OSCILLATION CRITERIA FOR SECOND ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS WITH DELAY

The oscillation criteria of different types of differential equations are often the topic of numerous scientific papers, because their application in nuclear physics, fluid mechanics, relativistic mechanics, the study of chemical reactions in the system and in general are large in science. In this paper, the oscillation criteria using averaging functions of the half-linear differential equation are generalized to the half-linear differential equation with delay, under the appropriate assumptions for the delay function. Suitable examples illustrate the application of set oscillation criteria.


INTRODUCTION
Part of the qualitative analysis of differential equations is particularly intense in the last thirty years. During this time, new testing methods were developed and important and useful results were obtained. Probably the highest-studied differential equation of the second order is the Sturm-Liouville linear differential equation of the second order: In the last decade of the last century, significant progress was made in determining the qualitative similarity of the solution of the equation (L) and the second-order half-linear differential equation: (HL) p (t) Φ u (t) + q (t) Φ (u (t)) = 0 where q ∈ C([t 0 , ∞)), p ∈ C 1 ([t 0 , ∞); (0, ∞)), Φ : R → R defined by Φ (s) := |s| α−1 s, α > 0 is a constant. Especially it is necessary to point out the articles of Mirzov (Mirzov, 1976) and Elbert (Elbert, 1979), who first established that the equations (L) and (HL) have similar properties describing the character of the oscillations solutions.
Between the large number of oscillation criteria shown using averaging functions, it can be noted that as a weight function is the most frequently used or positive, continuous differentiable function ρ, such that ρ is a nonnegative and decreasing function, or the function (t − s) α for α is a natural or real number greater than the unit, or product of these functions. Articles Philos (Philos, 1989) and Li (Li, 1995) on the oscillatory of a linear differential equation were given a positive answer to the question posed by the mathematicians who dealt with this problem -can a wider family of functions be used as a weight function?
Ten years later, J. Manojlović, who made an outstanding contribution to the whole theory of the oscillation of differential equations, announces the paper (Manojlović, 1999) in which one step further.
In the last years of the last century, the attention of the author has attracted a second-order differential equation known as halflinear differential equation with a delay of form: where α > 0 is a constant, and functions p(t), q(t), τ(t) which satisfy the conditions: By studying these equations, led to the conclusion that there is a certain qualitative similarity of its solutions and solutions of the equation (HL). The contribution to this study is given in the papers (Hsu & Yeh, 1996), (Kusano & Naito, 1997), (Kusano & Wang, 1995), (El-Sheikh & Sallam, 2000) and (Wang, 1997). In this paper, the given oscillatory criteria using averaging functions (HL) given in the paper (Manojlović, 1999), will be generalized to the equation (RHL).
Theorem 1. The equation (RHL) is an oscillatory if exist constant λ ∈ (0, 1) and function H ∈ H + (D) such that it is: Proof. Suppose the opposite, that there exists a nonoscillatory solution of u(t) of equation (RHL). According to Lemma 1.1. from (Bojičić, 2015), there exists We define in [T 0 , ∞) function w as: From here we have: According to Lemma 1.2. (Bojičić, 2015), for every µ ∈ (0, 1), we obtain , t T 0 . (2) If we multiply last inequality by H(t, s), and integrate it from T to t for T T 0 , we get: Using integration by parts, we have so that from equality (3) we obtain If we use a inequality Hardly, Littewood & Polya (Hardly et al., 1988) and put Therefore, for t T T 0 is valid inequality Since H ∈ H + , i.e. monotonically non-increasing by s, then for Therefore, from (5) we obtain From the last inequality it is obvious that: According to the condition (C 1 ), we obtain a contradiction. Hence, the equation (RHL) doesn't have nonoscillatory solutions, i.e. its equation is oscillatory.
Corollary 2. Equation (RHL) is oscillatory if exist function H ∈ H + (D) such that hold conditions: In order to illustrate the previously proven criteria, we consider the following example: Example 3. Consider the differential equation: where is ν, α, µ are arbitrary constants that satisfy the conditions µ > 0 and 0 ν < α 2. We check the conditions of Theorem 1: It remains to be determined whether the condition (C 1 ) is valid: where, due to the arbitrariness of constants λ ∈ (0, 1) we take λ = 1 α+1 α+1 .
Consequently, condition (C 1 ) is satisfied, hence from here follows equation (E 1 ) is oscillatory by Theorem 1.
Theorem 4. Suppose there is a function H ∈ H(D) such that the following condition is satisfied: If exist constant λ ∈ (0, 1) and function ϕ ∈ C([t o , ∞)) such that: for every T t 0 and Proof. We suppose that there exists a solution u(t) of equation (RHL) such that u(t) > 0, t T 0 . Defining the function w(t) as in the proof of Theorem 1, we get (4) and (5), for every t T T 0 Then, for (5), we have Therefore, it is true that: Now, we can conclude: We define functions Then, by (4) and (7) we see that It remains to be proved If we suppose that (9) fails, there exists T 1 > T 0 such that where µ > 0 is arbitrary number, and ξ is a positive constant, such that Then we have where with µ we denote µ = µ λ . By (11) there is a T 2 T 1 such thate H(t,T 1 ) ξ for all t T 2 , we conclude that G (t) µ, for all t T 2 .
Since µ = µ λ , and µ is arbitrary number, we get Consider now the number sequence {σ n } ∞ n=1 in (T 0 , ∞) such that Then, there exists a constant M such that for all sufficiently large n holds: Since (12) ensures that and (13) implies lim From equations (14) and (15), for sufficiently large n, we derive: i.e. F(σ n ) G(σ n ) > 1. Therefore, by using (15), we get: On the other hand, by Hölder's inequality, for every n ∈ N we have So, because of (16), we have and this is in contradiction with the condition (C 2 ). Therefore, (9) holds. Now, from (6), we obtain Since Theorem 4 can be applied in certain cases where it is not possible to apply Theorem 1, the two oscillatory criteria are independent of each other. One such case is described in the following example.
Example 5. Consider the differential equation for t t 0 , where k, ν, α are constants such that k > 0, ν < α, α > 2. If for the weight function we take the function H (t, s) = (t − s) 2 and constant λ = 1 α+1 α+1 , we can determine that the condition (C 1 ) does not apply. Indeed, for t t 0 , we have: