A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions

Abstract

We consider a new class of boundary value problems of nonlinear fractional differential equations with fractional separated boundary conditions. A connection between classical separated and fractional separated boundary conditions is developed. Some new existence and uniqueness results are obtained for this class of problems by using standard fixed point theorems. Some illustrative examples are also discussed.

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Bashir Ahmad 1 and Sotiris K. Ntouyas 2

Recommended by Shaher Momani

1, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2, Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 21 January 2012; Accepted 8 February 2012

1. Introduction

In this paper, we investigate the existence of solutions for a fractional boundary value problem with fractional separated boundary conditions given by [figure omitted; refer to PDF] where ^cDq denotes the Caputo fractional derivative of order q , f is a given continuous function, and _αi ,_βi ,_γi (i=1,2) are real constants, with _α1 ...0;0 .

Fractional calculus has recently gained much momentum as extensive and significant progress on theoretical and practical aspects of the subject can easily be witnessed in the literature. As a matter of fact, the tools of fractional calculus have been effectively applied in the modelling of many physical and engineering problems. The recent development in the theory and methods for fractional calculus indicates its popularity. For some recent work on fractional boundary value problems, See [1-16] and the references therein.

2. Linear Problem

Let us recall some basic definitions of fractional calculus [1, 3].

Definition 2.1.

For a continuous function g:[0,∞)[arrow right]... , the Caputo derivative of fractional order q is defined as [figure omitted; refer to PDF] where [q] denotes the integer part of the real number q .

Definition 2.2.

The Riemann-Liouville fractional integral of order q is defined as [figure omitted; refer to PDF] provided the integral exists.

To define the solution of the boundary value problem (1.1) we need the following lemma, which deals with linear variant of the problem (1.1).

Lemma 2.3.

For a given σ∈C([0,1],...) , the unique solution of the problem [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

It is well known [3] that the solution of fractional differential equation in (2.3) can be written as [figure omitted; refer to PDF] Using ^cDp b=0 ( b is a constant), ^cDp t=^t1-p /Γ(2-p) , ^cDpIq σ(t)=^Iq-p σ(t) , (2.6) gives [figure omitted; refer to PDF] From the boundary condition _α1 x(0)+_β1^(cDp x(0))=_γ1 , we have [figure omitted; refer to PDF] By the boundary condition _α2 x(1)+_β2 ( ^{D c p} x(1))=_γ2 , we get [figure omitted; refer to PDF] which, on inserting the value of _b1 , gives [figure omitted; refer to PDF] Using (2.5) in the above equation, we obtain [figure omitted; refer to PDF] Substituting the values of _b1 and _b2 in (2.6), we get (2.4).

Remark 2.4.

In the limit p[arrow right]^1- , it has been observed that the solution (2.4) of problem (2.3) is not reduced to the solution of the resulting problem given by [figure omitted; refer to PDF] The solution of (2.12) is [figure omitted; refer to PDF] where Δ=_α1 (_α2 +_β2 )-_α2β1 ...0;0 . However, we notice that the solution (2.4) of problem (2.3) does not depend on the parameter _β1 (appearing in the boundary conditions of (2.3)). Thus we conclude that the parameter _β1 is of arbitrary nature. Furthermore, it has been found that the solutions (2.4) and (2.13) coincide by taking _β1 =0 in (2.13). Hence, for a particular choice of _β1 =0 in problems (2.3) and (2.12), the two problems have the same solution.

3. Main Results

Let ...9E;=C([0,1],...) denotes the Banach space of all continuous functions from [0,1][arrow right]... endowed with the norm defined by ||x||=sup {|x(t)|,t∈[0,1]} .

In view of Lemma 2.3, we define an operator F:...9E;[arrow right]...9E; by [figure omitted; refer to PDF] Observe that the problem (1.1) has solutions if and only if the operator equation Fx=x has fixed points.

Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problem (1.1) is new.

Theorem 3.1.

Suppose that f:[0,1]×...[arrow right]... is a continuous function and satisfies the following assumption:

(A₁ ): |f(t,x)-f(t,y)|...4;L|x-y| , for all t∈[0,1], L>0 , x,y∈... .

Then the boundary value problem (1.1) has a unique solution provided [figure omitted; refer to PDF]

Proof.

Setting _{sup t∈[0,1]} |f(t,0)|=M<∞ and choosing r...5;(ΛM+N)/(1-LΛ) , where [figure omitted; refer to PDF] we show that F_Br ⊂_Br , where _Br ={x∈...9E;:||x||...4;r} . For x∈_Br , we have [figure omitted; refer to PDF] Now, for x,y∈...9E; and for each t∈[0,1] , we obtain [figure omitted; refer to PDF] As (L/Γ(q+1))(1+|_α2 |/|_ν1 |)+(|_β2 |/|_ν1 |)(L/Γ(q-p+1))<1 , therefore F is a contraction. Thus, the conclusion of the theorem followed by the contraction mapping principle (Banach fixed point theorem).

Example 3.2.

Consider the following fractional boundary value problem [figure omitted; refer to PDF] Here, q=3/2 , p=1/2 , _α1 =1 , _α2 =1/2 , _β2 =1/3 , _γ1 =1/2 , _γ2 =2 , _β1 is arbitrary, and f(t,x)=(1/^(t+2)2 )(|x|/1+|x|) . As |f(t,x)-f(t,y)|...4;(1/4)|x-y| , therefore, (_A1 ) is satisfied with L=1/4 . Further, _ν1 =1/2+2/3π , _ν2 =7/4 and [figure omitted; refer to PDF] Thus, by the conclusion of Theorem 3.1, the boundary value problem (3.6) has a unique solution on [0,1] .

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii's fixed point theorem [17].

Theorem 3.3 (Krasnoselskii's fixed point theorem).

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

Theorem 3.4.

Let f:[0,1]×...[arrow right]... be a jointly continuous function satisfying the assumption (_A1 ) . In addition one assumes that

(A₂ ): |f(t,x)|...4;μ(t) , for all (t,x)∈[0,1]×... , and μ∈C([0,1],^...+ ) .

Then the problem (1.1) has at least one solution on [0,1] if [figure omitted; refer to PDF]

Proof.

Letting _{sup t∈[0,1]} |μ(t)|=||μ|| , we choose a real number r¯ satisfying the inequality [figure omitted; refer to PDF] and consider _Br¯ ={x∈C:||x||...4;r¯} . We define the operators ...AB; and ...AC; on _Br¯ as [figure omitted; refer to PDF] For x,y∈_Br¯ , we find that [figure omitted; refer to PDF] Thus, ...AB;x+...AC;y∈_Br¯ . It follows from the assumption (A₁ ) together with (3.8) that ...AC; is a contraction mapping. Continuity of f implies that the operator ...AB; is continuous. Also, ...AB; is uniformly bounded on _Br¯ as [figure omitted; refer to PDF]

Now we prove the compactness of the operator ...AB; .

In view of (A₁ ), we define _{sup (t,x)∈[0,1]×Br¯} |f(t,x)|=f¯ , and consequently we have [figure omitted; refer to PDF] which is independent of x . Thus, ...AB; is equicontinuous. Hence, by the Arzelá-Ascoli Theorem, ...AB; is compact on _Br¯ . Thus, all the assumptions of Theorem 3.3 are satisfied. So the conclusion of Theorem 3.3 implies that the boundary value problem (1.1) has at least one solution on [0,1] .

Our next existence result is based on Leray-Schauder nonlinear alternative [18].

Lemma 3.5 (nonlinear alternative for single-valued maps).

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 3.6.

Let f:[0,1]×...[arrow right]... be a jointly continuous function. Assume that:

(A₃ ): there exist a function p∈C([0,1],^...+ ) and a nondecreasing function ψ:^...+ [arrow right]^...+ such that |f(t,x)|...4;p(t)ψ(||x||) , for all (t,x)∈[0,1]×... ;

(A₄ ): there exists a constant M>0 such that

[figure omitted; refer to PDF] Then the boundary value problem (1.1) has at least one solution on [0,1] .

Proof.

Consider the operator F:...9E;[arrow right]...9E; defined by (3.1). We show that F maps bounded sets into bounded sets in C([0,1],...) . For a positive number r , let _Br ={x∈C([0,1],...):||x||...4;r} be a bounded set in C([0,1],...) . Then [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Next we show that F maps bounded sets into equicontinuous sets of C([0,1],...) . Let t[variant prime],^{t[variant prime][variant prime]} ∈[0,1] with t[variant prime]<^{t[variant prime][variant prime]} and x∈_Br , where _Br is a bounded set of C([0,1],...) . Then we obtain [figure omitted; refer to PDF] Obviously the right hand side of the above inequality tends to zero independently of x∈_Br as ^{t[variant prime][variant prime]} -t[variant prime][arrow right]0 . As F satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that F:C([0,1],...)[arrow right]C([0,1],...) is completely continuous.

Let x be a solution. Then, for t∈[0,1] and using the computations in proving that F is bounded, we have [figure omitted; refer to PDF]

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

In view of (A₄ ), there exists M such that ||x||...0;M . Let us set [figure omitted; refer to PDF] Note that the operator F:U¯[arrow right]C([0,1],...) is continuous and completely continuous. From the choice of U , there is no x∈∂U such that x=λF(x) for some λ∈(0,1) . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.5), we deduce that F has a fixed point x∈U¯ which is a solution of the problem (1.1). This completes the proof.

In the special case when p(t)=1 and ψ(|x|)=κ|x|+N we have the following corollary.

Corollary 3.7.

Let f:[0,1]×...[arrow right]... be a continuous function. Assume that there exist constants 0...4;κ<1/_Λ1 , where _Λ1 =(1/Γ(q+1)(1+|_α2 |/|_ν1 |))+(|_β2 |/|_ν1 |Γ(q-p+1)) and _N1 >0 such that |f(t,x)|...4;κ|x|+_N1 for all t∈[0,1] , x∈C[0,1] . Then the boundary value problem (1.1) has at least one solution.

Example 3.8.

Consider the following boundary value problem: [figure omitted; refer to PDF] Here, q=3/2 , p=1/2 , _α1 =1 , _α2 =1/2 , _β2 =1/3 , _γ1 =1/2 , _γ2 =2 , _β1 is arbitrary, and [figure omitted; refer to PDF] Clearly _N1 =1 and [figure omitted; refer to PDF] Thus, all the conditions of Corollary 3.7 are satisfied, and consequently the problem (3.21) has at least one solution.

Acknowledgment

The research of Bashir Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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