Abstract
This paper is devoted to studying the half-linear functional dynamic equations of second-order on an unbounded above time scale \(\mathbb {T}\). We present some Nehari-type oscillation criteria for a class of second-order dynamic equations. The obtained results show that there is a substantial improvement in the literature on second-order dynamic equations. We include some examples illustrating the significance of our results.
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1 Introduction
In order to combine continuous and discrete analysis, Stefan Hilger [25] has proposed the theory of dynamic equations on time scales. In many applications, different types of time scales can be applied. The theory of dynamic equations includes the classical theories for the differential equations and difference equations cases, and other cases in between these classical cases. That is, we are worthy of considering the \(q-\)difference equations when \({\mathbb {T=}}q^{\mathbb {N}_{0}}:={\mathbb {\{}}q^{\lambda }:\) \(\lambda \in {\mathbb {N}}_{0}\) for \(q>1\}\) which has important applications in quantum theory (see [27]), and various types of time scales such as \({ \mathbb {T=}}h\mathbb {N}\), \({\mathbb {T=N}}^{2},\) and \(\mathbb {T=T}_{n},\) where \(\mathbb {T}_{n}\) is the set of the harmonic numbers, can also be considered. See [1, 9, 10] for more details of time scales calculus.
Oscillation phenomena take part in different models from real world applications; we refer to the papers [23, 30] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. The study of half-linear dynamic equations is dealt with in this paper because these equations arise in various real-world problems such as non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and in the study of \( p-\)Laplace equations; see, e.g., the papers [4, 7, 8, 12, 29, 37] for more details. Therefore, we are concerned with the behavior of the oscillatory solutions to the half-linear functional dynamic equation of second-order
on an arbitrary unbounded above time scale \(\mathbb {T}\), where \(t\in [t_{0},\infty )_{\mathbb {T}}:=[t_{0},\infty )\cap \mathbb {T}\), \(t_{0}\ge 0\), \(t_{0}\in \mathbb {T}\), \(\phi _{\beta }(u):=\left| u\right| ^{\beta -1}u\), \(\beta >0\), b is a positive rd-continuous function on \(\mathbb {T}\), \(k:\mathbb {T\rightarrow T}\) is a rd-continuous function satisfying \( \lim _{t\rightarrow \infty }k(t)=\infty \), and a is a positive rd-continuous function on \(\mathbb {T}\) such that \(a^{\Delta }\ge 0\) such that \(\int _{t_{0}}^{\infty }a^{-\frac{1}{\beta }}(\tau )\Delta \tau =\infty . \)
By a solution of Eq. (1) we mean a nontrivial real–valued function \( y\in \textrm{C}_{\textrm{rd}}^{1}[t_{y},\infty )_{\mathbb {T}}\) for some \( t_{y}\ge t_{0}\) with \(t_{0}\in \mathbb {T}\) such that \(y^{\Delta },a(t)\phi _{\beta }\left( y^{\Delta }(t)\right) \in \textrm{C}_{\textrm{rd} }^{1}[t_{y},\infty )_{\mathbb {T}}\) and y(t) satisfies Eq. (1) on \( [t_{y},\infty )_{\mathbb {T}},\) where \(\textrm{C}_{\textrm{rd}}\) is the space of right-dense continuous functions. It may be noted that in a particular case when \(\mathbb {T}=\mathbb {R}\) then
and the equation (1) becomes the half-linear differential equation
The oscillation properties of special cases of equation (2) are investigated by Nehari [32] as follows: every solution of the linear differential equation
is oscillatory if
We will show that our results not only extend some of the known oscillation results for differential equations, but we can also perform these results on other cases in which the oscillatory behaviour of solutions to these equations on various types of time scales is not known. Notice that, if\(\ { \mathbb {T}}=\mathbb {Z}\), then
and (1) becomes the half-linear difference equation
If \({\mathbb {T}}=h\mathbb {Z}\), \(h>0,\) thus
and (1) gets the half-linear difference equation
If
then
where \(t_{0}=q^{n_{0}}\), and (1) becomes the half-linear \(q-\)difference equation
If
then
and (1) converts to the half-linear difference equation
If \(\mathbb {T}=\{H_{n}:n\in \mathbb {N}_{0}\}\) where \(H_{n}\) is the harmonic numbers defined by
then
and (1) becomes the half-linear harmonic difference equation
For Nehari-type oscillation criteria of second-order dynamic equations, Erbe et al. [20] examined the nonlinear dynamic equation
where \(\beta \ge 1\) is a quotient of odd positive integers and \(k(t)\le t\) for \(t\in \mathbb {T}\) and showed that every solution of (10) is oscillatory, if
and
where \(l:=\liminf _{t\rightarrow \infty }\) t\(/{\sigma (}\) t\({)>0}\). Erbe et al. [21] investigated Nehari-type oscillation criterion for the half-linear dynamic equation
where \(0<\beta \le 1\) is a quotient of odd positive integers, \(a^{\Delta }\ge 0,\) and \(k(t)\le t\) for \(t\in \mathbb {T}\) and proved that every solution of (13) is oscillatory, if (11) holds,
and
where \(l:=\liminf _{t\rightarrow \infty }\) t\(/{\sigma (}\) t\({)>0.}\)
We refer the reader to associated results [2, 5,6,7,8, 11, 14, 16, 19, 22, 24, 31, 33,34,35, 38] and the references cited therein. It may be noted that the contributions of Nehari [32] strongly motivated research in this paper. The objective of this paper is to conclude some Nehari-type oscillation criteria for Eq. (1) in the cases where k(t\()\le \) t and k(t\()\ge \) t. Besides, we reference that, contrary to [20, 21], a restrictive condition (11) is not needed in our oscillation theorems, and also, our results can function for any positive real numbers \( \beta \). All functional inequalities deemed in the sequel are tacitly supposed to hold eventually. That is, they are satisfied for all sufficiently large t.
2 Main Results
We start this section with the following introductory lemmas.
Lemma 1
([12, Lemma 2.1]) Suppose that (14) holds. If y is a positive solution of Eq. (1) on \( [t_{0},\infty )_{{\mathbb {T}}}\), then
eventually.
Lemma 2
([12, Lemma 2.2]) If
then \(\dfrac{y(t)}{t-t_{0}}\) is strictly decreasing on \((t_{0},\infty )_{{ \mathbb {T}}}\).
In the sequel we will use the following notations \(l:=\liminf _{t\rightarrow \infty }\dfrac{t}{\sigma (t)}\) and
Theorem 1
Suppose that (14) holds. If \(l>0\) and
for sufficiently large \(T\in [t_{0},\infty )_{{\mathbb {T}}},\) then all solutions of Eq. (1) are oscillatory.
Proof
Assume y is a nonoscillatory solution of Eq. (1) on \([t_{0},\infty )_{{ \mathbb {T}}}.\) Then, let \(y(t)>0\) and \(y(k(t))>0\) on\(\ {[}\) t\(_{0},\infty )_{\mathbb {T}}\), without loss of generality. From Lemma 1, we see that
Define
By the rules of product and quotient, we get
From (1) and the definition of x(t), we have
Let \(t\in [t_{0},\infty )_{{\mathbb {T}}}\) be fixed. When \(k(t)\le t\) , in view of Lemma 2, \(\left( \dfrac{y(t)}{t-t_{0}}\right) ^{\Delta }<0\) on \((t_{0},\infty )_{{\mathbb {T}}}\), we obtain
Then there exists \(t_{\lambda }\in [t_{0},\infty )_{{\mathbb {T}}}\), for each \(0<\lambda <1\), such that
If \(k(t)\ge t\), then \(y(k(t))\ge y(t)>\lambda y(t)\) for \(t\ge t_{\lambda } \). In both cases, from the definition of \(\varphi (t)\) we have that
Therefore
Using the Pötzsche chain rule to get
If \(0<\beta \le 1,\) then
and if \(\beta \ge 1\), then
Note that y(t) is strictly increasing and \(a^{\frac{1}{\beta }}y^{\Delta }\) is strictly decreasing, we see that for \(\beta >0\) and \(t\in [t_{\lambda },\infty )_{\mathbb {T}},\)
Multiplying by \(\dfrac{t^{\beta +1}}{a(t)}\) and integrating from T to \( \sigma \left( t\right) \in [t_{\lambda },\infty )_{\mathbb {T}}\), we obtain
Now for any \(\varepsilon >0\), there exists \(t\ge t_{\lambda }\) such that
where
It follows from (22) that
Using integration by parts, we obtain
Utilizing the quotient rule and applying the Pötzsche chain rule, we see
Hence
Using the inequality
with \(A=\beta \left( l-\varepsilon \right) ^{\beta +1},\) \(B=\beta +1\) and \( u=\left( \dfrac{\tau ^{\beta }}{a(\tau )}x(\tau )\right) ^{\sigma }\), we obtain
Dividing by t, we get
Taking the \(\limsup \) as \(t\rightarrow \infty \) to get
where
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we get
Multiplying both sides of (21) by \(\dfrac{t^{\beta +1}}{a(t)}\), we obtain
Using (23) gives
Integrating (33) from T to \(\sigma \left( t\right) \in [t_{\lambda },\infty )_{ \mathbb {T}}\), we obtain
By integrating by parts, we conclude that
By using (27) , we get
Dividing both sides by t, we have
Taking the \(\lim \sup \) as \(t\rightarrow \infty \) and using (31) , we get
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we have
Substituting (32) into (35) , we achieve
which contradicts the condition (16) . The proof is completed. \(\square \)
Theorem 2
Suppose that (14) holds. If \(l>0\) and
for sufficiently large \(T\in [t_{0},\infty )_{{\mathbb {T}}},\) then all solutions of Eq. (1) are oscillatory.
Proof
Assume y is a nonoscillatory solution of Eq. (1) on \([t_{0},\infty )_{{ \mathbb {T}}}.\) Then, let \(y(t)>0\) and \(y(\lambda (t))>0\) on\(\ {[}\) t\( _{0},\infty )_{\mathbb {T}}\), without loss of generality. From Lemma 1 , we see that
As shown in the proof of Theorem 1, (30) and (34) hold for sufficiently large \(t\in [t_{0},\infty )_{{\mathbb {T}}}\). Dividing both sides of (30) by \(\sigma \left( t\right) ,\) we obtain
Taking the \(\limsup \) as \(t\rightarrow \infty \) to obtain
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we get inequality
Dividing both sides by of (34) \(\sigma \left( t\right) ,\) we obtain
Taking the \(\lim \sup \) as \(t\rightarrow \infty \) and utilizing (31), we see
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we have
Substituting (37) into (38) , we achieve
which contradicts the condition (36) . The proof is completed. \(\square \)
Example 1
Consider the dynamic equations of second-order for \(t\in [t_{0},\infty )_{\mathbb {T}},\)
and
where \(l=\liminf _{t\rightarrow \infty }\) t\(/{\sigma (}\) t\({)}>0\) and \( \alpha >0\) is a constant. It is not difficult to derive that all solutions of (39) and (40) are oscillatory if \(\alpha >\frac{1}{ l^{3}}\left( 1-\frac{l^{2}}{2}\right) \) or \(\alpha >\frac{1}{l^{2}}\left( 1- \frac{l}{2}\right) \) by using Theorems 1 and 2 respectively.
Theorem 3
Suppose that (14) holds. If \(l>0\) and
for sufficiently large \(T\in [t_{0},\infty )_{{\mathbb {T}}},\) then all solutions of Eq. (1) are oscillatory.
Proof
Assume y is a nonoscillatory solution of Eq. (1) on \([t_{0},\infty )_{{ \mathbb {T}}}.\) Then, let \(y(t)>0\) and \(y(k(t))>0\) on\(\ {[}\) t\(_{0},\infty )_{\mathbb {T}}\), without loss of generality. From Lemma 1, we see that
As shown in the proof of Theorem 1, (21) holds for sufficiently large \(t\in [t_{0},\infty )_{{\mathbb {T}}}\). Multiply (21) by \(\dfrac{t^{\beta +1}}{a(t)}\) and integrating from T to \(t\in [t_{\lambda },\infty )_{\mathbb {T}}\), we get
Progressing as in the proof of Theorem 1, we arrive that
Dividing both sides by t, we obtain
Taking the \(\limsup \) as \(t\rightarrow \infty ,\) we see
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we get inequality
Again, multiplying (21) by \(\dfrac{t^{\beta +1}}{a(t)}\), we get
Progressing as in the proof of Theorem 1, we arrive that
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we get
Substituting (43) into (45) , we achieve
which contradicts the condition (41) . The proof is completed. \(\square \)
Example 2
Consider a nonlinear dynamic equation of second-order for \( t\in [t_{0},\infty )_{\mathbb {T}}\),
where \(\gamma >0\) is a constant and \(l=\liminf _{t\rightarrow \infty }\) t\(/ {\sigma (}\) t\({)}>0\). Let \(b(t)=\gamma a(t)/{(}\) tk\({^{\beta }(}\) t )). Therefore,
Employment of Theorem 3 means that every solution of (46) is oscillatory if
Theorem 4
Suppose that (14) holds. If \(l>0\) and
for sufficiently large \(T\in [t_{0},\infty )_{{\mathbb {T}}},\) then every solution of Eq. (1) is oscillatory.
Proof
Assume y is a nonoscillatory solution of Eq. (1) on \([t_{0},\infty )_{{ \mathbb {T}}}.\) Then, let \(y(t)>0\) and \(y(k(t))>0\) on\(\ {[}\) t\(_{0},\infty )_{\mathbb {T}}\), without loss of generality. From Lemma 1, we see that
As shown in the proof of Theorem 3, (42) and (44) hold for sufficiently large \(t_{\lambda }\in [t_{0},\infty )_{{ \mathbb {T}}}\). Dividing both sides of (42) by \(\sigma \left( t\right) ,\) we obtain
Taking the \(\limsup \) as \(t\rightarrow \infty \) we get
Since \(0<\lambda <1\) and \(\varepsilon >0\) are arbitrary, we get inequality
Integrating the inequality (44) from t to \(t\in [t,\infty )_{\mathbb {T}}\) to obtain
By integrating by parts, we obtain
By using (27) , we get
Dividing both sides by \(\sigma \left( t\right) ,\) we have
Taking the \(\lim \sup \) as \(t\rightarrow \infty \) and utilizing (31) , we get
Since \(\varepsilon >0\) and \(0<\lambda <1\) are arbitrary, we see
Substituting (48) into (49) , we achieve
which contradicts the condition (47) . The proof is completed. \(\square \)
3 Discussions and Conclusions
-
(1)
In this paper, several new Nehari-type criteria are presented that can be applied to Eq. (1) are valid for various types of time scales, e.g., \(\mathbb {T}=\mathbb {R},\mathbb {T}=\mathbb {Z},\mathbb {T}=h \mathbb {Z}\) with \(h>0\), \(\mathbb {T}=q^{\mathbb {N}_{0}}\) with \(q>1\), etc. (see [9]).
-
(2)
The results in this paper are including the both cases and also we do not need to assume \(k(t)\ge t\) or \(k(t)\le t\), for all sufficiently large t.
-
(3)
We note that Theorems 2 and 3 improve Theorem 4, namely, conditions (36) and (41) improve (47) ; see the following details:
$$\begin{aligned} \frac{1}{\sigma \left( t\right) }\int _{T}^{\sigma \left( t\right) }\dfrac{ \tau ^{\beta +1}}{a(\tau )}\varphi (\tau )b(\tau )\Delta \tau\ge & {} \frac{1}{ \sigma \left( t\right) }\int _{T}^{t}\dfrac{\tau ^{\beta +1}}{a(\tau )} \varphi (\tau )b(\tau )\Delta \tau \\\le & {} \frac{1}{t}\int _{T}^{t}\dfrac{\tau ^{\beta +1}}{a(\tau )}\varphi (\tau )b(\tau )\Delta \tau \end{aligned}$$and
$$\begin{aligned} 1-\frac{l^{\beta }}{\beta +1}\le 1-\frac{l^{\beta +1}}{\beta +1}. \end{aligned}$$ -
(4)
It would be interesting to extend the sharp Nehari-type criterion that the solutions of the second-order Euler differential equation \(y^{\prime \prime }(t)+\dfrac{\gamma }{t^{2}}y(t)=0\) are oscillatory when \(\gamma >\dfrac{1}{4}\) to a second-order dynamic equation, see [32].
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Acknowledgements
This research has been funded by Scientific Research Deanship at University of Ha’il – Saudi Arabia through project number RG-21 011. R.A. El-Nabulsi would like to thank Jaume Giné for inviting him to submit a work to QTDS.
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Hassan, T.S., Elabbasy, E.M., El-Nabulsi, R.A. et al. Nehari-type Oscillation Theorems for Second Order Functional Dynamic Equations. Qual. Theory Dyn. Syst. 22, 13 (2023). https://doi.org/10.1007/s12346-022-00711-w
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DOI: https://doi.org/10.1007/s12346-022-00711-w