Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations
Rabha W. Ibrahim1, Adem Kılıçman2,, Faten H. Damag2
1Institute of Mathematical Sciences, University Malaya, Malaysia
2Department 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. https://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:
Definition 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.
Definition 2.2 In 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. ,
Definition 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
.


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. AssumptionsIn 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
![]() | (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 (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:
![]() | (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 (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:
![]() | (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 (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
![]() | (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.
References
[1] | “Bacteria.” UXL Encyclopedia of Science. 2002. Encyclopedia.com. 31 Mar. 2015 <https://www.encyclopedia.com>. | ||
![]() | |||
[2] | Stanescu, D., and Chen-Charpentier, B. M. (2009). Random coefficient differential equation models for bacterial growth. Mathematical and Computer Modelling, 50(5), 885-895. | ||
![]() | View Article | ||
[3] | Smith, H. L. (2007). Bacterial Growth. Retrieved on, 09-15. | ||
![]() | |||
[4] | Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). Academic press. | ||
![]() | |||
[5] | Miller, K. S., and Ross, B. An Introduction to the Fractional Calculus and Differential Equations. 1993. | ||
![]() | |||
[6] | Dhage, B. C. (2012). Basic results in the theory of hybrid differential equations with linear perturbations os second type. Tamkang Journal of Mathematics, 44(2), 171-186. | ||
![]() | View Article | ||
[7] | Lu, H., Sun, S., Yang, D., and Teng, H. (2013). Theory of fractional hybrid di_erential equations with linear perturbations of second type. Boundary Value Problems, 2013(1), 1-16. | ||
![]() | View Article | ||
[8] | Dhage, B. C., and Lakshmikantham, V. (2010). Basic results on hybrid differential equations. Nonlinear Analysis: Hybrid Systems, 4(3), 414-424. | ||
![]() | View Article | ||
[9] | Dhage, B. C., and Jadhav, N. S. (2013). Basic Results in the Theory of Hybrid Differential Equations with Linear Perturbations of Second Type. Tamkang Journal of Mathematics, 44(2), 171-186. | ||
![]() | View Article | ||
[10] | Dhage, B. C. (2014). Approximation methods in the theory of hybrid differential equations with linear perturbations of second type. Tamkang Journal of Mathematics, 45(1), 39-61. | ||
![]() | View Article | ||
[11] | Ibrahim R.W. (2012). Existence of deviating fractional differential equation, CUBO A Mathematical Journal, 14 (03), 127-140. | ||
![]() | |||
[12] | Ibrahim, R. W., Kiliçman, A., and Damag, F. H. (2015). Existence and uniqueness for a class of iterative fractional differential equations. Advances in Difference Equations, 2015(1), 1-13. | ||
![]() | View Article | ||
[13] | Ladde, G. S., Lakshmikantham, V., and Vatsala, A. S. (1985). Monotone iterative techniques for nonlinear differential equations (Vol. 27). Pitman Publishing. | ||
![]() | |||
[14] | Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using dfferential operators of Caputo type (Vol. 2004). Springer Science and Business Media. | ||
![]() | View Article | ||
[15] | Haubold, H. J., Mathai, A. M., and Saxena, R. K. (2011). Mittag-Leffer functions and their applications. Journal of Applied Mathematics, 2011. | ||
![]() | |||
[16] | Loverro A. (2004). Fractional Calculus: History, Deffnitions and Applications for the Engineer. | ||
![]() | |||
[17] | Kilbas, A. A. A., Srivastava, H. M., and Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204). Elsevier Science Limited. | ||
![]() | View Article | ||
[18] | Faten H. Damag, Adem Kılıcman and RabhaW. Ibrahim, Findings of Fractional Iterative Differential Equations Involving First Order Derivative, Int. J. Appl. Comput. Math. | ||
![]() | |||