Oscillation Criteria for Two-dimensional System of Non-linear Ordinary Differential Equations

New oscillation criteria are established for the system of non-linear equations u = g(t)|v| 1 α sgn v, v = −p(t)|u| α sgn u, R are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [0, +∞]. Among others, presented oscillatory criteria generalize well-known results of E. Hille and Z. Nehari and complement analogy of Hartman–Wintner theorem for the considered system.

(1.3)By a solution of system (1.1) on the interval J ⊆ [0, +∞[ we understand a pair (u, v) of functions u, v : J → R, which are absolutely continuous on every compact interval contained in J and satisfy equalities (1.1) almost everywhere in J.

Z. Opluštil
It was proved by Mirzov in [10] that all non-extendable solutions of system (1.1) are defined on the whole interval [0, +∞[.Therefore, when we are speaking about a solution of system (1.1), we assume that it is defined on [0, +∞[.Definition 1.1.A solution (u, v) of system (1.1) is called non-trivial if |u(t)| + |v(t)| = 0 for t ≥ 0. We say that a non-trivial solution (u, v) of system (1.1) is oscillatory if its each component has a sequence of zeros tending to infinity, and non-oscillatory otherwise.
In [10,Theorem 1.1], it is shown that a certain analogue of Sturm's theorem holds for system (1.1) if the function g is nonnegative.Especially, under assumption (1.2), if system (1.1) has an oscillatory solution, then any other its non-trivial solution is also oscillatory.Definition 1.2.We say that system (1.1) is oscillatory if all its non-trivial solutions are oscillatory.
Oscillation theory for ordinary differential equations and their systems is a widely studied and well-developed topic of the qualitative theory of differential equations.As for the results which are closely related to those of this section, we should mention [2,[4][5][6][7][8][9][11][12][13].Some criteria established in these papers for the second order linear differential equations or for two-dimensional systems of linear differential equations are generalized to the considered system (1.1) below.
Many results (see, e.g., survey given in [2]) have been obtained in oscillation theory of the so-called "half-linear" equation r(t)|u | q−1 sgn u + p(t)|u| q−1 sgn u = 0 (1.4) (alternatively this equation is referred as "equation with the scalar q-Laplacian").Equation (1.4) is usually considered under the assumptions q > 1, p, r : [0, +∞[ → R are continuous and r is positive.One can see that equation (1.4) is a particular case of system (1.1).Indeed, if the function u, with properties u ∈ C 1 and r|u | q−1 sgn u ∈ C 1 , is a solution of equation (1.4), then the vector function (u, r|u | q−1 sgn u ) is a solution of system (1.1) with g(t) := r 1 1−q (t) for t ≥ 0 and α := q − 1.
Moreover, the equation is also studied in the existing literature under the assumptions α ∈ ]0, 1] and p : R + → R is a locally integrable function.It is mentioned in [6] that if u is a so-called proper solution of (1.5) then it is also a solution of system (1.1) with g ≡ 1 and vice versa.Some oscillations and non-oscillations criteria for equation (1.5) can be found, e.g., in [6,7].Finally, we mention the paper [1], where a certain analogy of Hartman-Wintner's theorem is established (origin one can find in [3,14]), which allows us to derive oscillation criteria of Hille-Nehari's type for system (1.1). Let In view of assumptions (1.2) and (1.3), there exists t g ≥ 0 such that f (t) > 0 for t > t g and f (t g ) = 0. We can assume without loss of generality that t g = 0, since we are interested in behaviour of solutions in the neighbourhood of +∞, i.e., we have and, moreover, lim t→+∞ f (t) = +∞. (1.7) For any λ ∈ [0, α[ , we put Now, we formulate an analogue (in a suitable form for us) of the Hartman-Wintner's theorem for the system (1.1) established in [1].
One can see that two cases are not covered by Theorem 1.3, namely, the function c α (t; λ) has a finite limit and lim inf t→+∞ c α (t; λ) = −∞.The aim of this Section is to find oscillation criteria for system (1.1) in the first mentioned case.Consequetly, in what follows, we assume that lim

Main results
In this section, we formulate main results and theirs corollaries.
be satisfied, where B(α, µ) is the greatest root of the equation (2.12) Then system (1.1) is oscillatory.
Finally, we formulate an assertion for the case, when both conditions (2.6) and (2.10) are fulfilled.In this case we can obtain better results than in Theorems 2.6 and 2.7.
Moreover, the results of [6] obtained for equation (1.5) are in a compliance with those above, where we put g ≡ 1, λ = 0, and µ = 1 + α.Observe also that Corollary 2.3 and Theorems 2.6 and 2.7 extend oscillation criteria for equation (1.5) stated in [7], where the coefficient p is suppose to be non-negative.In the monograph [2], it is noted that the assumption p(t) ≥ 0 for t large enough can be easily relaxed to t 0 p(s)ds > 0 for large t.It is worth mentioning here that we do not require any assumption of this kind.Finally we show an example, where we can not apply oscillatory criteria from the above mentioned papers, but we can use Theorem 2.1 succesfully.
Example 2.10.Let α = 2, g(t) ≡ 1, λ = 0, and It is clear that the function p and its integral for t ≥ 0 change theirs sign in any neighbourhood of +∞.Therefore neither of results mentioned in Remark 2.9 can be applied.
On the other hand, we have for t > 0 and thus, the function c 2 (•, 0) has the finite limit Moreover, Consequently, according to Theorem 2.1, system (1.1) is oscillatory.

Auxiliary lemmas
We first formulate two lemmas established in [1], which we use in this section.
Remark 3.3.One can easily verify (see the proofs of Lemma 4.2 and Corollary 2.5 in [1]) that if (u, v) is a solution of system (1.1) satisfying with t u > 0 and the function c α (•; λ) has a finite limit (1.8), then where the number γ is defined by (2.8), and Moreover, according to Lemma 3.2, we have and one can show (see Lemma 4.1 and the proof of Corollary 2.5 in [1]) that where A(α, λ) denotes the smallest root of equation (2.9).
Proof.Let (u, v) be a non-oscillatory solution of system (1.1).Then there exists t u > 0 such that (3.1) holds.Define the function ρ by (3.4).Then we obtain from (1.1) that Multiplaying the last equality by f λ (t) and integrating it from t u to t, we get (3.9) Integrating the left-hand side of (3.9) by parts, we obtain where the function h is defined in (3.3).Hence, where Therefore, in view of relations (3.2) and (3.6), it follows from (3.10) that Hence, It is clear that if m = +∞, then (3.7) holds.Therefore, we suppose that m < +∞.

Z. Opluštil
In view of (2.6), (3.5), and (3.13), relation (3.12) yields that ), then 0 is a root of equation (2.9).Moreover, in view of Lemma 3.2 and the assumption λ < α, we see that the function x → α|x + γ| Then it follows from (3.12) that (3.17) On the other hand, the function x → α|x + γ| is non-decreasing on [0, +∞[.Therefore, by virtue of (3.5), (3.15), and (3.16), one gets from (3.17) that Since ε was arbitrary, the latter relation leads to the inequality α|m + γ| One can easily derive that the function y : Integrating the left-hand side of the last equality by parts, we get where According to Lemma 3.1, it follows from (3.20) that where , then it is not difficult to verify that ( µ 1+α ) α is a root of the equation (2.12) and the function x → α|x| Using the latter inequality in (3.25), we get where Consequently, where which, by virtue of the assumption α < µ and condition (1.7) and (3.24), yields that Since ε was arbitrary, the latter inequality leads to One can easily derive that the function y : x → α|x|

Proofs of main results
Proof of Theorem 2.1.Assume on the contrary that system (1.1) is not oscillatory, i.e., there exists a solution (u, v) of system (1.1) satisfying relation (3.1) with t u > 0. Analogously to the proof of Lemma 3.4 we show that equality (3.11) holds, where the functions h, ρ and the number γ are defined by (3.3), (3.4), and (2.8).Moreover, conditions (3.5) and (3.6) are satisfied.
Multiplying of (3.11) by g(t) f α−1−λ (t) and integrating it from t u to t, one gets Observe that Hence, it follows from (4.1) that On the other hand, according to (2.8), (3.3), and Lemma 3.1 with ω := α, the estimate holds for s ≥ t u .Moreover, in view of (1.2), (1.6), and (3.5), it is clear that Consequently, by virtue of the last inequality and (4.3), it follows from (4.2) that Hence, in view of (1.7), we get lim sup , which contradicts (2.1).

Z. Opluštil
Proof of Corollary 2.2.Observe that for t > 0, we have and Moreover, it is easy to show that On the other hand, by virtue of (2.2), from relation (4.5) one gets lim inf Therefore, in view of relation (1.7), it follows from (4.6) that lim inf Now, equality (4.4) and inequality (4.7) guarantee the validity of condition (2.1) and thus, the assertion of the corollary follows from Theorem 2.1.
Proof of Corollary 2.3.If assumption (2.3) holds, then it follows from (4.4) that condition (2.1) is satisfied and thus, the assertion of the corollary follows from Theorem 2.1.Let now assumption (2.4) be fulfilled.Observe that Therefore, in view of (2.4), we obtain lim inf On the other hand, it is clear that Hence, we have and consequently, by virtue of assumption (1.8) and condition (4.8), we get In view of (4.8), there exist ε > 0 and t ε > 0 such that Hence, it follows from (4.9) that Since ε > 0, by virtue of (1.7), from the last relation we derive inequality (2.1).Therefore, the assertion of the corollary follows from Theorem 2.1.
Proof of Theorem 2.5.Assume on the contrary that system (1.
is satisfied for t ≥ t u .Moreover, according to Lemma 3.1 with ω := µ, it is clear that where Consequently, by virtue of (1.7), relation (4.12) leads to a contradiction with assumption (2.5).

Z. Opluštil
Proof of Theorem 2.6.Suppose on the contrary that system (1.1) is not oscillatory.Then there exists a solution (u, v) of system (1. Since ε was arbitrary, in view of (1.7), from the latter inequality we get which contradicts assumption (2.7).
Proof of Theorem 2.7.Assume on the contrary that system (1.1) is not oscillatory, i.e., there exists a solution (u, v) of system (1.1) satisfying relation (3.1) with t u > 0. Analogously to the proof of Lemma 3.4 we show that equality (3.12) holds, where the number γ and the functions h, ρ are defined by (2.8), (3.3), and (3.4).
On the other hand, according to Lemma 3.5, estimate (3.19) is fulfilled, where B(α, µ) is the greatest root of equation (2.12).Let ε > 0 be arbitrary.Then there exists t ε ≥ t u such that In view of the last inequality, (1.2), (1.6) and (3.5), it follows from (3.12) that Since ε was arbitrary, we get Proof of Theorem 2.8.Suppose on the contrary that system (1.1) is not oscillatory.Then there exists a solution (u, v) of system (1.1) satisfying relation (3.1) with t u > 0. Put i.e., m denotes the smallest root of equation (2.9) and M is the greatest root of equation (2.12).According to Lemmas 3.4 and 3.5, we have where the function ρ and the number γ are defined in (3.4) and (2.8).
Analogously to the proof of Theorem 2.5 we show that relation (4.10) holds for t ≥ t u , where the number δ(t u ) and the function h are defined by (3.21) and (3.3).
In view of (2.6), one can easily show that the function y :