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Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations

Rabha W. Ibrahim, Adem Kılıçman*, Faten H. Damag

Department of Mathematics, University Putra Malaysia, Serdange, Malaysia

Abstract: This paper highlights the mathematical model of biological experiments, that have an effect on our lives. We suggest a mathematical model involving fractional differential operator, kind of hybrid iterative fractional differential equations. Our technique is based on monotonous iterative in the nonlinear analysis. The monotonous sequences described extremal solutions converging for hybrid monotonous fractional iterative differential equations. We apply the monotonous iterative method under appropriate conditions to prove the existence of extreme solutions. The tool relies on the Dhage fixed point Theorem. This theorem is required in biological studies in which increasing or decreasing know freshly split bacterial and could control.

Keywords: fractional differential equation, fractional differential operator, fractional calculus, monotonous sequences, extreme solution

Citation: Rabha W. Ibrahim, Adem Kılıçman, Faten H. Damag, Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations. Turkish Journal of Analysis and Number Theory, 4 (3). 60-66. January 2016. doi:10.12691/tjant-4-3-2

1. Introduction

A class of mathematical models based on differential equations plays a big role in all areas of life, such as physics, engineering, agriculture, and medicine, etc. In this research, we use biological mathematical models in the studying of the growth and reproduction of bacteria or decay. It is well known that the bacteria are tiny bodies, generally made of a cell that does not have chlorophyll. But for the virus, it is the little things of life on Earth. Bacteria are likely to multiply very quickly under favorable conditions, the formation of settlements of millions or even billions of organisms in such a tiny space like a drop of water. Any bacteria have a particular temperature rank that can survive. For a particular rather bacteria, this range can be too high, too low, or someplace in between, although it is still a narrow range [1].

Results extension and predictions of mathematical models must be considered for further understanding of the different processes. These results are essential in more fields like science and engineering. An example is the application of differential equations for growth and bacterial cell division [2]. Humans have learned to exploit bacteria and other microbe beneficial uses, such as genetically engineered human insulin. Right now it is more convenient to insert the human insulin gene in bacteria and let produced in large industrial enzymes. It is therefore important achieve bacterial growth [3].

A class of fractional differential equations manipulates a big role in almost all sciences such as engineering, medicine, economics, social, linguistic and physics. This area is the more general field of mathematical analysis. It includes, fractional order that is, the value of This field is not new, but older age ranges between 200 and 300 years, but now has become the subject of studies and concerns because it gives effective results and more general, and therefore, we find a lot of scholars working in this field ([4,5]). The significance of differential equations was that polls hybrid entails a number of dynamic systems treated as special cases ([6,7]). Dhage and Lakshmikantham [8], Dhag and Jadhav [9], showed some of the main results of the hybrid linear differential equations of the first order and disturbances of second type. [10]. A good mathematical model for growth bacteria is described by iterative differential equation. Ibrahim [11] established the existence of a class of fractional iterative differential equation (Cauchy type) utilizing the non-expansive operator technique. This class is generated by the authors [12].

This paper deals with the mathematical model of biological experiments, that have an affect on our lives. We impose a mathematical model involving fractional differential operator, type hybrid iterative fractional differential equations. Our method is based on monotonous iterative in the nonlinear analysis. The monotonous sequences described extremal solutions converging for hybrid monotonous fractional iterative differential equations. We employ the monotonous iterative method under appropriate conditions to show that the existence of extreme solutions. The tool relies on the Dhage fixed point Theorem. This theorem is required in biological studies in which increasing or decreasing know freshly split bacterial and could control.

2. Preliminaries

Recall the following preliminaries:

Definition 2.1 The definition of fractional (arbitrary) order derivative function of order for Riemann-Liouville is

[picture] | (1)

in which is a whole number and is a real number.

Definition 2.2 In which the function is defined such as fractional (arbitrary) order derivative function of order for the Caputo is

[picture] | (2)

in which is a whole number and is a real number.

Definition 2.3 The fractional (arbitrary) integral of order for Riemann-Liouville is defined by the formula

[picture] | (3)

Based on the Riemann-Liouville differential operator, we impose the following useful definitions:

Definition 2.4 We said that is a function which is a lower solution for the equation introduced on if

1. is continuous, and

2. ,

Definition 2.5 We said that is a function which is an upper solution for the equation introduced on I if

1. is continuous, and

2. ,

Definition 2.6 [14] Let The function introduced by

[picture] | (4)

provided that the series converging be called the Mittag-Leffer function of order

Remark 2.1 The exponential function corresponding to Figure 1 shows the Mittag Leffer function for different values of . More general class of functions follows:

Definition 2.7 Let The function introduced by

[picture] | (5)

provided that the series converges is called Mittag- Leffer function two parameters with parameters and .

Figure 1. The Mittag-Leffler Function for different

Remark 2.2 Clearly the Mittag-Leffer functions a parameter can be defined in terms of their counterparts in two parameters using the relationship in

Cases. [15] Some special cases of the Mittag-Leffer function as follow:

1.

2.

3.

4.

5.

Definition 2.8 Assume the closed period bounded interval in ( the real line), for some The initial value problem of fractional iterative hybrid differential equations (FIHDE) can be formulated as

[picture] | (6)

with where are continuous. A solution of the FIHDE (6) can be defined by

1. is a function which is continuous and

2. contented the equations in (6). In which space is of real-valued continuous functions defined on I.

Also, our definition of the fractional iterative of hybrid equation integrated FIHIE as following:

[picture] | (7)

Note that the fractional hybrid differential equations can be found in [16].

2.1. Assumptions

In the following assumptions relating to function is very important in the studying of Eq(6).

(a0) The function is injective in

(b0) is a bounded real-valued function on

(a1) The function is increasing in for all

(a2) There is a constant so that

and

(b1) There is a constant o that and

(b2) The function is nondecreasing in

(b3) There is a real number so that

and with

(b4) There is a constant so that

for each and

3. Main Results

In this section, our purpose is to discuss the technique of monotone iterative to FIHDE(6) some under appropriate conditions are charged solutions proving the existence of extreme.

Lemma 3.1 [17] Let and

1. The equality achieves

2. The equality

exerted on nearly throughout in I.

Lemma 3.2 Suppose the assumptions (a0)-(b0) are achieved. Thereafter for any and is function which is a solution of the FIHDE

[picture] | (8)

and

[picture] | (9)

if and only if v must be the solution of the fractional iterative of hybrid equation integrated FIHIE

[picture] | (10)

Proof. Firstly we suppose that v is a solution for the Cauchy problem (6) defined on I. Since the Riemann-Liouville fractional integral operator is a monotonous, then operating Eq.(6) by . In view of Lemma 3.1, we obtain

followed by (9), we obtain

namely

Therefore, (10) is satisfied.

On the contrary, suppose that the function v fulfills the Eq(10) in I. Thus the application on either side of (10), (8) holds. Again, substituting in (10) by income

the map is increasing in the map is injective in and This completes the proof.

Theorem 3.1 Let and be the lower and upper solutions respectively fulfilling on I and let the assumptions (a1)-(a2) and (b3)-(b4) achieved. Then there exist monotone sequences and so that and regularly on I, in which is minimal and is maximal solutions for the FIHDE(6) on I and

[picture] | (11)

Proof. For any with on I, deem fractional iterative hybrid differential equation,

[picture] | (12)

Presently the Eq(12) is tantamount the issue

Integrating factor using the equation above can be formed

where is called Mittag-Leffer function see to some special cases above and by Lemma 3.2 the iterative differential equation up is equivalent to hybrid FIHIE(7)

By hypothesis (b4), there is one solution of the FIHDE(6) defined in I due to the principle of Banach contraction.

Define the map W on by Such a mapping will utilize for defining sequences and . Let us presently show that

1. and

2. W is monotonous operator in this sector

To show (1), set in which be a unique solution of the Eq(12) on I with . Indicate as follows:

[picture] | (13)

for . Then , and

[picture] | (14)

This proves that

and than from (14), we get

Since assumption (a1) achieves, or, equivalently, Similarly, we can show that

To show (2), let be such that on I . Therefore, we obtain

[picture] | (15)

for all .

Assume that and and set

for certain Then,

and

[picture] | (16)

As above, the foregoing inequality yields that on I which implicates that prove (2). Presently we defined both sequences and by

for . The monotony of the operator W implies that

It is plain to demonstrate that the sequences and are regularly bounded and equi-continuous on I: Clearly, and are the solutions in from

[picture] | (17)

and

[picture] | (18)

To demonstrate that and are solution of extremes for FIHDE(6) on I, we should check that if v is any another solution for FIHDE(6), so that then

Presume that for some on I and set

for some

and

[picture] | (19)

This yields that for the whole In the same way, it proves that for the whole Since on I , we have, by induction precept which on I for all Taking the limit as , we conclude which on I. Thus and are straightly the minimal and maximal solutions for the FIHDE(6) on I.

Corollary 3.1 Let and are straightly the solutions of lower and upper for the FIHDE(6) on I fulfilling on I and that all conditions of Theorem 3.1 are fulfilled assumptions (b4) substituted for (b2). And the FIHDE(6) provides extreme solutions on I.

Corollary 3.1 comes from Theorem 3.1 by substituting the constant M assumptions (b3) with M = 0. Therewith, we discuss the case when is non-increasing in nearly v throughout for Let and are straightly the solutions of lower and upper for the FIHDE(6) on I. Now consider the two sequences and iterations definite in the following way:

[picture] | (20)

and

[picture] | (21)

for

Next we shown that every sequences of and having two alternating sequences converging uniformly and monotonously with the solutions of extremal for FIHDE(6) on I. We need the following result, which can be found in [18]

Lemma 3.3 Let be lower and upper solutions of FIHDE(6) satisfying , and let the assumptions (a1)-(a2) and (b1) achieved. Then, there is a solution v(s) of (6), in the closed set satisfying

Theorem 3.2 Let the assumptions (a1)-(a2), (b1) and (b4) achieved. Then either,

1. iterates presented by Eq(20) and a unique solution v of FIHDE(6) introduced in I fulfill:

[picture] | (22)

for every if Moreover, the sequence and and monotonically converge uniformly toward and fulfilling for all or

2. iterates given by Eq(21) and a unique solution v of FIHDE(6) posed in I fulfill:

[picture] | (23)

for every if In addition, the sequence and converge uniformly monotonically toward and fulfilling for all

Actually as extreme solutions are unique, and on I fulfilling

Proof. By Lemma 3.3, there exist a lower solution an upper solution and a solutio v for the FIHDE(6), such that

We will just demonstrate that case (1), since the demonstrate of case(2) followers with similar arguments. Presume on I. First we demonstrate that

[picture] | (24)

Set as follows:

[picture] | (25)

for Next,

and Hence,

In view of the assumption (a1), we obtain on I. Let

[picture] | (26)

for Consequently, we have

and Hence,

Since assumption (a1) achieved, one has on I. By employing similar procedures, we may prove respectively that

In order to demonstrate Eq(22), the induction principle is applied, i.e suppose that Eq(22) is true for some t and demonstrate that it is true for (t + 1). Let

Next, using the monotonicity of , which

and This proves and thus The repetition of arguments similar we can obtain

on I. Since Eq(22) is right for t = 1, it is following by principle of induction which Eq(22) achieved for all t. Obviously, that the sequences are equicontinuous and uniformly bounded; thus in virtue of the Arzela-Ascoli Theorem, they are uniformly converges and monotonously to respectively and that on I. Case (2) yields with similar arguments. This completes the proof.

Corollary 3.2 Let the hypothesis (a1) - (a2), (b1) and (b2) satisfied. Moreover, let

where on I. Then on I.

Observed which in the show of Theorem 3.2, and are in quasi-solutions for the FIHDE(6), since we have that

[picture] | (27)

and

[picture] | (28)

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors jointly worked on deriving the results and approved the final manuscript.

References

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