**Turkish Journal of Analysis and Number Theory**

## Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations

**Rabha W. Ibrahim**^{1}, **Adem Kılıçman**^{2,}, **Faten H. Damag**^{2}

^{1}Institute of Mathematical Sciences, University Malaya, Malaysia

^{2}Department of Mathematics, University Putra Malaysia, Serdange, Malaysia

Abstract | |

1. | Introduction |

2. | Preliminaries |

3. | Main Results |

Competing Interests | |

Authors’ Contributions | |

References |

### 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

Received April 27, 2016; Revised June 17, 2016; Accepted June 28, 2016

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- 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*. Vol. 4, No. 3, 2016, pp 60-66. http://pubs.sciepub.com/tjant/4/3/2

- Ibrahim, Rabha W., Adem Kılıçman, and Faten H. Damag. "Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations."
*Turkish Journal of Analysis and Number Theory*4.3 (2016): 60-66.

- Ibrahim, R. W. , Kılıçman, A. , & Damag, F. H. (2016). Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations.
*Turkish Journal of Analysis and Number Theory*,*4*(3), 60-66.

- Ibrahim, Rabha W., Adem Kılıçman, and Faten H. Damag. "Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations."
*Turkish Journal of Analysis and Number Theory*4, no. 3 (2016): 60-66.

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### At a glance: Figures

### 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:

**De****fi****nition 2.1 **The definition of fractional (arbitrary) order derivative function* ** *of order* ** *for *Riemann-Liouville* is

(1) |

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

**De****f****i****nition 2.2 I**n which the function is defined such as* ** *fractional (arbitrary) order derivative function of order* ** *for the *Caputo* is

(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

(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. ,

**De****fi****nition 2.6 **^{[14]} Let* ** *The function* ** *introduced by

(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

(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

(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:

(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 (*a*0)*-*(*b*0) are achieved. Thereafter for any* ** *and * ** *is function which is a solution of the *FIHDE*

(8) |

and

(9) |

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

(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

(11) |

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

(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:

(13) |

for . Then , and

(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

(15) |

for all* *.

Assume that* ** *and* ** *and set

for certain Then,

and

(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

(17) |

and

(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

(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 (*b*4) substituted for (*b*2). And the *FIHDE*(6) provides extreme solutions on *I*.

Corollary 3.1 comes from Theorem 3.1 by substituting the constant *M *assumptions (*b*3) 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:

(20) |

and

(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 (*a*1)*-*(*a*2) and (*b*1) achieved. Then, there is a solution *v*(*s*) of (6), in the closed set satisfying

**Theorem 3.2 **Let the assumptions (*a*1)-(*a*2)*, *(*b*1) and (*b*4) achieved. Then either,

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

(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:

(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

(24) |

Set as follows:

(25) |

for * *Next,

and Hence,

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

(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 (*a*1)* - *(*a*2)*, *(*b*1) and (*b*2) 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

(27) |

and

(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.

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