EXISTENCE, STABILITY AND BOUNDEDNESS OF PERIODIC SOLUTIONS FOR CERTAIN NONLINEAR BOUNDARY VALUE PROBLEM OF FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS
Abstract
In this paper, some new conditions on the existence, stability and boundedness of periodic solutions for certain nonlinear boundary value problems were investigated. Construction of a complete Lyapunov function for higher order differential equations were not all that easy and therefore a new method for investigating and proving the existence, stability and boundedness was considered. The method of Leray-Schauder fixed point theorem provided existence of periodic solutions which depended on the availability of suitable boundedness results. In some cases, boundedness results were very difficult to establish due to the nature of the Lyapunov function involved and the method of the integrated equation was used as a mode for estimating apriori bounds for the fourth order differential equation. The aim of using integrated equation was to amelioriate the technical problems arising from the construction of Lyapunov function for higher order differential equations which was considered to be cumbersome and complex. However, our results generalize and complement some existing results in literature.
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References
Afuwape A.U (1981). Ultimate boundedness results for a certain system of third order non-linear differential equation. Journal of Mathematical Analysis and Applications, 97:140-150.
Bockelman, B (2006). Schauder fixed point theorem. Advanced ordinary differential equation II, Avasion Publications, 2nd edition, 97:101
Cartwright M.L (1956). On the stability of solution of certain differential equation of the fourth order. Journal of Mechanical Application Mathematics, 9:185-194.
Ezeilo J.O.C (1962). A property of the phase-space trajectories of a third order nonlinear differential equation. Journal of Mathematical Society London :1-37.
Ezeilo J.O.C (1962) On the boundedness and the stability of solution of some fourth order equation. Journal of Mathematical Analysis and Application, 5:136-146.
Ezeilo J.O.C (1963). An extension of the property of the phase space trajectories of a third order differential equation. Journal of the Department of Mathematics University of Ibadan Nigeria, 63(1):387-397.
Ezeilo J.O.C (1966). An estimate for the solution of a certain system of differential equations. Nigeria Journal of Science Association, 1:5-10.
Ezeilo J.O.C and Onyia J.O (1984). Non-resonance oscillations for some third order differential equation. Journal of Nigerian Mathematical Society, 83-96.
Mawhin J and Ward J.R (1982). Periodic solutions of some Lienard differential equation. Resonance Archives of Mathematics, 58:153-168.
Njoku F.I and Omari P (2003). Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions. Applications of Mathematics Computer, 135:471-490
Ressig R (1972). Periodic solution of third order nonlinear differential equation. Analysis of Mathematical Pure Applications, 4(92):193-198.
Seyranian A.P and Wang Y. (2011). On stability of periodic solution of Duffing equation. Journal of National Academy of Science of Amenia, 111(4):211-232.
Tiryaki A and Tune C. (1995). Construction of Lyapunov function for certain fourth order autonomous differential equation. Indian Journal of Pure Application of Mathematics, 26:225-232.
Voitovich, M.V (2011). Existence of bounded solutions for a class of nonlinear fourth order equation. Differential equations and Applications, 3(2):247-268.
Zanoline F and Liu B (1998). Boundedness of solution of nonlinear differential equations. Journal of Differential equations, 144:66-98.
Ezeilo J.O.C (1960). On the existence of periodic solutions of a certain third order differential equation. Pro. Camb. Phi. Soc. 56:381-387.
Harrow M. (1967) Further results for certain third and fourth order differential equations. Proceedings of Cambridge Philes Society, 63:147-153
Tejumola, H.O (1969) On boundedness and periodicity of solutions of a certain third order nonlinear differential equation. Ann. Mat. Pura. Appl. 83:195-212.
Eze E.O, Obasi, U.E, Osu, B.O and Ogbobo, C.P (2021) Existence and stability of periodic solutions of Airy’s equation with elastic coefficients. Nigeria Journal of Pure and Applied Sciences, 34(1):3631-3640.
Eze E.O and Udaya C.O.D (2020). On the existence and uniqueness of 2π-periodic solution of Duffing’s equation using Abstract implicit function theorem in Banach spaces. Journal of the Mathematical Association of Nigeria, 47(1):156-160.
Osu B.O, Eze E.O, Obasi U.E and Ukomah H.I (2020). Existence of solutions of some boundary value problems with stochastic volatility, 6(2):1-12.
Eze E.O, Obasi U.E, Udaya C.O and Daniel F (2019). On the Krasnoselskii’s fixed point theorem and the existence of periodic solution for a damped and forced Duffing oscillator. American Journal of Mathematics and Statistics, 9(3):131-135.
Eze E.O, Obasi U.E and Ezeh S.I (2019). Boundedness and stability of periodic solutions of a Hill equation with six independent arbitrary parameters. American Journal of Mathematics and Statistics, 9(2):51-56.
Eze E.O, Udaya C.O and Agwu E.U (2019). On existence and uniqueness of solutions of non-linear first order differential equation using implicit function theorem in Banach space. Journal of the Nigerian Association of Mathematical Physics, 50:9-12.
Eze E.O, Obasi U.E and Agwu E.U (2019). Stability analysis of periodic solutions of some Duffing equation. Open Journal of Applied Sciences, 9:198-124
Eze E.O, Ezeora J and Obasi U.E (2018). A study of periodic solution of Duffing equation using implicit function theorem. Open Access Journal of Applied Sciences, 8(10):459-464.
Eze E.O, Obasi U.E and Egenkonye V.C (2017). Stability properties of zero solution of a damped Duffing oscillator. Journal of Nigerian Association of Mathematical Physics, 39:105-110.
Okereke R.N and Eze E.O (2017). Application of lyapunov functions to some nonlinear second order differential equations. Transactions of the Nigerian Association of Mathematical Physics, 3:17-24.
Eze E.O and Obasi U.E (2017) Stability analysis of cubic Duffing oscillator (The Hard spring model). Transactions of the Nigerian Association of Mathematical Physics, 3:45-54.
Osisiogu U.A, Eze E.O and Obasi U.E (2016). Existence and stability of periodic solution for a class of second order nonlinear differential equation. Journal of Nigerian Association of Mathematical Physics, 36(2):37-42.
Osisiogu U.A, Eze E.O and Obasi U.E (2016). Comparative analysis of some methods of lyapunov constructions for the cubic Duffing’s oscillator- the hard spring model. Journal of Nigerian Association of Mathematical Physics, 32(2):43-48.
Eze E.O, Ukeje E and Ogbu H.M (2015). Stable, bounded and periodic solution in a nonlinear second order ordinary differential equation. American Journal of Engineering, 4(8):14-18.
Eze E.O and Aja R.O (2014). On application of Lyapunov and Yoshizawa’s thorems on stability, asymptotic stability, boundedness and periodic solution of a Duffing’s equation. Asian Journal of Applied Sciences, 2(6):970-975.
Eze E.O, Udaya C. and Ogbu, H.M (2013). On application of Leray-Schauder fixed point and Scheafer’s lemma in solution of Duffing equation. Proceedings of September Annual National Conference, Mathematical Association of Nigeria, 495-503
Guoshan Z and Peizhao Y. (2018) Lyapunov method for stability of descriptor second order and high order system. Journal of Industrial and Management Optimization, 14(2):673-686.
Bernstein D.S and Bhat S.P (1994). Lyapunov stability, semistability and asymptotic stability of matrix second order systems. Journal of Vibration and Acoustic, 117:145-153.
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