OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS OF MIXED TYPE

Some oscillation criteria are established for the second order nonlinear neutral differential equations of the form ((x(t) + ax(t �1) bx(t + �2)) � ) 00 = q(t)x � (t �1) + p(t)x � (t + �2), and (x(t) ax(t �1) + bx(t + �2) � ) 00 = q(t)x � (t �1) + p(t)x � (t + �2) whereandare the ratios of odd positive integers with � � 1. Examples are provided to illustrate the main results.

Let θ = max{τ 1 , σ 1 }.By a solution of equation (1.1) or (1.2), we mean a real valued function x(t) defined for all t ≥ t 0 − θ, and satisfying the equation (1.1) or (1.2) for all t ≥ t 0 .A nontrivial solution of equation (1.1) or (1.2) is said to be oscillatory, if it has arbitrarily large zeros; otherwise it is called nonoscillatory.
The problem of determining oscillation and nonoscillation of second order delay and neutral type differential equations has received great attention in recent years, see for example , and the references cited therein.If a = 0 or b = 0 and either q(t) ≡ 0 or p(t) ≡ 0, then the oscillatory behavior of solutions of equations (1.1) and (1.2) are studied in [1, 4, 6, 8-10, 14-18, 20].In particular if α = β = 1 and α = β, then the oscillatory behavior of solutions of equations (1.1) and (1.2) are discussed in [2, 3, 5, 7, 11-13, 19, 21].Motivated by this observation, in this paper we establish some new sufficient conditions for the oscillation of all solutions of equations (1.1) and (1.2) when β ≥ 1.
In Section 2, we present some sufficient conditions for the oscillation of all solutions of equations (1.1) and (1.2).Examples are provided in Section 3 to illustrate the main results.

Oscillation Results
In this section we shall obtain some sufficient conditions for the oscillation of all solutions of the equations (1.1) and (1.2).Before proving the main results we state the following lemma which will be useful in proving the main results.
First we study the oscillation of all solutions of equation (1.1).
> 0, and q(t) and p(t) be positive and nonincreasing for all t ≥ t 0 .Assume that the differential inequalities y ′′ (t) − p(t) and where ,have no eventually positive increasing solution, no eventually positive decreasing solution,and no eventually positive solution, respectively.Then every solution of equation (1.1) is oscillatory.
Proof.Assume that x(t) is a nonoscillatory solution of equation (1.1).Without loss of generality we may assume that there exists t 1 ≥ t 0 such that x(t − θ) > 0 for all Then z ′′ (t) = q(t)x β (t − σ 1 ) + p(t)x β (t + σ 2 ) > 0 for all t ≥ t 1 .Therefore, both z(t) and z ′ (t) are of one sign for all t ≥ t 1 .We shall prove that z(t) > 0 eventually.
That is Using the above inequality in equation (1.1), we have Hence u(t) is a positive solution of the inequality (2.5), a contradiction.Therefore z(t) > 0 eventually.Now we define a function y(t) as Then (2.7) Using the monotonicity of q(t) and p(t) and the inequality (2.1) in (2.7), we get Now using z(t) > 0 for t ≥ t 1 , and the inequality (2.2) in the above inequality, we obtain which implies that both y(t) and y ′ (t) are of one sign, eventually.We shall prove that y(t) > 0 eventually.If not, then . Using the last inequality in (2.8), we obtain Therefore v(t) is a positive solution of (2.5), contradiction.Thus y(t) > 0, eventually.Next we consider the following two cases: We claim that y ′ (t) < 0 for all t ≥ t 2 .If not, then y(t) > 0, y ′ (t) > 0 and y ′′ (t) > 0 imply that lim t→∞ y(t) = ∞.On the other hand, z(t) > 0 and z ′ (t) < 0 imply that lim t→∞ z(t) = c < ∞.Applying limit on both the sides of equation (2.6) we obtain a contradiction.Thus y ′ (t) < 0 for all t ≥ t 2 .
Using the monotonicity of z(t), we obtain The above inequality together with (2.8) implies that Thus y(t) is a positive decreasing solution of the inequality (2.4), which is a contradiction.
Case:2.Let z ′ (t) > 0 for all t ≥ t 2 .Now we consider the following two subcases: EJQTDE, 2012 No. 75, p. 5 Subcase (i): Assume that y ′ (t) < 0 for all t ≥ t 2 .Proceeding as in Case 1, and using the monotonicity of z(t), we obtain Using the last inequality in (2.8) and the monotonicity of y(t), we obtain and once again y(t) is a positive decreasing solution of the inequality (2.4), which is a contradiction.

Subcase (ii):
Assume that y ′ (t) > 0 for all t ≥ t 2 .Then we have y(t+) ≤ (1 + a β − b β 2 β−1 )z(t + τ 2 ), and this with (2.8) implies That is, y(t) is a positive increasing solution of the inequality (2.3), which is a contradiction.The proof is now complete.Next we present a ready to verify conditions for the oscillation of all solutions of equation (1.1).
lim sup and lim inf then every solution of equation (1.1) is oscillatory.
Hence we have y(t) > 0, y ′ (t) > 0 and y ′′ (t) ≤ 0, for t ≥ t 1 .Then we obtain From (2.5) and the monotonicity of y(t),we have Combining the last two inequalities,we obtain Let w(t) = y ′ (t).Then we see that w(t) is a positive solution of EJQTDE, 2012 No. 75, p. 7 which is a contradiction by condition(2.11)and Theorem 2.1.1 in [15].Hence (2.5) has no eventually positive solution.More over condition (2.9) is sufficient for the inequality (2.3) to have no positive increasing solution and condition (2.10) is sufficient for the inequality (2.4) to have no positive decreasing solution,see [1, Lemma 2.2.12].Then the proof follows from Theorem 2.2.
Next we consider the equation (1.2), and present sufficient conditions for the oscillation of all solutions.
Theorem 2.4.Assume that σ i > τ i for i = 1, 2, q(t) and p(t) are positive and nondecreasing functions for t ≥ t 0 .If the differential inequality has no positive increasing solution, the differential inequality y ′′ (t) − q(t) has no positive decreasing solution, and the differential inequality where , has no positive solution, then every solution of equation (1.2) is oscillatory.
Proof.Let x(t) be a nonoscillatory solution of equation (1.2).Without loss of generality, we may assume that there exists a t 1 ≥ t 0 such that x(t − θ) > 0 for all t ≥ t 1 .By setting we have z ′′ (t) = q(t)x β (t − σ 1 ) + p(t)x β (t + σ 2 ) > 0 for all t ≥ t 1 .Therefore, both z(t) and z ′ (t) are of one sign for all t ≥ t 1 .We shall prove that z(t) > 0 for all t ≥ t 1 .If not, then z(t) < 0 and which implies that From equation (1.2), we obtain Thus u(t) is a positive solution of the inequality (2.14), which is a contradiction.
Hence z(t) > 0 for all t ≥ t 1 .Now define a function y(t) by Differentiating (2.15) twice, and using the equation (1.2), we obtain EJQTDE, 2012 No. 75, p. 9 Using the monotonicity of q(t) and p(t) in the above inequality, we obtain Now using the inequalities (2.1), (2.2) and z(t) > 0 for all t ≥ t 1 , we obtain (2.17) Therefore, both y(t) and y ′ (t) are of one sign eventually.We prove that y(t) > 0, eventually.If not, then y(t) < 0 and . Using the last inequality in (2.17), we obtain 0 ≥ v ′′ (t) + q(t) Thus v(t) is a positive solution of the inequality (2.14),a contradiction.Hence y(t) > 0, eventually.Now we consider the following two cases.
Hence y ′ (t) < 0 for all t ≥ t 2 .Now using the monotonicity of z(t), we get EJQTDE, 2012 No. 75, p. 10 Using the last inequality in (2.17) and the monotonicity of y(t), we have Thus y(t) is a positive decreasing solution of the inequality (2.13), a contradiction.
Case:2 Let z ′ (t) > 0 for all t ≥ t 2 ≥ t 1 .Now we consider the following two subcases.
Subcases (i): Assume that y ′ (t) < 0 for all t ≥ t 2 .Then proceeding as in Case 1 and using the monotonicity of z(t), we obtain Using last inequality in (2.17) and the monotonicity of y(t), we get Thus y(t) is a positive decreasing solution of the inequality (2.13), a contradiction.Subcases (ii): Assume that y ′ (t) > 0 for all t ≥ t 2 .Then using the monotonicity of z(t), we have Using the last inequality in (2.17), we obtain Therefore y(t) is a positive increasing solution of the inequality (2.12), a contradiction.This completes the proof. and Then every solution of equation (1.2) is oscillatory.
Proof.The proof is similar to that of Corollary 2.3, and hence the details are omitted.

Examples
Now we present some examples to illustrate the main results.
Example 3.1.Consider the differential equation (3.1)  Example 3.3.Consider the differential equation  We conclude this paper with the following remark.
Remark: It would be interesting to study the oscillatory behavior of all solutions of equations (1.1) and (1.2) when β < 1.

Remark 2 . 1 .
Theorem 2.2 permits us to get various oscillation criteria for equation(1.1).Also we are able to study the asymptotic properties of solutions of equation (1.1) even if some of the assumptions of Theorem 2.2 are not satisfied.If the differential inequality (2.3) has eventually positive increasing solution then the conclusion of Theorem 2.2 will be replaced by "every solution x(t) of equation (1.1) is either oscillatory or x(t) tends to ∞ as t → ∞." EJQTDE, 2012 No. 75, p. 6

2 and α = β = 1 .
Then one can see that all the conditions of Corollary 2.3 are satisfied.Hence all the solutions of equation (3.2) are oscillatory.In fact x(t) = sint is one such solution of equation (3.2), since it satisfies the equation (3.2).
Let σ i > τ i , for i = 1, 2, and β = α.Assume .Then it is easy to check that condition (2.9) of Corollary 2.3 is not satisfied.Therefore equation (3.1) has a nonoscillatory solution.In fact x(t) = t is one such nonoscillatory solution, since it satisfies the equation (3.1).