### Journal Menu >> Oscillation Criteria of second Order Non-Linear Differential EquationsHishyar Kh. Abdullah Dept. of Mathematics, University of Sharjah, Sharjah, U.A.E. hishyar@sharjah.ac.ae Abstract—In this paper we are concerned with the oscillation criteria of second order non-linear homogeneous differential equation. Example have been given to illustrate the results. Keywords-component; Oscillatory, Second order differential equations, Non-Linear. 1. Introduction The purpose of this paper is to establish a new oscillation criteria for the second order non-linear differential equation with variable coefficients of the form (1) where is a fixed real number and f(x) and g(x) are continuously differentiable functions on the interval . The most studied equations are those equivalent to second order differential equations of the form , (2) where h(x)>0 is a continuously differentiable functions on the interval .Oscillation criteria for the second order nonlinear differential equations have been extensively investigated by authors(for example see, ,,, ,,  and the authors there in). Where the study is done by reducing the problem to the estimate of suitable first integral. Definition1:A solution x(t) of the differential equation (2) is said to be "nontrivial " if x(t)Į0 for at least one tęę Definition2:A nontrivial solution x(t) of the differential equation (2) is said to be oscillatory if it has arbitrarily large zeros on [t΋,Ğ), otherwise it said to be " non oscillatory ". Definition3:We say that the differential equation (1) oscillatory if an equivalent differential equation (2) is oscillatory. 2. Main Results In  the author considered a class of systems equivalent to the second order non-linear differential equation (1). The standard equivalent system (3) while he worked on a wider class of systems of the form (4) If ĮW!then (4) is equivalent to a differential equation of the type (1). This allows to choose a modified system in order to be able to cope with different problems related to (2).Ta king() g,where . One obtains . (5) System (5) cab be transformed into , (6) where , which is equivalent to (2) where sufficient conditions for solutions of differential equation (1) to oscillate are given. Remark: Assume that f(x(t)) and Let us set Since , for all is invertible on I, we define the transformation u=׋(x(t)), acting on I. Accordion to Lemma1 in  any solution x(t) of (3) is a solution is a solution of (6). Theorem1: Let h(x) be continuous and continuously differentiable on (-Ğ,0)Ĥ(0,Ğ) with and lett , (7) then any solution of the differential equation (2) is either oscillatory or tends monotonically to zero as tėĞ . Proof: Suppose that x(t) is non-oscillatory solution of (2), and assume x(t)>0 for some. From (2) we get Identify applicable sponsor/s here. (sponsors) Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology120 Copyright © 2012 SciRes. . Put Then , then Since h’(x(t))>0 then By hypothesis (7)we have this means we obtain for some constant k>0 Integrating from tЅ to t for tЅ>0 we get (8) The right hand side is negative, since x(tЅ)>0, x(t) is positive. From (8) we conclude Thus x(t) is oscillatory or tends monotonically to zero as tėĞTheorem2: If In addition to hypotheses (7) we assume that for some x(t)>0 (9) Then every solution x(t) of the differential equation(2) is oscillatory. Proof: As in theorem 1,we want to show that x(t) doesn't tend monotonically to zero as tėĞ. Assume x(t)>0 for a>0 on . Since from (8) we have then there exists a positive real number m such that , This means is bounded below by a finite positive number, then by hypothesis (9), x(t) doesn't tend monotonically to zero as tėĞ.Then x(t) is oscillatory. Theorem3: Assume that h(x) satisfies then every solution x(t) of (2) is oscillatory. Proof: Let x(t) be non-oscillatory solution of (2), which without loss of generality, may be assumed to be positive for large t. Define then or (10) Integrating (10) from Į to t we get (11) Since and from (11) we get w(t)<0 from which we get x’(t)<0 for large twhich is a contradiction (by lamma1II.I.8,) where x(t)>0 and thenx’(t)>0 for large t. This completes the proof of the theorem. EXAMPLESConsider the second order nonlinear order differential (12) for this differential equation we have( and . Then the equivalent second order differential equation to (12) is (13) where . To show the applicability of Theorem 1, the hypothesis is satisfied as follows Copyright © 2012 SciRes.121 Therefore the Theorem implies that the differential equation is oscillatory. To show the applicability of Theorem 2 it is clear that the hypothesis is satisfied hence and Hence Theorem 2 is applicable. To show the applicability of Theorem 3 the hypothesis is satisfied as follows And Hence Theorem 3 is applicable. 3. Acknowledgment I would like to extend my thanks to the University of Sharjah for its support. REFERENCES 1. Al-Ashker, M.M.: Oscillatory properties of certain second order nonlinear differential equations, (1993) M.Sc. theses university of Jordan.  2.Cakmak D.: Oscillation for second order nonlinear differential equations with damping, (2008) Dynam. Systems Appl., vol.17, No.1 139-148.  3.Close W. J.:Oscillation criteriafor nonlinear second order equations,(1969) Ann. Mat. 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