Abstract and Applied Analysis

Volume 2012 (2012), Article ID 163648, 16 pages

doi:10.1155/2012/163648

Research Article

Existence Results for the Distributed Order Fractional Hybrid Differential Equations

Hossein Noroozi, Alireza Ansari, and Mohammad Shafi Dahaghin

Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran

Received 22 July 2012; Accepted 7 October 2012

Academic Editor: Yongfu Su

Copyright © 2012 Hossein Noroozi et al. 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.

We introduce the distributed order fractional hybrid differential equations (DOFHDEs) involving the Riemann-Liouville differential operator of order with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved via a fixed point theorem in the Banach algebras under the mixed Lipschitz and Caratheodory conditions.

1. Introduction

The differential equations involving Riemann-Liouville differential operators of fractional order are very important in the modeling of several physical phenomena [1, 2]. In recent years, quadratic perturbations of nonlinear differential equations and first-order ordinary functional differential equations in Banach algebras, have attracted much attention to researchers. These type of equations have been called the hybrid differential equations [3–8]. One of the important first-order hybrid differential equations (HDE) is defined as [4, 9]

where is a bounded interval in for some and with . Also, and , such that is the class of continuous functions and is called the Caratheodory class of functions which are Lebesgue integrable bounded by a Lebesgue integrable function on . Moreover

(i)the map is measurable for each , (ii)the map is continuous for each .

For the above hybrid differential equation, Dhage and Lakshmikantham [9] established existence, uniqueness, and some fundamental differential inequalities. Also, they stated some theoretical approximation results for the extremal solutions between the given lower and upper solutions [10]. Later, Zhao. et al. [11] developed the following fractional hybrid differential equations involving the Riemann-Liouville differential operators of order ,

where is bounded in for some and , .

They established the existence, uniqueness, and some fundamental fractional differential inequalities to prove existence of the extremal solutions of (1.2). Also, they considered necessary tools under the mixed Lipschitz and Caratheodory conditions to prove the comparison principle.

Now, in this article in view of the distributed order fractional derivative [12–14], we develop the distributed order fractional hybrid differential equations (DOFHDEs) with respect to a nonnegative density function.

In this regard, in Section 2 we introduce the distributed order fractional hybrid differential equation. Section 3 is about some main theorems which are used in this paper. In Section 4, we prove the existence theorem for this class of equations, and we express some special cases for the density function used in the distributed order fractional hybrid differential equation. Finally, the main conclusions are set.

2. The Fractional Hybrid Differential Equation of Distributed Order

In this section, we recall some definitions which are used throughout this paper.

Definition 2.1 (see [1, 2]). The fractional integral of order with the lower limit for the function is defined as

Definition 2.2 (see [1, 2]). The Riemann-Liouville derivative of order with the lower limit for the function can be written as

Definition 2.3. The distributed order fractional hybrid differential equation (DOFHDEs), involving the Riemann-Liouville differential operator of order with respect to the nonnegative density function , is defined as

Moreover, the function is continuous for each , where is bounded in for some . Also, and .

3. The Main Theorems

In this section, we state the existence theorem for the DOFHDE (2.3) on . For this purpose, we define a supremum norm of in as

and for

is a multiplication in this space. We consider is a Banach algebra with respect to norm and multiplication (3.2). Moreover the norm for is defined by

Now, for expressing the existence theorem for the DOFHDE (2.3), we state a fixed point theorem in the Banach algebra.

Theorem 3.1 (see [15]). Let be a nonempty, closed convex, and bounded subset of the Banach algebra and let and be two operators such that

(a) is Lipschitz constant , (b) is completely continuous, (c) for all , (d), where .

Then the operator equation has a solution in .

At this point, we consider some hypotheses as follows.

  The function is increasing in almost everywhere for .   There exists a constant such that

   for all and .   There exists a function and a real nonnegative upper bound such that

for all and .

Theorem 3.2 (Titchmarsh theorem [16]). Let be an analytic function which has a branch cut on the real negative semiaxis. Furthermore, has the following properties:

for any sector , where . Then, the Laplace transform inversion can be written as the Laplace transform of the imaginary part of the function as follows:

Definition 3.3. Suppose that be a metric space and let . Then, is equicontinuous if for all there exists such that for all and

Theorem 3.4 (Arzela-Ascoli theorem [17]). Let be a compact metric space and let . Then, is compact if and only if is closed, bounded, and equicontinuous.

Theorem 3.5 (Lebesgue dominated convergence theorem [18]). Let be a sequence of real-valued measurable functions on a measure space . Also, suppose that the sequence converges pointwise to a function and is dominated by some integrable function in the sense that

for all numbers in the index set of the sequence and all points in . Then, is integrable and

4. Existence Theorem for the DOFHDEs

We apply the following lemma to prove the main existence theorem of this section.

Lemma 4.1. Assume that hypothesis () holds in pervious section, then for any and , the function is a solution of the DOFHDE (2.3) if and only if satisfies the following equation

such that and

Proof. Applying the Laplace transform on both sides of (2.3) and letting

we have

Since , we have

and hence,

such that

Now, using the inverse Laplace transform on both sides of (4.6) and applying the convolution product, we get

or equivalently

Since is an analytic function which has a branch cut on the real negative semiaxis, according to the Titchmarsh Theorem 3.2 we get

which by the Laplace transform definition, (4.1) is held. Conversely, let satisfies (4.1), therefore, satisfies the equivalent equation (4.9). By in (4.1), we have

According to hypothesis , the map is injective in and hence . Next, with dividing (4.9) by and using the Laplace transform operator on both sides of this equation, (4.6) also holds. Since , we obtain (4.4) and by applying the inverse Laplace transform, (2.3) also holds.

Theorem 4.2. Suppose that hypothesis ()–() hold. Further, if

then, the DOFHDE (2.3) has a solution defined on J.

Proof. We set as a Banach algebra and define a subset of by

such that

It is obvious that is closed and if , then and , also by properties of the norm, we get

Therefore, is a convex and bounded and by applying Lemma 4.1, DOFHDE (2.3) is equivalent to (4.1).

Define operators and by

thus, from (4.1), we obtain the operator equation as follows:

If operators and satisfy all the conditions of Theorem 3.1, then the operator equation (4.17) has a solution in . For this paper, let which by hypothesis we have

and if for all take a supremum over , then we have

Therefore, is a Lipschitz operator on with the Lipschitz constant , and the condition (a) from Theorem 3.1 is held. Now, for checking the condition (b) from this theorem, first, we shall show that is continuous on .

Let be a sequence in such that

with . By applying the Lebesgue-dominated convergence Theorem 3.5 for all , we get

Thus, is a continuous operator on . In next stage, we shall show that is a compact operator on . For this paper, we shall show that is a uniformly bounded and eqicontinuous set in . Let , then by hypothesis for all we have

Let such that . Then by the existence Laplace transform theorem [19], there exists a constant such that for a constant that ,

Hence, we find an upper bound for the integral of (4.22) as follows:

such that

Finally, with respect to the inequality (4.22) we obtain

which by applying supremum over , we get for all

Thus, is uniformly bounded on .

In this stage, now we show that is an equicontinuous set in . Let , with . In this respect, we have for all

If we set and , then by Laplace transform definition and (4.23), for and we can write

Therefore, we have

Also, by (4.24) we have

Finally, with respect to (4.28), (4.30), and (4.31) we obtain

Hence, for , there exists such that if , then for all and all we have

which implies that is an equicontinuous set in and according to the Arzela-Ascoli Theorem 3.4, is compact. Therefore is continuous and compact operator on into and is a completely continuous operator on and the condition (b) from the Theorem 3.1 is held.

For checking the condition (c) of Theorem 3.1, let and be arbitrary such that . Then, by hypothesis we get

Therefore,

which by taking a supremum over , we obtain

Thus, the condition (c) of Theorem 3.1 is satisfied. If we consider

the hypothesis (d) of Theorem 3.1 is satisfied.

Hence, all the conditions of Theorem 3.1 are satisfied and therefore the operator equation has a solution in . As a result, the DOFHDE (2.3) has a solution defined on and proof is completed.

5. Some Special Cases

In this section, we discuss some special cases of the density function for the DOFHDE (2.3) and we find the operators and which introduce in Theorem 4.2. In proof of Lemma 4.1, the following equation is equivalent to the DOFHDE (2.3),

such that,

(1) Let . Then we have

Thus,

where is the exponential integral defined by

Therefore, for this case, the DOFHDE (2.3) is

and it is equivalent to the following equation:

such that the operators and in Theorem 4.2 are

(2) Two-term equation: Let , which also, and are nonnegative constant coefficients and is the Dirac delta function. Then by the following inverse Laplace transform [2]:

where is the Mittag-Leffler function in two parameters

we get the DOFHDE (2.3) as

It is equivalent to the following equation

such that the operators and in Theorem 4.2 are

(3) Three-term equation: Let,

which and , , and are nonnegative constant coefficients and is the Dirac delta function. Then, by virtue of [2]

where

and is the th derivative of the Mittag-Leffler function in two parameters

We get the DOFHDE (2.3) as

It is equivalent to the following equation:

such that the operators and in Theorem 4.2 are

(4) General Case: -term equation: suppose that

which and for are nonnegative constant coefficients. Therefore, by the following inverse Laplace transform [2], we have

where

Thus, for this case, the DOFHDE (2.3) is

It is equivalent to the following equation:

and the operators and in Theorem 4.2 are given by

6. Conclusions

In this paper, we introduced a new class; the fractional hybrid differential equations of distributed order and stated an existence theorem for it. We pointed out a fixed point theorem in the Banach algebra for the existence of solution. Basis of this theorem is on finding two operator equations which in special cases for multiterms fractional hybrid equations are given with respect to the derivatives of Mittag-Leffler function.

Acknowledgment

The authors have been partially supported by the Center of Excellence for Mathematics, Shahrekord University.

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