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Oscillation Criteria of second Order Non-Linear Differential Equations Hishyar 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[2], [3],[4],[5], [6],[8], [9] 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) Į 0 for at least one t ę ę 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 [7] 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 ĮW!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 [7] 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 let t , (7) then any solution of the differential equation (2) is either oscillatory or tends monotonically to zero as t ėĞ . 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 Technology 120 Cop y ri g ht © 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 t ėĞ Theorem2: 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 t ėĞ . 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 t ėĞ .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,[1]) where x(t)>0 and then x’(t)>0 for large t. This completes the proof of the theorem. EXAMPLES Consider 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 Cop y ri g ht © 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] 1. Al-Ashker, M.M.: Oscillatory properties of certain second order nonlinear differential equations, (1993) M.Sc. theses university of Jordan. [2] 2.Cakmak D.: Oscillation for second order nonlinear differential equations with damping, (2008) Dynam. Systems Appl., vol.17, No.1 139-148. [3] 3.Close W. J.:Oscillation criteriafor nonlinear second order equations,(1969) Ann. Mat. 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