J. Korean Math. Soc. 49 (2012), No. 5, pp. 1017–1030 http://dx.doi.org/10.4134/JKMS.2012.49.5.1017 INTERVAL CRITERIA FOR FORCED OSCILLATION OF DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN AND NONLINEARITIES GIVEN BY RIEMANN-STIELTJES INTEGRALS Taher S. Hassan and Qingkai Kong Abstract. We consider forced second order differential equation with p-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of ( p(t)φ γ ( x ′ (t) )) ′ + q 0 (t) φ γ (x(t)) + ∫ b 0 q (t, s) φ α(s) (x(t)) dζ (s)= e(t), where φ α (u) := |u| α sgn u, γ, b ∈ (0, ∞) ,α ∈ C [0,b) is strictly increasing such that 0 ≤ α (0) <γ<α (b-), p, q 0 ,e ∈ C ([t 0 , ∞), R) with p (t) > 0 on [t 0 , ∞), q ∈ C ([0, ∞) × [0,b)), and ζ : [0,b) → R is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature. 1. Introduction We are concerned with the oscillatory behavior of forced second order dif- ferential equations with p-Laplacian and nonlinearities given by a Riemann- Stieltjes integrals in the form of (1.1) (p(t)φ γ (x ′ (t))) ′ + q 0 (t) φ γ (x(t)) + ∫ b 0 q (t,s) φ α(s) (x(t)) dζ (s)= e(t), where φ α (u) := |u| α sgn u, γ, b ∈ (0, ∞) ,α ∈ C [0,b) is strictly increasing such that 0 ≤ α (0) <γ<α (b−), p, q 0 ,e ∈ C ([t 0 , ∞), R) with p (t) > 0 on [t 0 , ∞), q ∈ C ([0, ∞) × [0,b)), and ζ : [0,b) → R is nondecreasing. Our interest is to establish oscillation criteria for Eq.(1.1) without assuming that q 0 (t), q (t,s), and e(t) are of definite sign. Here ∫ b 0 f (s)dζ (s) denotes the Riemann-Stieltjes integral of the function f on [0,b) with respect to ζ . Received March 30, 2011; Revised December 20, 2011. 2010 Mathematics Subject Classification. 34C10, 34C15. Key words and phrases. interval criteria, forced oscillation, p-Laplacian, nonlinear differ- ential equations. The second author is supported by the NNSF of China (No. 10971231). c ⃝2012 The Korean Mathematical Society 1017