An equation of the form (1)
is called half-linear differential equation, where r, c are continuous functions and r (t) > 0 was introduced for the first time in . During the last decades, these equations have been widely studied in the literature. The name half-linear equation was introduced in . This term is motivated by the fact that the solution space of these equations is homogeneous (likewise in the linear case) but not additive. Since the linear Sturmian theory extends verbatim to half-linear case (for details, we refer to Section 1.2 in ), we can classify Eq.(1) as oscillatory or nonoscillatory. It is well known that oscillation theory of Eq.(1) is very similar to that of the linear Sturm-Liouville differential equation, which is the special case of Eq.(1) for p = 2 ( see ).
Actually, we are interested in the conditional oscillation of half-linear differential equations with different periodic coefficients. We say that the equation (2)
with positive coefficients is conditionally oscillatory if there exists a constant γ0 such that Eq.(2) is oscillatory for all γ > γ0 and nonoscillatory for all γ < γ0. The constant γ0 is called an oscillation constant (more precisely, oscillation constant of c with respect to r)of this equation.
Considerable effort has been made over the years to extend oscillation constant theory of half linear differential equation (1), see [5-10] and reference therein. According to our knowledge, the first attempt to this problem was made by Kneser in , where the oscillation constant for Cauchy-Euler differential equation (3)
The conditional oscillation of linear equations is studied, e.g., in [9, 10]. In [9, 12], the oscillation constant was obtained for linear equation with periodic coefficients. Using the notion of the principle solution, the main result of  was generalized in , where periodic half-linear equations were considered.
In  the half-linear Euler differential equation of the form (4)
and the half-linear Riemann-Weber differential equation of the form (5)
In , the half-linear differential equation of the form (6)
was considered for r, c being α—periodic, positive functions and it was shown that Eq.(6) is oscillatory if γ > K and nonoscillatory if γ < K, where K is given by
for p and q are conjugate numbers, i.e., If the functions r, c are positive constants, then Eq.(6) is reduced to the half-linear Euler equation (4), whose oscillatory properties were studied in detail [4, 7] and references given therein.
was considered positive, α—periodic functions r; c and d; which are defined on [0, ∞) and it was shown that Eq.(6) is oscillatory if and only if γ ≤ γrc , where γrc is given by
and in the limiting case γ = γrc Eq.(7) is nonoscillatory if μ < μrd and it is oscillatory if μ > μrd where μrd is given by
If the functions r, c, and d are positive constants, then Eq.(7) is reduced to the half-linear Rimann-Weber equation (5), whose oscillatory properties are studied in detail [4, 7] and references given therein.
In , the half-linear differential equation of the form (8)
was considered for r: [a, ∞) → ℝ ,(a > 0) where r is a continuous function for which mean value exists and for which
and c: [a, ∞) → ℝ ,(a > 0), be a continuous function having mean value and it was shown that Eq.(8) is oscillatory if M (c) > Γ and nonoscillatory if M (c) > Γ, where Γ is given by
Our goal is to find the explicit oscillation constant for Eq.(7) with periodic coefficients having different periods. We point out that the main motivation of our research comes from the paper , where the oscillation constant was computed for Eq.(7) with the periodic coefficients having the same α—period. But in that paper the oscillation constant wasn’t obtained for the periodic functions having different periods and consequently for the case when the least common multiple of these periodic coefficients is not defined. Thus in this paper we investigate the oscillation constant for Eq.(7) with periodic coefficients having different periods. For the sake of simplicity, we usually use the same notations as in the paper .
This paper is organized as follows. In section 2, we recall the concept of half-linear trigonometric functions and their properties. In section 3 we compute the oscillation constant for Eq.(7) with periodic coefficients having different periods. Additionally, we show that if the same periods are taken, then our result compiles with the known result in  or if the same period α, given in  can be chosen as the least common multiple of periods of the coefficients (which have different periods), then our result coincides with the result of . Thus, our results extend and improve the results of . Finally in the last section, we give an example to illustrate the importance of our result.
and denote by x = x (t) its solution given by the initial conditions x (0) = 0, x′(0) = 1: We see that the behavior of this solution is very similar to the classical sine function. We denote this solution by sinp t and its derivative as (sinp t)′ =cosp t. These functions are 2πp—periodic; where and satisfy the half-linear Pythagorean identity (10)
All solutions of Eq.(9) are of the form x (t) = C sinp (t + φ), where C, φ are real constants. All these solutions and their derivatives are bounded and periodic with the period 2πp. The function u = Φ (cosp t) is a solution of the reciprocal equation to Eq.(9);
which is an equation of the form as in Eq.(9), so the functions u and u' are also bounded.
Let x be a nontrivial solution of Eq.(1) and we consider the half-linear Prüfer transformation which is introduced using the half-linear trigonometric functions (11)
where and Prüfer angle φ is a continuous function defined at all points where x (t)≠0.
Let then w(t) is the solution of Riccati equation (12)
associated with Eq.(1).
At the same time, using the fact that w solves Eq.(12), we obtain
Combinig the last equation with Eq.(12) we get
Multiplying both sides of this equation by and using the half-linear Pythagorean identity Eq.(10), we obtain the equation
which will play the fundamental role in our investigation.
It is well-known that the nonoscillation of Eq.(7) is equivalent to the boundedness from above of the Prüfer angle φ given by Eq.(11) (see [6, 10]). Next, we briefly mention the principal solution of nonoscillatory equation Eq.(1) in , which is defined via the minimal solution of the associated Riccati equation Eq.(12). Nonoscillation of Eq.(1) implies that there exist T ∈ ℝ and a solution of Eq.(12) defined on some interval is called the minimal solution of all solutions of Eq.(12) and it satisfies the inequality where w is any other solution of Eq.(12) defined on some interval [Tw, ∞) and then is the principal solution of Eq.(1) via the formula
We finish this section with a lemma without proof, to be used in the next section.
Eq.(1) is non-oscillatory if and only if there exists a positive differentiable function h such that for large t.
Let then for every positive ∈,the linear differential equation is oscillatory.
3 Main results
To prove the main result, we need the following lemmas.
Let φ = φ1 + φ2 + φ3, + φ4 + Μ, (Μ is a suitable constant) be a solution of the equation
with r, c and d are positive defined functions having different periods β1, β2, and β3, respectively and let
where ξ is one of the periods β1, β2, or β3 Then θ is a solution of
and φ (τ) — θ (t) = o (1) as t → ∞.
Taking derivative of θ (t ), we have
Using integration by parts, we get
By the fact that, ds for any T–periodic function and half-linear Pythagorean identity, the expressions
are bounded. Thus we get
We can rewrite this equation as
Similarly as in  if we use integration by parts, we get
and by using the definition of R, C and D we get
And using the half-linear trigonometric functions, we have
By the mean value theorem we can write
for t1∈ [t, t + β1], t2 ∈ [t, t + ξ], t3 ∈ [t, t + β2], t4 ∈ [t, t + β3]. Thus
This implies that
Hence we get □
The computation of oscillation constant μ in Eq.(7) is based on the following lemma
Let and θ is a solution of the differential equation (13)
where R,C, D are as in Lemma 3.1.
We rewrite Eq.(13) in the form
This is the equation for Prüfer angle θ, which corresponds to the differential equation
which is the same (using the formula (1 + x)α = 1 + αx + ο (x) as x →0) as the equation (14)
i.e., the same as
First, suppose that
Let ε > 0 be sufficiently small and let Τ be so large that
for t ≥ T. Then, the equation (15)
We will show that the function satisfies (16)
for t →∞, where Η is a real constant. At the same time, by a direct computation,
Now suppose that
Let ε > 0 be again sufficiently small and let Τ be so large that
for t ≥ T. Then the equation (17)
Suppose, by contradiction, that Eq.(17) is nonoscillatory and let w be the minimal solution of its associated Riccati equation
where we denote
Let and denote and v = hp (w – wh).
Then by a direct computation v is a solution of the equation (18)
Denote now ƒ (t) := (1 + R(t)q-1, and Then we have
At this extremal point
Substituting f (t) = (1 + R(t))q-1into F (v*) and using again the formula (1 + x)α = 1 +αx + o(x)as x → 0,
Consequently, for sufficiently large t, where η is any positive constant. Using this estimate in Eq.(18)
and we see that
for large t, where the constant ζ > 0 depends on ε and η and μρ + δ – ζ > 0 if ε, η are sufficiently small. Hence v' (t) < 0 for large t, which means that there exists the limit v0= limt→∞ v (t). This limit is finite, because of Lemma 2.1, w (t) > 0, i.e., v (t) = hp (t) (w (t) – wh (t)) ≥ -hp (t) wh(t) = –λρ.
Next, we show that v0 = 0. If v0 ≠0, then i.e.,
for large t. From Eq.(18), using the fact that
i.e., since v (t) > – λρ
for sufficiently large Τ. When t → ∞, we obtain the convergence of the integral which is a contradiction, so necessarily ω must be equal to zero, which means that v0 = 0. Using second order Taylor’s expansion , we have
for |v| sufficiently small. Hence, for t sufficiently large
The last inequality is the Riccati type inequality associated with the linear second order differential equation (19)
The main results of this paper are as follows
In the limiting case γ = γ*, Eq.(7) is nonoscillatory if
and it is oscillatory if μ > μ*, where
The statement (i) is proved in  when γ ≠ γ*.
By the help of Lemma 3.1, θ is a solution of
holds, then θ (t)⊒∞as t ⊒∞and by the help of Lemma 3.1, φ is unbounded. Then from Eq.(11), Eq.(7) is oscillatory. This result also shows that Eq.(6) is nonoscillatory in the limiting case γ = γ* since this case corresponds to Eq.(7) with μ = 0 < μ*. The proof is now complete. □
In a similar way it is easy to see that if there exists a 1cm (β1, β2, β3) and the period a, given in  is chosen as the number 1cm (β1, β2, β3), then there exist some natural numbers m,l, and s such as α = mβ1 = lβ2 = sβ3 and by the help of periodic functions properties we get
Consider the equation (20)
which is Eq.(7) for and d (t) = 2 + cos (ax + b),(a,b ∈ ℝ). In this case defined for all t ∈ ∞ with period is positive defined for all t ∈ R with period and d (t) = 2 + cos (ax + b) > 0 can be considered as positive defined function with period
and equation Eq.(20) is nonoscillatory if and only if This computation shows that we can compute the oscillation constant μ* for every period If the functions r (t) , c (t) , and d(t) are having an period it is well known that there exists some m, l and s natural numbers such as In this case we use the fact of the remark 3.4 and we can apply Theorem 3.1 in  to the above example and we get oscillation constant as
Finally, as a future work this paper can be improved if we replace the periodic coefficient functions, having different periods, with asymptotically almost periodic coefficients or having different mean values coefficients functions. The conditional oscillation of half-linear equations with asymptotically almost periodic coefficient or coefficients having mean values are studied in [13-15].
The authors would like to express their sincere gratitude to the referees for a number of valuable comments and suggestions which led to significant improvement of the final version of the paper.
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About the article
Published Online: 2017-05-04
Conflict of interestsThe authors declare that there is no conflict of interest regarding the publication of this paper.
Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 548–561, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0046.
© 2017 Misir and Mermerkaya. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0