ON NECESSARY AND SUFFICIENT CONDITIONS FOR THE OSCILLATION OF ALL SOLUTIONS OF ANOTHER SPECIAL CASE OF OF x^'+∑_(i=1)^n▒〖p_i x(t-τ_i ) 〗=0

Bala Maaji Abdulhamid, Muktari Muhammad Bamaina

Abstract


In this paper, we establish that all solutions of  x' (t)+p_1 x(t-τ_1 )+p_2 x(t-4τ_1 )=0   are Oscillatory if and only if  1/τ_1  logξ+p_1 ξ+p_2 ξ^4>0 where ξ=(-k^(1/2)+√(k-2U))/2 and u=[(k^(2/3)-q)/k^(1/3) ], p_1 and p_2>0  and τ_1 is the delay


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References


Arino, O; Jawhari, A & Gÿori, I (1984). Oscillation Criteria in Delay Equations. Journal of Differential Equations, 53, 115-123.

Arino, O; Ladas, G & Sficas Y. G. (1987). On Oscillation of Some Retarded Differential Equations. SIAM Journal of Mathematical Analysis, 18(1), 64-73

Bamaina, M. M. (2004). Conditions for Oscillation and Non-oscillation of First Order Linear Delay Differential Equations, Unpublished M.Sc. Project Abubakar Tafawa Balewa University.

Sesay, M. S, Haruna, Y. & Bamaina, M. M(2005). Oscillatory and Non-oscillatory Conditions of First Order Linear Delay Differential Equations. Science Forum; Journal of Pure and Applied Sciences, 8(1), 41-51.

Grove, E. A; Ladas, G. & Schinas, J. (1988). Sufficient Conditions for the Oscillation of Delay and Neutral Equations. Canad. Math. Bull, 31(4), 459-466.

Gÿori, I. (1989). Oscillations of Retarded Differential Equations of the Neutral and the Mixed Types, Journal of Mathematical Analysis and Applications, 141(1), 1-20.

Gÿori, I; Ladas, G. & Pakula, L. (1989). On Oscillation of Unbounded Solutions. Proceedings of the American Mathematical Society, 106(3), 785-792.

Gÿori I; Ladas, G & Pakula, L (1990). Oscillation Theorems for Delay Differential Equations Via Transforms. Canad. Math. Bull, 33(3), 323-326.

Gÿori I. & Ladas G. (1991). Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press Oxford.

Hunt, B. R. & Yorke, J. A. (1984). When all solution of x^'=-∑_(i=1)^∞▒〖q_i (t)x(t-τ_i (t) ) 〗 Oscillates. Journal if Differential Equations, 53,139-145.

Kusano, T. (1982). On Even Order Functional Differential Equations with Advanced and Retarded Arguments. Journal Differential Equations, 45, 75-84.

Ladas, G. (1979). Sharp Conditions for Oscillation Caused by Delays. Applicable Anal., 9, 93-98.

Ladas, G. & Stavroulakis, P. I. (1982a). On Delay Differential Inequalities of First Order. Funcciaj Ekvacioj 25, 105-113.

Ladas,G. & Stavroulakis, P. I. (1982b). Oscillations Caused by Retarded and Advanced Arguments. Journal of Differential Equations, 44, 134-152.

Ladas,G; Sficas, Y. G & Stavroulakis, P. I. (1983). Necessary and Sufficient Conditions for Oscillations, American Mathematical Monthly, 90, 637-640.

Ladas,G. & Sficas, Y. G. (1986). Oscillations of Higher Order Neutral Equations. Journal Austral. Math. Soc. Ser. B, 27, 502-511.

Ladde, G. S (1977). Oscillations Caused by Retarded Perturbations of First Order Linear Differential Equations. Atti. Accad. Naz. Lincei Rend. Cl.Sci.Fis. Mat. Natur. 351-359.

Ӧǧȗnmez H. & Ӧclan Ӧ. (2013). Oscillation of Higher Order Systems of Differential Equations. International Journal of Mathematical Analysis, 7(15), 735-740.

Philos, C. G. (1990). On the Oscillation of Differential Equations with Periodic Coefficients. American Mathematical Society, 111(2), 433-441.

Tramov, M. I. (1975). Conditions for Oscillatory Solutions of First Order Differential Equations with a Delayed Argument, Isv.VysÅ¡. UÄebn. Zaved. Mat 19(3) 92-96.

Wider, D. V. (1971). An Introduction to Transform Theory, Academic Press.


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