ON SOME QUALITATIVE BEHAVIORS OF CERTAIN DIFFERENTIAL EQUATIONS OF FOURTH ORDER WITH MULTIPLE RETARDATIONS

Erdal Korkmaz; Cemil Tunc

doi:10.11948/2016026

2016 Volume 6 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2016 > 6(2): 336-349

Next Article Previous Article

Erdal Korkmaz, Cemil Tunc. ON SOME QUALITATIVE BEHAVIORS OF CERTAIN DIFFERENTIAL EQUATIONS OF FOURTH ORDER WITH MULTIPLE RETARDATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 336-349. doi: 10.11948/2016026

Citation:

Erdal Korkmaz, Cemil Tunc. ON SOME QUALITATIVE BEHAVIORS OF CERTAIN DIFFERENTIAL EQUATIONS OF FOURTH ORDER WITH MULTIPLE RETARDATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 336-349. doi: 10.11948/2016026

ON SOME QUALITATIVE BEHAVIORS OF CERTAIN DIFFERENTIAL EQUATIONS OF FOURTH ORDER WITH MULTIPLE RETARDATIONS

Erdal Korkmaz¹,
Cemil Tunc²

1 Dep. of Mathematics, Mus Alparslan University, Mus, 49100 Mus, Turkey;
2 Dep. of Mathematics, Yuzuncu Yil University, Van, 65080, Turkey

Abstract

Abstract

In this paper, we give sufficient conditions to guarantee the asymptotic stability and boundedness of solutions to a kind of fourth-order functional differential equations with multiple retardations. By using the LyapunovKrasovskii functional approach, we establish two new results on the stability and boundedness of solutions, which include and improve some related results in the literature.
- Boundedness /
- stability /
- Lyapunov functional
MSC: 34D20;34C11

FullText(HTML)

References

[1]	A. M. A. Abou-El-Ela and A. I. Sadek, A stability result for certain fourth order differential equations, Ann. Differential Equations, 6(1990)(1), 1-9. Google Scholar
[2]	A. M. A. Abou-El-Ela, A. I. Sadek and A. M. Mahmoud, On the stability of solutions of certain fourth-order nonlinear non-autonomous delay differential equation, Int. J. Appl. Math., 22(2009)(2), 245-258. Google Scholar
[3]	A. M. A. Abou-El-Ela, A. I. Sadek and A. M. Mahmoud, On the stability of solutions to a certain fourth-order vector delay differential equation, Ann. Differential Equations, 28(2012)(1), 1-10. Google Scholar
[4]	O. A. Adesina and B. S.Ogundare, Some new stability and boundedness results on a certain fourth order nonlinear differential equation, Nonlinear Stud., 19(2012)(3), 359-369. Google Scholar
[5]	S. Ahmad and M. Rama Mohana Rao, Theory of ordinary differential equations, With applications in biology and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi, 1999. Google Scholar
[6]	H. Bereketoglu, Asymptotic stability in a fourth order delay differential equation, Dynam. Systems Appl., 7(1998)(1), 105-115. Google Scholar
[7]	L. I. Burganskaja, The stability in the large of the zero solution of certain nonlinear fourth order differential equations, Differencial'nye Uravnenija, 6(1970), 1314-1317. Google Scholar
[8]	M. L. Cartwright, On the stability of solutions of certain differential equations of the fourth order, Quart. J. Mech. Appl. Math., 9(1956), 185-194. Google Scholar
[9]	K. E. Chlouverakis and J. C. Sprott, Chaotic hyperjerk systems, Chaos Solitons Fractals, 28(2006)(3), 739-746. Google Scholar
[10]	E. N. Chukwu, On the stability of a nonhomogeneous differential equation of the fourth order, Ann. Mat. Pura Appl., 92(1972)(4), 1-11. Google Scholar
[11]	J. Cronin-Scanlon, Some mathematics of biological oscillations, SIAM Rev., 19(1977)(1), 100-138. Google Scholar
[12]	R. Eichhorn, S. J. Linz and P. Hänggi, Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows, Phys Rev E, 58(1998), 7151-7164. Google Scholar
[13]	Z. Elhadj and J. C. Sprott, Boundedness of certain forms of jerky dynamics, Qual. Theory Dyn. Syst., 11(2012)(2), 199-213. Google Scholar
[14]	J. O. C. Ezeilo, A stability result for solutions of a certain fourth order differential equation, J. London Math. Soc., 37(1962), 28-32. Google Scholar
[15]	J. O. C. Ezeilo, On the boundedness and the stability of solutions of some differential equations of the fourth order, J. Math. Anal. Appl., 5(1962), 136-146. Google Scholar
[16]	J. O. C. Ezeilo, Stability results for the solutions of some third and fourth order differential equations, Ann. Mat. Pura Appl., 66(1964)(4), 233-249. Google Scholar
[17]	J. O. C. Ezeilo and H. O. Tejumola, On the boundedness and the stability properties of solutions of certain fourth order differential equations, Ann. Mat. Pura Appl., 95(1973)(4), 131-145. Google Scholar
[18]	M. Harrow, Further results on the boundedness and the stability of solutions of some differential equations of the fourth order, SIAM J. Math. Anal., 1(1970), 189-194. Google Scholar
[19]	Chao Yang Hu, The stability in the large for certain fourth order differential equations, Ann. Differential Equations, 8(1992)(4), 422-428. Google Scholar
[20]	H. Kang and L. Si, Stability of solutions to certain fourth order delay differential equations, Ann. Differential Equations, 26(2010(4), 407-413. Google Scholar
[21]	H. Kaufman and M. Harrow, A stability result for solutions of certain fourth order differential equations, Rend. Circ. Mat.Palermo, 20(1971)(2), 186-194. Google Scholar
[22]	V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar
[23]	E. Korkmaz and C. Tunc, Stability and boundedness to certain differential equations of fourth order with multiple delays, Filomat, 28(2014)(5), 1049-1058. Google Scholar
[24]	N. N. Krasovskiĭ, Stability of motion, Applications of Liapunov's second method to differential systems and equations with delay. Stanford, Calif.:Stanford University Press, 1963. Google Scholar
[25]	B. S. Lalli and W. A. Skrapek, On the boundedness and the stability of some differential equations of the fourth order, SIAM J. Math. Anal., 2(1971), 221-225. Google Scholar
[26]	B. S. Lalli and W. A. Skrapek, Some further stability and boundedness results of some differential equations of the fourth order, Ann. Mat. Pura Appl., 90(1971)(4), 167-179. Google Scholar
[27]	B. S. Lalli and W. A. Skrapek, Further stability and boundedness results for the solutions of some differential equations of the fourth order, Ann. Mat. Pura Appl., 95(1973)(4), 293-301. Google Scholar
[28]	Shi-zhong Lin, Zheng-rong Liu and Yuan-hong Yu, Stability for certain fourth order nonlinear differential equations, Demonstratio Math., 31(1998)(1), 87-96. Google Scholar
[29]	S. J. Linz, On hyperjerky systems, Chaos Solitons Fractals, 37(2008)(3), 741-747. Google Scholar
[30]	B. S. Ogundare and G. E. Okecha, Boundedness and stability properties of solution to certain fourth order non-linear differential equation, Nonlinear Stud., 15(2008)(1), 61-70. Google Scholar
[31]	A. I. Ogurcov, On the stability in the large of the solutions of third and fourth order non-linear differential equations, Izv. Vysš. Učebn. Zaved. Matematika, 1(1958)(2), 124-129. Google Scholar
[32]	A. I. Ogurcov, On the stability of the solutions of certain non-linear differential equations of third and fourth order, Izv. Vysš. Učebn. Zaved. Matematika, 3(1959)(10), 200-209. Google Scholar
[33]	O. E. Okoronkwo, On stability and boundedness of solutions of a certain fourthorder delay differential equation, Internat. J. Math. Math. Sci., 12(1989)(3), 589-602. Google Scholar
[34]	L. L. Rauch, Oscillation of a third order nonlinear autonomous system, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, Princeton University Press, Princeton, N. J., (20)(1950), 39-88.. Google Scholar
[35]	R. Reissig, G. Sansone and R. Conti, Non-linear differential equations of higher order, Translated from the German. Noordhoff International Publishing, Leyden, 1974. Google Scholar
[36]	A. I. Sadek, On the stability of solutions of certain fourth order delay differential equations, Appl. Math. Comput., 148(2004)(2), 587-597. Google Scholar
[37]	H. Smith, An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. Google Scholar
[38]	A. S. C. Sinha, On stability of solutions of some third and fourth order delaydifferential equations, Information and Control, 23(1973), 165-172. Google Scholar
[39]	A. S. Skidmore, On the stability of solutions of a differential equation of fourth order, J. London Math. Soc., 41(1966), 649-661. Google Scholar
[40]	W. A. Skrapek and B. S. Lalli, On the boundedness and stability of a fourth order differential equation, Ann. Mat. Pura Appl., 123(1980)(4), 1-9. Google Scholar
[41]	H. O. Tejumola, Further results on the boundedness and the stability of certain fourth order differential equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 52(1972)(8), 16-23 Google Scholar
[42]	H. O. Tejumola and B. Tchegnani, Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations, J. Nigerian Math. Soc., 19(2000), 9-19. Google Scholar
[43]	C. Tunc, On the stability and the boundedness properties of solutions of certain fourth order differential equations,Istanbul Univ. Fen Fak. Mat. Derg., 54(1995), 161-173. Google Scholar
[44]	C. Tunc, A note on the stability and boundedness results of solutions of certain fourth order differential equations,Appl. Math. Comput., 155(2004)(3), 837-843. Google Scholar
[45]	C. Tunc, Some stability and boundedness results for the solutions of certain fourth order differential equations, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 44(2005), 161-171. Google Scholar
[46]	C. Tunc, Stability and boundedness of solutions to certain fourth order differential equations, Electron. J. Differential Equations, 35(2006), 10. Google Scholar
[47]	C. Tunc, On stability of solutions of certain fourth order delay differential equations, Chinese translation appears in Appl. Math. Mech., 27(2006)(8), 994-1000. Appl. Math. Mech. (English Ed.), 27(2006)(8), 1141-1148. Google Scholar
[48]	C. Tunc, About stability and boundedness of solutions of certain fourth order differential equations, Nonlinear Phenom. Complex Syst., 9(2006)(4), 380-387. Google Scholar
[49]	C. Tunc, Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order, Comput. Appl. Math., 26(2007)(1), 1-17. Google Scholar
[50]	C. Tunc, Some stability results for the solutions of certain fourth order delay differential equations,Differential equations and applications. Vol. 4, 133-140, Nova Sci. Publ., New York, 2007. Google Scholar
[51]	C. Tunc, A boundedness criterion for fourth order nonlinear ordinary differential equations with delay, Int. J. Nonlinear Sci., 6(2008)(3),195-201. Google Scholar
[52]	C. Tunc, On the stability of solutions to a certain fourth order delay differential equation, Nonlinear Dynam., 51(2008)(1-2), 71-81. Google Scholar
[53]	C. Tunc, On the stability of solutions of non-autonomous differential equations of fourth order with delay, Funct. Differ. Equ., 17(2010)(1-2), 195-212. Google Scholar
[54]	C. Tunc, On the stability and boundedness of solutions in a class of nonlinear differential equations of fourth order with constant delay, Vietnam J. Math., 38(2010)(4), 453-466. Google Scholar
[55]	C. Tunc, The boundedness to nonlinear differential equations of fourth order with delay, Nonlinear Stud., 17(2010)(1), 47-56. Google Scholar
[56]	X. Wu and K. Xiong, Remarks on stability results for the solutions of certain fourth order autonomous differential equations, Internat. J. Control, 69(1998)(2), 353-360. Google Scholar
[57]	J. Yang, Stability and bifurcation analysis of a certain fourth-order delay differential equation, Math. Appl., 26(2013)(3), 526-537. Google Scholar