that problem (1)-(2) has solutions if and only if the operator F has fixed points.

For the sake of convenience, we put [figure omitted; refer to PDF]

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

Theorem 4.

Let f:[1,e]×R[arrow right]R be a continuous function satisfying the assumption that

(H₁ ): there exists a constant L>0 such that |f(t,x)-f(t,y)|...4;L|x-y| , for each t∈[1,e] and x,y∈R .

If [figure omitted; refer to PDF] where Φ is given by (15), then the boundary value problem (1)-(2) has a unique solution on [1,e] .

Proof.

We transform the problem (1)-(2) into a fixed point problem, x=Fx , where the operator F is defined by (14). By using Banach's contraction mapping principle, we will show that F has a fixed point which is a unique solution of problem (1)-(2).

We set _{sup...t∈[1,e]} |f(t,0)|=M<∞ and choose [figure omitted; refer to PDF]

Now, we show that F_Br ⊂_Br , where _Br ={x∈C:||x||...4;r} . For any x∈_Br , we have [figure omitted; refer to PDF] It follows that F_Br ⊂_Br .

For x,y∈C and for each t∈[1,e] , we have [figure omitted; refer to PDF] The above result implies that ||Fx-Fy||...4;LΦ||x-y|| . As LΦ<1 , therefore F is a contraction. Hence, by the Banach contraction mapping principle, we deduce that F has a fixed point which is the unique solution of the problem (1)-(2).

Next, we give the second existence and uniqueness result by using nonlinear contractions.

Definition 5.

Let E be a Banach space and let F:E[arrow right]E be a mapping. F is said to be a nonlinear contraction if there exists a continuous nondecreasing function Ψ:^R+ [arrow right]^R+ such that Ψ(0)=0 and Ψ(θ)<θ for all θ>0 with the property [figure omitted; refer to PDF]

Lemma 6 (see [29]).

Let E be a Banach space and let F:E[arrow right]E be a nonlinear contraction. Then F has a unique fixed point in E .

Theorem 7.

Let f: [1,e]×R[arrow right]R be a continuous function satisfying the assumption

(H₂ ): |f(t,x)-f(t,y)|...4;h(t)(|x-y|/(^H* +|x-y|)) , t∈[1,e] , x,y...5;0 , where h:[1,e][arrow right]^R+ is continuous and a constant ^H* is defined by [figure omitted; refer to PDF]

Then the boundary value problem (1)-(2) has a unique solution.

Proof.

We define the operator F:C[arrow right]C as (14) and a continuous nondecreasing function Ψ:^R+ [arrow right]^R+ by [figure omitted; refer to PDF] Note that the function Ψ satisfies Ψ(0)=0 and Ψ(θ)<θ for all θ>0 .

For any x,y∈C and for each t∈[1,e] , we have [figure omitted; refer to PDF] This implies that ||Fx-Fy||...4;Ψ(||x-y||) . Therefore F is a nonlinear contraction. Hence, by Lemma 6 the operator F has a fixed point which is the unique solution of the problem (1)-(2).

Next, we give an existence result by using Krasnoselskii's fixed point theorem.

Lemma 8 (Krasnoselskii's fixed point theorem [30]).

Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) Ax+By∈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 = Az + Bz.

Theorem 9.

Assume that f: [1,e]×R[arrow right]R is a continuous function satisfying assumption (_H1 ) . In addition we suppose that

(H₃ ): |f(t,x)|...4;κ(t) , ∀(t,x)∈[1,e]×R and κ∈C([1,e],^R+ ) .

If [figure omitted; refer to PDF] then the boundary value problem (1)-(2) has at least one solution on [1,e] .

Proof.

We define _{sup...t∈[1,e]} |κ(t)|=||κ|| and choose a suitable constant r¯ as [figure omitted; refer to PDF] where Φ is defined by (15). Furthermore, we define the operators P and Q on _Br¯ ={x∈C:||x||...4;r¯} as [figure omitted; refer to PDF] For x,y∈_Br¯ , we have [figure omitted; refer to PDF] This shows that Px+Qy∈_Br¯ . It follows from assumption (_H1 ) together with (24) that Q is a contraction mapping. Since the function f is continuous, we have that the operator P is continuous. It is easy to verify that [figure omitted; refer to PDF] Therefore, P is uniformly bounded on _Br¯ .

Next, we prove the compactness of the operator P . Let us set _{sup...(t,x)∈[1,e]×Br¯} |f(t,x)| =f¯<∞ ; consequently we get [figure omitted; refer to PDF] which is independent of x and tends to zero as _t2 [arrow right]_t1 . Thus, P is equicontinuous. So P is relatively compact on _Br¯ . Hence, by the Arzelá-Ascoli theorem, P is compact on _Br¯ . Thus, all the assumptions of Lemma 8 are satisfied. So the boundary value problem (1)-(2) has at least one solution on [1,e] . The proof is completed.

Remark 10.

In the above theorem we can interchange the roles of the operators P and Q to obtain a second result replacing (24) by the following condition: [figure omitted; refer to PDF]

Now, our last existence result is based on Leray-Schauder's nonlinear alternative.

Theorem 11 (nonlinear alternative for single-valued maps [31]).

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and 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=λF(u) .

Theorem 12.

Assume that f:[1,e]×R[arrow right]R is a continuous function. In addition we suppose that

(H₄ ): there exists a continuous nondecreasing function ψ:[0,∞)[arrow right](0,∞) and a function p∈C([1,e],^R+ ) such that [figure omitted; refer to PDF]

(H₅ ): there exists a constant N>0 such that [figure omitted; refer to PDF]

: where Φ is defined by (15).

Then the boundary value problem (1)-(2) has at least one solution on [1,e] .

Proof.

Firstly, we will show that the operator F , defined by (14), maps bounded sets (balls) into bounded sets in C . For a positive number R , let _BR ={x∈C:||x||...4;R} be a bounded ball in C . Then for t∈[1,e] , we have [figure omitted; refer to PDF] Therefore, we conclude that ||Fx||...4;K .

Secondly, we show that F maps bounded sets into equicontinuous sets of C . Let _{sup...(t,x)∈[1,e]×BR} |f(t,x)|=^f* <∞ , _ν1 ,_ν2 ∈[1,e] with _ν1 <_ν2 and x∈_BR . Then we have [figure omitted; refer to PDF] Obviously, the right hand side of the above inequality tends to zero independently of x∈_BR as _ν2 [arrow right]_ν1 . Therefore it follows from the Arzelá-Ascoli theorem that F:C[arrow right]C is completely continuous.

Let x be a solution. Then, for t∈[1,e] , following the similar computations as in the first step, we have [figure omitted; refer to PDF] Consequently, we have [figure omitted; refer to PDF] In view of (H₅ ), there exists N such that ||x||...0;N . Let us set [figure omitted; refer to PDF] Note that the operator F:U¯[arrow right]C is continuous and completely continuous. From the choice of U , there is no x∈∂U such that x=θFx for some θ∈(0,1) . Consequently, by nonlinear alternative of Leray-Schauder type (Theorem 11) we deduce that F has a fixed point in U¯ , which is a solution of the boundary value problem (1)-(2). This completes the proof.

4. Examples

Example 1.

Consider the following boundary value problem for Hadamard fractional differential equation: [figure omitted; refer to PDF]

Here q=3/2 , _λ1 =2 , _λ2 =1/5 , _λ3 =3 , _α1 =1/4 , _α2 =3/2 , _α3 =2 , _η1 =5/4 , _η2 =9/5 , _η3 =15/7 , _μ1 =1 , _μ2 =5 , _μ3 =-2 , _β1 =2/3 , _β2 =9/7 , _β3 =11/4 , _ξ1 =10/7 , _ξ2 =2 , _ξ3 =9/4 , and f(t,x)=(log...^t5 |x|)/(^et (t+2⁾² (3+|x|)) . Since [figure omitted; refer to PDF] then (H₁ ) is satisfied with L=5/27e . We can show that [figure omitted; refer to PDF] Hence, by Theorem 4, the boundary value problem (38) has a unique solution on [1,e] .

Example 2.

Consider the following boundary value problem for Hadamard fractional differential equation: [figure omitted; refer to PDF]

Here q=7/4 , _λ1 =1/4 , _λ2 =-2/3 , _λ3 =-2 , _α1 =6/7 , _α2 =3 , _α3 =5/2 , _η1 =7/3 , _η2 =7/5 , _η3 =2 , _μ1 =4 , _μ2 =11/4 , _β1 =5 , _β2 =3/4 , _ξ1 =11/5 , _ξ2 =16/13 , and f(t,x)=(^et |x|)/((t+1⁾² (2+|x|)) . We choose h(t)=^et /4 and that [figure omitted; refer to PDF] Clearly, [figure omitted; refer to PDF] Hence, by Theorem 7, the boundary value problem (41) has a unique solution on [1,e] .

Example 3.

Consider the following boundary value problem for Hadamard fractional differential equation: [figure omitted; refer to PDF]

Here q=6/5 , _λ1 =1 , _λ2 =-3 , _λ3 =-10 , _λ4 =6 , _λ5 =14/3 , _α1 =4 , _α2 =9/4 , _α3 =1/5 , _α4 =7/2 , _α5 =5 , _η1 =3/2 , _η2 =2 , _η3 =7/4 , _η4 =5/2 , _η5 =11/9 , _μ1 =3 , _μ2 =-7 , _μ3 =4/3 , _β1 =3/2 , _β2 =3 , _β3 =5/3 , _ξ1 =11/7 , _ξ2 =17/13 , _ξ3 =2 , and f(t,x)=(2sin(x/4))/(5π+(^ex +1⁾² )+(2+cos...(πt))/(10π+3) . Clearly, [figure omitted; refer to PDF] Choosing p(t)=2+cos...(πt) and ψ(|x|)=(|x|+1)/(10π) , we can show that [figure omitted; refer to PDF] which implies that N>0.1623483851 . Hence, by Theorem 12, the boundary value problem (44) has at least one solution on [1,e] .

Acknowledgments

The research of Phollakrit Thiramanus is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The research of Jessada Tariboon is supported by King Mongkut's University of Technology North Bangkok, Thailand. 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.

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