On the convergence behaviour of solutions of certain system of second order nonlinear delay differential equations. (English) Zbl 1532.34080
Summary: Convergence criteria for the solutions of certainsystem of two nonlinear delay differential equations with con-tinuous deviating argument \(\varrho (t)\) using a suitable Lyapunov-Krasovskii’s functional are established in this study. The newresult attained extends and updates some results mentioned in the literature. A numerical illustration is given to show the va-lidity of the result as well geometric analysis to describe thebehavior of solutions of the system.
MSC:
| 34K40 | Neutral functional-differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
Keywords:
convergence of solution; system of nonlinear delay differential equations; second order; Lyapunov-Krasovskii’s functionalReferences:
| [1] | A. U. Afuwape, Ultimate boundedness results for a certain system of third-order nonlinear differential equations, Journal of Mathematical Analysis and Applica-tions 97(1) 140-150, 1983. · Zbl 0537.34031 |
| [2] | A. U. Afuwape and M. O. Omeike, Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations, Acta Univ., Palacki Olomuc. Fac.rer.nat. Mathematica 43 7-20, 2004. · Zbl 1068.34052 |
| [3] | A. U. Afuwape and M. O. Omeike, Convergence of solutions of certain system of third order nonlinear ordinary differential equations, Ann of Diff. Eqns. 21(4) 533-540, 2005. · Zbl 1103.34036 |
| [4] | T. A. Burton and Z. Shunian, Unified boundedness, periodicity and stability in ordinary and functional Differential equations, Ann. Mat. Pura. Appl. 145(5), 129-158, 1986. · Zbl 0626.34038 |
| [5] | E. N. Chukwu, On the boundedness of solutions of third order differential equa-tions, Ann. Math. Pur. Appl. 104 123-149, 1975. · Zbl 0319.34027 |
| [6] | J. Cronin, Some mathematics of biological oscillations, SIAM Review 19 100-137, 1997. · Zbl 0366.92001 |
| [7] | J. O. C. Ezeilo, n-dimensional extensions of boundedness and stability theorems for some third order differential equations, J. Math. Anal. Appl. 18 394-416, 1967. · Zbl 0173.10302 |
| [8] | J. O. C. Ezeilo, On the stability of solutions of certain system of ordinary differ-ential equation, J. Math. Anal. Appl. 18 17-26, 1966. · Zbl 0145.33402 |
| [9] | A. L. Olutimo, Convergence results for solutions of certain third order nonlinear vector differential equations, Indian J. Math. 54(3) 299-311, 2012. · Zbl 1291.34087 |
| [10] | A. L. Olutimo, Result on the convergence behaviour of solutions of certain third order differential equations, Intl. J. Modern Nonlinear Theory and Appl. 5(1) 48-58, 2016. |
| [11] | A. L. Olutimo, S. A. Iyase and H. I. Okagbue, Convergence behavior of solutions of a kind of third order nonlinear differential equations, Advances in Differential Equations and Control Processes 23(1) 1 -19, 2020. · Zbl 1479.34095 |
| [12] | M. O. Omeike, A. A. Adeyanju, D. O. Adams and A. L. Olutimo, Boundedness of solutins of certain system of second order ordinary differential equations , Kragu-jevac Journ. of Math. 45(5) 787-796, 2021. · Zbl 1499.34236 |
| [13] | L. L. Rauch, Oscillations of a third-order nonlinear autonomous system in con-tributions to the theory of nonlinear oscillations, Annals of Mathematics Studies 20 39-88, 1950. · Zbl 0039.09802 |
| [14] | H. O. Tejumola, Boundedness theorems for some systems of two differential equa-tions, Atti. Accad. Naz. Lincei Rend. Cl, Sci. Fis. Mat. Natur. 51(4) 472-476, 1971. · Zbl 0271.34043 |
| [15] | C. Tunc, Convergence of solutions of certain third order differential equation, Dis-crete Dyn. Nat. Soc., Vol. 2009, Article ID 863178, 12pp. Doi: 10.1155/2009/863178 · Zbl 1183.34068 · doi:10.1155/2009/863178 |
| [16] | C. Tunc and E. Korkmaz, On the of convergence of solutions of some nonlinear differential equations of third order, J. Comput. Anal. Appl. 13(3) 470 -470, 2011. · Zbl 1223.34075 |
| [17] | E. Korkmaz and C. Tunc, On the of convergence of solutions of some nonlinear differential equations of fourth order, Nonlinear Dyn. Syst. Theory 14(3) 313 -322, 2014. · Zbl 1310.34058 |
| [18] | C. Tunc and M. Gozen, Convergence of solutions to a certain vector differential equation of third order, Abstract and Applied Analysis, Vol. 2014, Article ID 424512. https://doi.org/10.1155/2014/424512 · Zbl 1469.34025 · doi:10.1155/2014/424512 |
| [19] | C. Tunc, O. Tunc, Y. Wang and J-C Yao, Qualitative analysis of differential sys-tems with time-varying delays via Liapunov-krasovskii approach, Σ Mathematics 9(11) 1196, 2021. https://doi.org/10.3390/math9111196 · doi:10.3390/math9111196 |
| [20] | O. Tunc, C. Tunc and Y. Wang, Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems, Axioms 2021, 10(3), 138. https://doi.org/10.3390/axioms10030138 · doi:10.3390/axioms10030138 |
| [21] | DEPARTMENT OF MATHEMATICS, LAGOS STATE UNIVERSITY, OJO, NIGERIA E-mail address: akinwale.olutimo@lasu.edu.ng |
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