Qualitative Behavior for Fourth-Order Nonlinear Differential Equations
Engineering Mathematics
Volume 2, Issue 2, December 2018, Pages: 63-67
Received: Oct. 1, 2018; Accepted: Oct. 16, 2018; Published: Nov. 5, 2018
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Authors
Omar Bazighifan, Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout, Yemen
Elmetwally Elabbasy, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
Osama Moaaz, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
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Abstract
In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.
Keywords
Oscillation, Fourth-Order, Delay Differential Equations
To cite this article
Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz, Qualitative Behavior for Fourth-Order Nonlinear Differential Equations, Engineering Mathematics. Vol. 2, No. 2, 2018, pp. 63-67. doi: 10.11648/j.engmath.20180202.12
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
R. Agarwal, S. R. Grace, P. Wong, On the bounded oscillation of certain fourth order functional differential equations, Nonlinear Dyn. Syst., 5 (2005), 215-227.
[2]
R. Agarwal, S. Grace and D. O'Regan, Oscillation theory for difference and functional differential equations, Kluwer Acad. Publ., Dordrecht (2000).
[3]
R. Agarwal, M. Bohner, T. Li, C. Zhang, Oscillation theorems for fourth-order half- linear delay dynamic equations with damping, Mediterr. J. Math., 11 (2014), 463-475.
[4]
B. Baculikova, J. Džurina, I. Jadlovska, Oscillation of solutions to fourth-order tri- nomial delay differential equations, Electron. J. Differential Equations, 70 (2015), 1-10.
[5]
O. Bazighifan, Oscillation criteria for nonlinear delay differential equation, Lambert Academic Publishing, Germany, (2017).
[6]
O. Bazighifan, Oscillatory behavior of higher-order delay differential equations, General Letters in Mathematics, 2 (2017), 105-110.
[7]
Džurina and Jadlovskă, Oscillation theorems for fourth order delay differential equations with anegative middle term, Math. Meth. Appl. Sci., 42 (2017), 1-17.
[8]
E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation Solution for Higher-Order Delay Differential Equations, Journal of King Abdulaziz University, 29 (2017), 45-52.
[9]
E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation of Fourth-Order Advanced Differential Equations, Journal of Modern Science and Engineering, 3 (2017), 64-71.
[10]
E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation Criteria for Fourth-Order Nonlinear Differential Equations, International Journal of Modern Mathematical Sciences, 15 (2017), 50-57.
[11]
S. Grace and B. Lalli, Oscillation theorems for nth order nonlinear differential equations with deviating arguments, Proc. Am. Math. Soc., 90 (1984), 65-70.
[12]
S. Grace, R. Agarwal and J. Graef, Oscillation theorems for fourth order functional differential equations, J. Appl. Math. Comput., 30 (2009), 75--88).
[13]
I. Gyori and G. Ladas, Oscillation theory of delay differential equations with applications, Clarendon Press, Oxford (1991).
[14]
I. Kiguradze and T. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Kluwer Acad. Publ., Drodrcht (1993).
[15]
G. Ladde, V. Lakshmikantham and B. Zhang, Oscillation theory of differential equations with deviating arguments, Marcel Dekker, NewYork, (1987).
[16]
T. Li, B. Baculikova, J. Dzurina and C. Zhang, Oscillation of fourth order neutral differential equations with p-Laplacian like operators, Bound. Value Probl., 56 (2014), 41-58.
[17]
O. Moaaz, E. M. Elabbasy and O. Bazighifan, On the asymptotic behavior of fourth-order functional differential equations, Adv. Difference Equ., 261 (2017), 1-13.
[18]
O. Moaaz, Oscillation properties of some differential equations, Lambert Academic Publishing, Germany, (2017).
[19]
O. Moaaz, E. M. Elabbasy and E. Shaaban, Some oscillation criteria for a class of third order nonlinear damped differential equations, International Journal of Modern Mathematical Sciences, 15 (2017), 206-218.
[20]
C. Tunc and O. Bazighifan, Some new oscillation criteria for fourth-order neutral differential equations with distributed delay, Electronic Journal of Mathematical Analysis and Applications, 7 (2019), 235-241.
[21]
C. Philos, On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay, Arch. Math. (Basel), 36 (1981), 168-178.
[22]
Ch. Hou, S. S. Cheng, Asymptotic dichotomy in a class of fourth-order nonlinear delay differential equations with damping, Abstr. Appl. Anal., 26 (2009), 1-7.
[23]
C. Zhang, T. Li, R. Agarwal, M. Bohner, Oscillation results for fourth-order nonlinear dynamic equations, Appl. Math. Lett., 25(2012), 2058-2065.
[24]
C. Zhang, T. Li, B. Sun and E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett., 24 (2011), 1618-1621.
[25]
C. Zhang, T. Li and S. Saker, Oscillation of fourth-order delay differential equations, J. Math. Sci., 201 (2014), 296-308.
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