Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: https://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2016, 4(3), 60-66
DOI: 10.12691/tjant-4-3-2
Open AccessArticle

Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations

Rabha W. Ibrahim1, Adem Kılıçman2, and Faten H. Damag2

1Institute of Mathematical Sciences, University Malaya, Malaysia

2Department of Mathematics, University Putra Malaysia, Serdange, Malaysia

Pub. Date: August 09, 2016

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

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

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  “Bacteria.” UXL Encyclopedia of Science. 2002. Encyclopedia.com. 31 Mar. 2015 <http://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.
 
[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.
 
[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.
 
[8]  Dhage, B. C., and Lakshmikantham, V. (2010). Basic results on hybrid differential equations. Nonlinear Analysis: Hybrid Systems, 4(3), 414-424.
 
[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.
 
[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.
 
[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.
 
[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.
 
[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.
 
[18]  Faten H. Damag, Adem Kılıcman and RabhaW. Ibrahim, Findings of Fractional Iterative Differential Equations Involving First Order Derivative, Int. J. Appl. Comput. Math.