International Journal of Pure and Applied Mathematics Volume 113 No. 1 2017, 73-91 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v113i1.8 P A ijpam.eu OSCILLATION OF SUBLINEAR AND SUPERLINEAR SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS A.K. Tripathy 1 , A.K. Sethi 2 1,2 Department of Mathematics Sambalpur University Sambalpur, 768019, INDIA Abstract: In this work, we establish the necessary and sufficient conditions for oscillation of a class of functional differential equations of the form: (r(t)(x(t)+ p(t)x(τ (t))) ′ ) ′ + q(t)G(x(σ(t))) + v(t)H(x(η(t))) = 0 under the assumption that  ∞ 0 dt r(t) = ∞ for various ranges of p(t). AMS Subject Classification: 34K, 34C10 Key Words: oscillation, nonoscillation, neutral, delay, nonlinear 1. Introduction Consider the nonlinear neutral delay differential equations of the form: (r(t)(x(t)+ p(t)x(τ (t))) ′ ) ′ + q(t)G(x(σ(t))) + v(t)H (x(η(t))) = 0, (1) where r, q, v, τ, σ,η ∈ C (R + , R + ),p ∈ C (R + , R) such that τ (t) ≤ t, σ(t) ≤ t, η(t) ≤ t with lim t→∞ τ (t)= ∞ = lim t→∞ σ(t)= ∞ = lim t→∞ η(t) and G, H ∈ C (R, R) satisfying the property yG(y) > 0, uH (u) > 0 for y,u = 0. The Received: November 2, 2016 Revised: February 22, 2017 Published: February 28, 2017 c  2017 Academic Publications, Ltd. url: www.acadpubl.eu