New oscillation criteria of special type second-order non-linear dynamic equations on time scales

By the Riccati transformation technique, we study some new oscillatory properties for the second-order dynamic equation on an arbitrary time scale $$\mathbb {T}.$$T. We also establish the Kamenev-type and Philos-type oscillation criteria. At the end, we give examples which illustrate our main results.


Introduction
In [25], Kubyshkin and Moryakova considered a secondorder differential-difference equation of delay type € xðtÞ þ A _ xðtÞ þ xðtÞ þ Kðxðt À hÞÞ þ Wð _ xðt À hÞÞ ¼ 0; and T ¼ S m2Z fm þ 1 n : n 2 Ng [ Z, etc. Throughout this paper, we obtain the sufficient conditions of oscillation for the dynamic equation (1.2). To the best of our knowledge, no work has been done regarding the oscillatory behavior of (1.2) so far.
Following Stefan's landmark [22], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete, quantum and continuous calculus to an arbitrary time scale calculus. A time scale is an arbitrary non-empty closed subset of the real numbers which have the topology that inherits from the real numbers with the standard topology. It has applications in electrical engineering, quantum mechanics, population dynamic and economics etc [3,11]. In particular, a time scale q Z [ f0g; q [ 1 is used in quantum physics, see in [6,7]. Many authors have worked on various aspects of new theory, see in [2,6,7,14,17,19,20,21,34] and the references therein. These literatures summarize and organize much of time scale calculus.
Stefan's theory has attracted the attention of many researchers on oscillation of second-order linear and nonlinear dynamic equation on time scales. In recent years, many researchers have focused on oscillation and nonoscillation criteria of second-order ordinary dynamic equations on time scales. Several authors have studied the oscillation criteria by employing the Riccati transformation technique as well as established the Kamenev-type and Philos-type oscillation criteria. For more details on such criteria, we refer the reader to the papers [1,12,13,15,20,28,[30][31][32][33] and reference therein. To establish the oscillation criteria, in [24], Kamenev considered a second-order differential equation and studied the sufficient conditions for oscillation. They have assumed that there exist two positive rd-continuous functions p and q such that 1. r : T ! R is a positive and rd-continuous function and c 2 N is odd , 2. p; q : T ! R are rd-continuous functions such that pðtÞ À qðtÞ [ 0 for t 2 T, 3. F : T Â R ! R and G : T Â R 2 ! R are functions such that uFðt; uÞ [ 0 and uGðt; u; vÞ [ 0 for all u 2 R nf0g; v 2 R; t 2 T, 4. jFðt; uÞj ! pðtÞjuj c ; jGðt; u; vÞj qðtÞjuj c 8u 2 R n f0g; v 2 R; t 2 T.
In [18], Graef  In this paper, we first deal with two functions Kðyðt À hÞÞ and Wðy D ðt À hÞÞ; which play an important role in our analytical findings. As we see in the above assumption (4), the absolute value of functions F and G are related with the absolute value of the unknown function u(t), with the functions p(t) and q(t), respectively. In Eq. To establish oscillation criteria for (1.2), we need jKyðtÞj ! pðtÞjyðtÞj for yðtÞ 6 ¼ 0 such that yðtÞKðyðtÞÞ [ 0: Moreover, there exist a function f 2 CðR; RÞ such that jf ðWðy D ðtÞÞÞj ! MðtÞjy D ðtÞj as well as yðtÞf ðWðy D ðtÞÞÞ [ 08yðtÞ 6 ¼ 0 in R; where M(t) is a nonnegative rd-continuous function defined on T. Now we choose the real coefficients w j ; k i such that For simplicity, throughout this paper, we denote ½a1Þ T ¼ ½a1Þ T T: In addition, we also need the following assumptions, as follows: • ðO 1 Þ Assume a; p : ½t 0 ; 1Þ T ! R are positive rd-continuous functions such that 0\pðtÞ k 1 \1 and qðtÞ :¼ pðtÞ lðtÞ w 1 .
[ 0 8t 2 T: Let us recall that, a solution y(t) of (1.2) is a non-trivial or yðtÞ 6 ¼ 0; such that yðtÞ 2 C D 2 rd ð½t y ; 1Þ T Þ for certain t y ! t 0 : If it is eventually positive or eventually negative, then it must be non-oscillatory, otherwise oscillatory, i.e., it is oscillatory if there exists a real sequence, say fa n g such that a n ! 1 as n ! 1 and yða n Þ ¼ 08n 2 N: Our attention is restricted to those solutions of (1.2) which exist on the half-line ½t y ; 1Þ T and satisfy supfjyðtÞj : t [ t Ã g 6 ¼ 0 for any t y t Ã and sup T ¼ 1: This paper is organized as follows: In Sect. 2, we give basic definitions and present some necessary Lemmas. In the next section, we establish the sufficient conditions of oscillation of our Eq. (1.2). We further establish the Kamenev-type and Philos-type oscillation criteria. Some remarks for the particular case are also discussed. At the end in Sect. 4, to validation our results, we give an example. We also discuss the cases when the time scale is of a particular form.

Preliminaries
In this section, we present some basic definitions, useful Theorems and basic facts of time scales.
Definition 2.1 [6]. For t 2 T; forward and backward jump operators r; q : T ! T are defined by rðtÞ :¼ inffs 2 T : s [ tg and qðtÞ :¼ supfs 2 T : s\tg; respectively. The classification of points of time scale T: For t 2 T; t is called right-scattered if t\rðtÞ; and right dense if for all t\supT such that t ¼ rðtÞ: Similarly, t is left-scattered if t [ qðtÞ; and left dense if for all t [ infT such that t ¼ qðtÞ: The graininess operator l : T ! ½0; 1Þ is defined by lðtÞ ¼ rðtÞ À t.
where ; is an empty set.

Definition 2.3 [6]
A function f : T ! R is called rd-continuous provided it is continuous at all right-dense points in T and its left-sided limit exist (finite) at all left-dense points in T, which is denoted by C rd ¼ C rd ðTÞ ¼ C rd ðT; RÞ: if T has a left-scattered maximum n; and T j ¼ T; otherwise.
Definition 2.4 [6] For a function f : T ! R and t 2 T j , we define f D ðtÞ, to be a number (provided it exists) with the property that for any given [ 0, there exists a neigh- ½f ðrðtÞÞ À f ðrÞ À f D ðtÞ½rðtÞ À r jrðtÞ À rj 8r 2 Z: Thus, we call f D ðtÞ the D or Hilger derivative of f at t.
Theorem 2.5 [6] For the functions g; f : T ! R and t 2 T j . The following statements are true: 1. If f is differentiable at t, then f is continuous at t; 2. If f is continuous at t and t is right-scattered, then f is D-derivative at t and f D ðtÞ ¼ f ðrðtÞÞÀf ðtÞ 5. If f and g both are differentiable at t, then a product fg : T ! R is differentiable at t and hence,for t 2 T such that a t b; 8a; b 2 T; we have the following facts ð2:3Þ where g h ðzÞ is the cylinder transformation, which is defined by Before going to our main section, we first introduce some necessary lemmas which are crucial for our proofs. For sufficient large t, we derive a contradiction to (3.1), as the left-hand side of (3.10) finite, which completes the proof of our theorem.  The next result immediately follows from Theorem 3.1 by different choices of dðtÞ. In particular, we take dðtÞ as positive constant (say C [ 0), we establish the following corollary. For sufficiently large t, we derive a contradiction to (3.21), as the left-hand side finite,which completes the proof of our theorem. h In view of Theorem 3.5, we immediately obtain following corollary. To present our next theorems, we first introduce Saker's result [29] as follows ððt À sÞ N Þ D s À N ðt À rðsÞÞ N À1 0 for N [ 1 and rðsÞ t: ð3:32Þ By using an integral averaging technique of Kamenev-type, we present some new oscillation criteria of (1.2). Proof Assume to the contrary that (1.2) has a non-oscillatory solution. Let y(t) be a non-oscillatory solution of (1.2). Then, without loss of generality, we assume that y(t) is an eventually positive function, i.e., there exists t 0 such that yðtÞ [ 08t 2 ½t 0 1Þ T : A similar argument holds also for the case when y(t) is eventually negative. From Eq. (3.9), we have F ðtÞ À w D ðtÞ; for 2s 0 t: Our next aim to establish the Philos-type oscillation criteria for (1.2). We define some elementary assumptions as follows: For any number g 2 R; we define positive and negative parts, g þ and g À , respectively, of g by g þ :¼ maxf0; gg and g À :¼ maxf0; Àgg: Assume that the rd-continuous functions H; h : D ! R; where D ¼ fðt; sÞ; t 0 s 0 tg such that