Abstract and Applied Analysis

Volume 2012 (2012), Article ID 267108, 12 pages

doi:10.1155/2012/267108

Research Article

Existence Results for Solutions of Nonlinear Fractional Differential Equations

Ali Yakar¹ and Mehmet Emir Koksal²

¹Department of Statistics, Gaziosmanpasa University, 60250 Tokat, Turkey

²Department of Primary Mathematics Education, Mevlana University, 34528 Konya, Turkey

Received 15 February 2012; Accepted 1 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Ali Yakar and Mehmet Emir Koksal. 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.

Abstract.

This paper deals with theoretical and constructive existence results for solutions of nonlinear fractional differential equations using the method of upper and lower solutions which generate a closed set. The existence of solutions for nonlinear fractional differential equations involving Riemann-Liouville differential operator in a closed set is obtained by utilizing various types of coupled upper and lower solutions. Furthermore, these results are extended to the finite systems of nonlinear fractional differential equations leading to more general results.

1. Introduction

Fractional derivative, introduced around the 17th century, was developed almost until the 19th century. Although the introduction of the concept of fractional calculus involving fractional differentiation and integral is a few centuries old, it was realized only a few decades ago that these functional operations play an important role in various fields of science and engineering [1–8]. As a reason, since the significance of the fractional calculus has been more clearly perceived, many quality researches have been put forward on this branch of mathematical analysis in the literature (see [9–11] and the references therein), and many physical phenomena, chemical processes, biological systems, and so forth have described with fractional derivatives. In this framework, fractional differential equations have been gaining much interest and attracting the attention of many researchers. Some recent contributions on fractional differential equations can be seen in [9–20] and the references within.

On the other hand, the study for solutions of fractional ordinary and partial differential equations has received great interest by scientists. Especially, in the last decade, there are noteworthy works on the analytical and numerical solutions of fractional partial differential equations (see [21–28] and the references there in).

The attention drawn to basic theoretical concepts like the theory of existence and uniqueness of solutions to nonlinear fractional-order differential equations is obvious. Recently, there have been many paper investigating the existence and uniqueness of fractional-order differential equations [29–34].

An interesting and fruitful technique for providing existence results for nonlinear problems is the method of upper and lower solutions. This technique permits us to establish the existence results in a closed set, namely, the ordered interval, generated by upper and lower solutions. Thus, in this context, we are concerned with the existence of solutions of the following nonlinear fractional-order initial value problem (IVP):

where and is Riemann-Liouville (R-L) fractional derivative of order , .

The corresponding Volterra fractional integral equation of (1.1) is defined as

In recent years, Lakshmikantham and Vatsala investigated the existence theory and established a Peanos type local existence theorem for (1.1) by using integral inequalities and perturbation techniques [35]. McRae also studied an important existence result utilizing the method of upper and lower solutions [36], by means of which, monotone iterative and quasilinearization techniques are developed to fractional differential equations [37–40].

In this paper, we utilize the technique of upper and lower solutions and establish some existence results in terms of various types of coupled upper and lower solutions. Then we will extend this idea to the finite systems of nonlinear fractional differential equations.

The organization of this paper is as follows. In Section 2, we provide necessary background. In Section 3, we focus on the existence of solutions of nonlinear fractional differential equations in a sector. Finally, in Section 4, we extend our results to the finite systems of fractional differential equations.

2. Preliminaries

Now, we present some basic definitions and theorems which are used throughout the paper.

Definition 2.1. Let , then a function is said to be a   function if where

If we replace in (1.1) by the sum of two functions such that where then problem (1.1) takes the following form:

We give a variety of possible definitions of upper and lower solutions relative to (2.2).

Definition 2.2. Let , be locally Hölder continuous with exponent exist, and , then and are said to be as follows:

(i)natural upper and lower solutions of (2.2), respectively, if

(ii)coupled upper and lower solutions of type I of (2.2), respectively, if

(iii)coupled upper and lower solutions of type II of (2.2), respectively, if

(iv)coupled upper and lower solutions of type III of (2.2), respectively, if

where and .

Lemma 2.3. Let be locally Hölder continuous with exponent and for any , and one has

then it follows that

Proof. For the proof, see [16].

Remark 2.4. A dual result for Lemma 2.3 is valid.

The explicit solution of the following nonhomogeneous linear fractional differential equation

involving R-L fractional differential operator of order , is necessary for further development of our main results. In (2.9), is a real number and .

When we apply the method of successive approximations [16] to find the solution explicitly for the given nonhomogeneous IVP (2.9), we obtain

where denotes the two-parameter Mittag-Leffler function and .

If , we get

as the solution of the corresponding homogeneous IVP for (2.9).

We next give a Peano’s type local existence result and then an existence result in a special closed set generated by upper and lower solutions.

Theorem 2.5. Assume that and on where and and , then IVP (1.1) possesses at least one solution on where .

For the proof of the theorem, see [35].

If the existence of upper and lower solutions such that for IVP (1.1) is known, the existence of solutions can be proved in the closed set

Theorem 2.6. Let and be natural upper and lower solutions of IVP (1.1), which are locally Hölder continuous with exponent such that on and , then there exists a solution of IVP (1.1) satisfying .

For the detailed proof of the above theorem, see [36].

3. Existence Theorems

We are in position to give existence of solutions in the closed set.

Theorem 3.1. Let and be coupled upper and lower solutions of type I of (2.2) such that and Moreover, assume that is nonincreasing in for each , then there exists a solution of (2.2) satisfying on .

Proof. Let be defined by

Then defines a continuous extension of to which is also bounded since is bounded on . Employing Theorem 2.5, we get the following equation:

having a solution on We wish to prove that on . For this purpose, consider the following equations:

where and . This implies that

which gives where . It follows that in view of the upper and lower definitions of and . It is enough to show that

which proves the claim of the theorem as . First, suppose that the inequality on is not true, then there would exist a such that

Hence, , therefore and . If we construct , we get and . Employing Lemma 2.3,we obtain which gives a contradiction

Here, we have used the nonincreasing property of in for each and the fact that .

Similarly, it can be proved that the other side of the inequality (3.5) is valid for . To do this, we must show that on. Suppose that it is not true and so there exists a such that and for , then and . Hence, . If we put , we get and . Employing Lemma 2.3, we find . Since the nonincreasing property of in for each and the fact that , we arrive at the contradiction

Consequently, we have on , and letting , we get on proving the theorem.

Theorem 3.2. Let and be coupled upper and lower solutions of type II of (2.2) such that and Moreover, assume that is nonincreasing in for each , then there exists a solution of (2.2) satisfying on .

Proof. Let be defined by

Then defines a continuous extension of to which is also bounded since is bounded on . Therefore, by Theorem 2.5,

has a solution on We intend to show that on . For this purpose, consider the following equations:

where and are defined as before. It follows that . It is enough to show that

Suppose that it is not true. Thus, there would exist a such that

Hence, ; therefore, we get and . Setting , we have and , . Employing Lemma 2.3, we obtain which yields a contradiction

Here, we have used the nonincreasing property of in for each and the fact that . Thus, we get on .

After utilizing the similar procedure, the other case can be proved easily. Consequently, we have on , and letting , we get on which proves the theorem.

Theorem 3.3. Let and be coupled upper and lower solutions of type III of (2.2) such that and Moreover, assume that both and are nonincreasing in for each , then there exists a solution of (2.2) satisfying on .

Proof. In a similar manner in previous theorems, the existence of the solution can be proved. Thus, we omit the details.

4. Extensions to the Systems of Differential Equations

We can generalize the result of Theorem 2.6 to finite systems of fractional differential equations. Consider the following fractional differential system:

where , and is the fractional derivative of and .

At this point, we shall need an important property, known as quasimonotone nondecreasing relative to systems of inequalities.

Definition 4.1. A function is said to possess quasimonotone nondecreasing property if , for some ,then .

Here, we shall be using vectorial inequalities, which are understood to mean the same inequalities hold between their corresponding components.

Next, we give the following existence result for systems of differential equations.

Theorem 4.2. Let and be upper and lower solutions of (4.1), respectively, such that and where . Moreover, assume that is quasimonotone nondecreasing in for each , then there exists a solution of (4.1) satisfying on .

The proof of this theorem is a special case of the following theorem in which we choose not to be quasimonotone nondecreasing in provided that we strengthen the notion of upper and lower solutions of (4.1) as follows:

For each ,

We state and prove the following existence result relative to the definition of upper and lower solutions in (4.2).

Theorem 4.3. Let and be upper and lower solutions of (4.1), respectively, satisfying the relations given in (4.2), which are also locally Hölder continuous with exponent such that and , then there exists a solution of (4.1) satisfying on .

Proof. Let given by

then defines a continuous extension of to which is also bounded since is bounded on . Therefore, by Theorem 2.5,

has a solution on . For and consider and where . It is clear that . We wish to show that on . Suppose that it is not true, then there exists an index and a such that

Thus, we have and . Setting , it follows that

Applying Lemma 2.3 to the component , we get or which yields a contradiction

Now, letting , we arrive at on which proves the conclusion of the theorem.

Sometimes, we can have arbitrary coupling relative to upper and lower solutions. Let and be nonnegative integers for each  , so that we can split the vector into . Then the system (4.1) can be written as

where .

Definition 4.4. A function is said to possess a mixed quasimonotone property if for each , is monotone nondecreasing in and monotone nonincreasing in .

Definition 4.5. The functions and are said to be coupled upper and lower quasisolutions of (4.8) if they satisfy

for each  .

Next, we give an existence result which is also a special case of Theorem 4.3.

Theorem 4.6. Let be coupled upper and lower quasisolutions of (4.8) and . If possesses a mixed quasimonotone property, then there exists a solution of (4.8) such that on .

It should be noted that if satisfies a mixed quasimonotone property, then (4.2) holds for coupled upper and lower quasisolutions given by (4.9). Therefore, Theorem 4.3 includes Theorem 4.6 as a special case.

5. Conclusion

In this work, some existence theorems have been established for nonlinear fractional-order differential equations relative to coupled upper and lower solutions. The differential operator is taken in the Riemann-Liouville sense. For the further developments in applications of dynamical systems, we have generalized these results to the finite systems of nonlinear fractional differential equations. Being defined by a suitable differential operator, the process of finding a solution between upper and lower solutions generating a closed set could be applied to various types of linear and nonlinear fractional partial differential equations as a future work.

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