- Research
- Open access
- Published:
Distributions of zeros of solutions to first order delay dynamic equations
Advances in Difference Equations volume 2017, Article number: 205 (2017)
Abstract
This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences.
1 Introduction
The oscillation and distributions of zeros of solutions of first order delay differential and difference equations are studied widely in the literature; see [1–18] and the references therein. However, there are only a few papers considering the distribution of zeros of solutions of first order delay and advanced dynamic equations on time scales (see [8, 19]). In [12], Zhou considered the first order delay differential equation
where
and established lower and upper bounds for the quotient \(x ( t-\tau ) /x ( t ) \). In particular the author proved that \(x ( t-\tau ) /x ( t ) \geq f_{n} ( \rho ) \) and \(x ( t-\tau ) /x ( t ) < g_{m} ( \rho ) \), where the sequences \(f_{n} ( \rho ) \) and \(g_{n} ( \rho ) \) are defined by
and using these sequences the author studied the distribution of zeros of solutions of (1.1). In [13], Zhang and Zhou considered the first order delay differential equation
and studied the distribution of zeros of solutions using the two sequences \(f_{n} ( \rho ) \) and \(g_{m} ( \rho ) \) where
and
Zhang and Lian in [19] initiated the study of the distribution of zeros of dynamic equations on time scales and in particular, they considered the first order delay dynamic equation
on a time scale \(\mathbb{T}\), where \(p\in\mathbf{C}_{\mathrm{rd}} ( \mathbb{T},\mathbb{R}^{+} ) \) is a non-negative rd-continuous function, \(\tau\in\mathbf{C}_{\mathrm{rd}} ( \mathbb{T},\mathbb{T} ) \) is strictly increasing, \(\tau ( t ) < t\) for \(t\in\mathbb{T}\) and \(\lim_{t\rightarrow\infty}\tau ( t ) =\infty\). In [19] the authors established lower and the upper bounds for the quotient \(x ( \tau ( t ) ) /x ( t ) \) using the sequences \(f_{n}\) and \(g_{m}\) where
and
and where \(M<(1-\rho)/2\) and \(0\leq\rho<1\) satisfies the condition
where \(E= \{ \lambda:\lambda>0, 1-\lambda p ( t ) \mu ( t ) >0 \} \); \(\zeta_{\mu ( s ) }\) and \(\mu ( s ) \) will be defined later.
Motivated by these papers, we study the distribution of zeros of oscillatory solutions of the delay dynamic equation (1.5) on a time scale \(\mathbb{T}\) by considering new sequences \(f_{n}\) and \(g_{m}\). In the next section, we present some basic ideas on time scales. In Section 3, we establish lower and upper bounds for \(x ( \tau ( t ) ) /x ( t )\) and in Section 4, we study the distribution of zeros of solutions of (1.5).
2 Some preliminaries and lemmas
In this section, we present some preliminaries; see [20, 21]. A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers \(\mathbb{R}\). The forward and backward jump operators are defined by
with \(\inf \emptyset=\sup \mathbb{T}\) and \(\sup \emptyset =\inf \mathbb{T}\). The graininess function μ on a time scale \(\mathbb{T}\) is defined by \(\mu ( t ) :=\sigma(t)-t\). For a function \(f:\mathbb{T\rightarrow R}\) the (delta) derivative is defined by
if f is continuous at t and t is right-scattered. If t is not right-scattered then the derivative is defined by
provided this limit exists. A function f is said to be Δ-differentiable if its Δ-derivative exists. A useful formula is \(f^{\sigma}=f ( \sigma ( t ) ) =f ( t ) +\mu ( t ) f^{\Delta} ( t ) \). We will make use of the following product and quotient rules for the derivative of the product fg and the quotient \(f/g\) (where \(gg^{\sigma}\neq0\), and here \(g^{\sigma }=g\circ\sigma\)) of two Δ-differentiable functions f and g:
Let \(f:\mathbb{R\rightarrow R}\) be continuously differentiable and suppose \(g:\mathbb{T\rightarrow R}\) is delta differentiable. Then \(f\circ g:\mathbb{T\rightarrow R}\) is delta differentiable and the chain rule,
holds. A special case of (2.1) is
For \(s, t\in\mathbb{T}\), a function \(F:\mathbb{T\rightarrow R}\) is called an antiderivative of \(f :\mathbb{T\rightarrow R}\) provided \(F^{\Delta }=f(t)\) holds for all \(t\in\mathbb{T}\). In this case we define the integral of f by \(\int_{s}^{t}f(\tau)\Delta\tau=F(t)-F(s)\). For \(a, b\in \mathbb{T}\), and a Δ-differentiable function f, the Cauchy integral of \(f^{\Delta}\) is defined by \(\int_{a}^{b}f^{\Delta}(\tau )\Delta\tau=f(b)-f(a)\). The integration by parts formula reads
and infinite integrals are defined as
A function p: \(\mathbb{T\rightarrow R}\) is called regressive if \(1+\mu ( t ) p ( t ) \neq0\) for \(t\in\mathbb{T}\). A function \(p:\mathbb{T\rightarrow R}\) is called positively regressive (we write \(p\in\mathcal{R}^{+}\)) if it is rd-continuous function and satisfies \(1+\mu ( t ) p ( t ) >0\) for all \(t\in \mathbb{T}\). Hilger in [20] showed that for \(p ( t ) \) rd-continuous and regressive, the solution of the initial value problem
is given by the generalized exponential function \(e_{p} ( t,t_{0} ) \), which is defined by
where \(t_{0}\), \(t\in\mathbb{T}\), and the cylinder transformation \(\zeta _{h} ( z ) \) is defined by
where \(z\in\mathbb{R}\) and \(h\in\mathbb{R}^{+}\).
The next lemma can be found in [11].
Lemma 2.1
Assume \(t_{0}\), \(t\in\mathbb{T}\).
(i) For a non-negative φ with \(-\varphi\in\mathcal{R}^{+}\), we have the following inequality:
(ii) If φ is rd-continuous and non-negative, then
Lemma 2.2
Assume that \(\mathbb{T}\) is a time scale with \(t_{0}\in \mathbb{T} \). If \(f ( t ) >0\) on \([ t_{0},\infty ) _{\mathbb{T}}\), then
Proof
Fix t. We consider two cases: (i) \(f^{\Delta} ( t ) \leq0\) and (ii) \(f^{\Delta} ( t ) \geq0\).
In the first case, we see that
Now recall \(f ( \sigma ( t ) ) =f ( t ) +\mu ( t ) f^{\Delta} ( t )\) so \(h\mu ( t ) f^{\Delta} ( t ) +f ( t )=h f ( \sigma ( t ) )+(1-h) f ( t )>0\) and as a result
Apply the chain rule (2.1), and we get (note \(f^{\Delta } ( t ) \leq0\))
In the second case, we see that
Applying the chain rule (2.1), we get
Thus, we deduce in both cases that
The proof is complete. □
Lemma 2.3
Assume that \(\mathbb{T}\) is a time scale with \(t_{0}\in \mathbb{T} \). If \(f(t)>0\) and \(f^{\Delta} ( t ) \geq0\) for \(t\in [ t_{0},\infty ) _{\mathbb{T}}\), then for \(\alpha>0\)
Proof
Since \(f ( t ) >0\) and \(f^{\Delta} ( t ) \geq0\) for \(t\in [ t_{0},\infty ) _{\mathbb{T}}\), we have for \(h\in ( 0,1 ) \)
Applying the chain rule (2.1) and using (2.4), we see that
The proof is complete. □
3 Lower and upper bounds for \(x ( \tau (t ) ) /x ( t ) \)
In this section, we establish lower and upper bounds for \(x ( \tau ( t ) ) /x ( t ) \) where \(x(t)\) is a solution of equation (1.5). We use the notation \(\tau ^{0} ( t ) =t\) and inductively define the iterates of \(\tau ^{-i} ( t ) \) by
where \(\tau^{-1} ( t ) \) is the inverse function of \(\tau ( t )\). From the definition it is clear that
To find the lower bound for \(x ( \tau ( t ) ) /x ( t ) \) we define for \(0<\rho<1\) a sequence \(f_{n} ( \rho ) \) by
We note some properties of \(f_{n} ( \rho )\) for the reader’s interest (see [9] or use an elementary argument using \(\frac{x}{x+1-e^{(1-\rho)x} }\)). For \(0<1-\rho\leq1/e\), we have
so there exists a function \(f ( \rho ) \) such
where \(f ( \rho ) \) satisfies
If \(( 1-\rho ) >1/e\), then either \(f_{n} ( \rho ) \) is nondecreasing and \(\lim_{n\rightarrow\infty}f_{n} ( \rho ) =+\infty\) or \(f_{n} ( \rho ) \) is negative or \(f_{n} ( \rho ) \) is ∞ after a finite numbers of terms.
Theorem 3.1
Assume that \(\mathbb{T}\) is a time scale and \(t^{\prime}\), \(t_{0}\), \(t_{1} \in\mathbb{T}\), \(t_{0}\geq t^{\prime}\), \(t_{1}\geq\tau^{-3} ( t_{0} ) \), \(x(t)\) is a solution of (1.5) on \([t^{\prime},\infty)_{\mathbb{T}}\), \(x(t)\) is positive on \([ t_{0},t_{1} ] _{\mathbb{T}}\) and there exists \(\rho\in(0,1)\) with \(\infty>f_{n} ( \rho )> 0 \) for \(n\in\{2,3,\ldots\}\) and
here \(E= \{ \lambda:\lambda >0, 1-\lambda p ( t ) \mu ( t ) >0\textit{ for }t\in [ \tau^{-2} ( t_{0} ) ,t_{1} ] _{\mathbb{T}} \}\). Then for \(n\geq0\) when \(\tau^{- ( 2+n )} ( t_{0} ) \leq t_{1}\) we have
where \(f_{n} ( \rho ) \) is defined in (3.1).
Proof
From (1.5), we see that
so since \(x ( t ) \) is nonincreasing on \([ \tau ^{-1} ( t_{0} ) , t_{1} ] _{\mathbb{T}}\) we have
Note (3.5) and the fact that x is positive on \([ t_{0},t_{1} ] _{\mathbb{T}}\), so for \(t\in [ \tau ^{-2} ( t_{0} ) , t_{1} ] _{\mathbb{T}}\) we have (note \(x ( \sigma ( t ) )>0\) since \(\sigma(t) \geq t \geq\tau^{-2} ( t_{0} ) >t_{0}\))
Hence \(1-\mu ( t ) p ( t ) >0\) for \(t\in [ \tau^{-2} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\) so \(-p\in\mathcal{R}^{+}\) on the interval \([ \tau ^{-2} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\). Using Lemma 2.1 (with the time scale \([ \tau^{-2} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\)) and (3.3), we have for \(t\in [ \tau^{-3} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\) (note \(\tau(t) \in [ \tau^{-2} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\))
Integrating (1.5) from \(\tau ( t ) \) to t, we get
and hence, for \(t\in [ \tau^{-3} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\), we get
so
When \(\tau^{-4} ( t_{0} ) \leq t_{1}\), note, for \(t\in [ \tau^{-4} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\) and \(\tau(t) \leq s\leq t\), that
so from Lemma 2.2 we have
which implies that
and so using (3.5), we have (note \(\xi\in [ \tau ^{-2} ( t_{0} ) ,t_{1} ] _{ \mathbb{T}}\) since \(\tau(t) \leq s \leq t\) and \(t\in [ \tau^{-4} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\))
we write \(f_{0} ( \rho )\) (which is of course 1 here) to indicate the general procedure. Now applying Lemma 2.3 and using (3.6), (3.7) and (3.10), we get (here \(t\in [ \tau^{-4} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\))
Thus, for \(t\in [ \tau^{-4} ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\), we get
Repeating the above procedure, when \(\tau^{- ( 2+n ) } ( t_{0} ) \leq t_{1}\) we get for \(t\in [ \tau ^{- ( 2+n ) } ( t_{0} ) ,t_{1} ] _{\mathbb{T}}\)
The proof is complete. □
Remark 3.1
From the proof of Theorem 3.1 notice in the statement of Theorem 3.1 we could replace \(\infty>f_{n} ( \rho )> 0 \) for \(n\in\{2,3, \ldots\}\) with \(\infty>f_{n} ( \rho )> 0 \) for \(n\in\{2,3,\ldots, N-2\}\) if \(\tau^{- ( 2+N ) } ( t_{0} )< t_{1} < \tau^{- ( 3+N ) } ( t_{0} )\) or \(\infty>f_{n} ( \rho )> 0 \) for \(n\in \{2,3,\ldots, N-3\}\) if \(\tau^{- ( 2+N ) } ( t_{0} )=t_{1} < \tau^{- ( 3+N ) } ( t_{0} )\).
To establish the upper bound for \(x ( \tau ( t ) ) /x ( t ) \), we define a sequence \(g_{m} ( \rho ) \) by
where \(0\leq\rho<1\), \(m=1,2,3,\ldots\) , and \(0\leq M<(1-\rho)/2\).
We note some properties of \(g_{m} ( \rho )\) for the reader’s interest. Note \(g_{m+1} ( \rho ) < g_{m} ( \rho )\), for \(m=1,2,3,\ldots\) , and trivially
More generally when \(0<1-\rho\leq1/e\) using an induction argument (i.e. assuming \(g_{m} ( \rho )>\frac{\rho }{ ( 1-\rho ) ^{2}-2M ( 1-\rho ) }\)) then
thus \(g_{k} ( \rho )>\frac{\rho}{ ( 1-\rho ) ^{2}-2M ( 1-\rho ) }\) where \(k=1,3,\ldots\) . Then there exists a function \(g ( \rho ) \) with
for \(0<1-\rho\leq1/e\) (note \(2 ( 2M-1 ) -4 ( M-1 ) \rho-\rho^{2}>0\) if \(0<1-\rho\leq1/e\)).
Theorem 3.2
Assume that \(\mathbb{T}\) is a time scale and \(t^{\prime }, t_{0} \in\mathbb{T}\), \(t_{0}\geq t^{\prime}\), \(x(t)\) is a solution of (1.5) on \([t^{\prime},\infty)_{\mathbb{T}}\), there exists a positive integer \(N \geq4\) such that \(x ( t ) \) is positive on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\) and there exists \(\rho\in(0,1)\) with \(g_{m} ( \rho )> 0 \) for \(m\in\{2,3,\ldots, N-3\}\) and
where \(E= \{ \lambda:\lambda >0,1-\lambda p ( t ) \mu ( t ) >0\textit{ for }t\in [ \tau^{-2} ( t_{0} ) , \tau^{-N} ( t_{0} ) ] _{\mathbb{T}} \}\) and
Then for \(m\in\{1,\ldots,N-3\}\) we have
where \(g_{m} ( \rho ) \) is defined in (3.11).
Proof
From (1.5), we see that
and as in Theorem 3.1 notice \(1-\mu ( t ) p ( t ) >0\) for \(t\in [ \tau^{-2} ( t_{0} ) , \tau ^{-N} ( t_{0} ) ] _{\mathbb{T}}\) so \(-p\in \mathcal{R}^{+}\) on the interval \([ \tau^{-2} ( t_{0} ) , \tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\). From Lemma 2.1 (with the time scale \([ \tau^{-2} ( t_{0} ) , \tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\)) and (3.12), we have for \(t\in [ \tau^{-3} ( t_{0} ) , \tau^{-(N-1)} ( t_{0} ) ]_{\mathbb{T}}\) (note \(\tau^{-1}(t) \leq\tau^{-N} ( t_{0} )\))
Let \(t\in [ \tau^{-3} ( t_{0} ) , \tau ^{-(N-1)} ( t_{0} ) ] _{\mathbb{T}}\) and consider
Note \(G: [ t,\tau^{-1} ( t ) ] \rightarrow \mathbb{R}\) is nondecreasing, \(G ( t ) =-1+\rho<0\), and
If \(G ( \tau^{-1} ( t ) ) =0\), then
whereas if \(G ( \tau^{-1} ( t ) ) >0\) then \(G ( t ) <0<G ( \tau^{-1} ( t ) )\).
In either case (from the intermediate value theorem [20]) there exists \(t^{\ast}\in [ t,\tau^{-1} ( t ) ] _{ \mathbb{T}}\) with \(\sigma ( t^{\ast} ) \in [ t,\tau ^{-1} ( t ) ] _{ \mathbb{T}}\) such that \(G(t^{\ast }) G(\sigma ( t^{\ast} )) \leq0\) and so
Integrating both sides of (1.5) from t to \(\sigma ( t^{\ast} ) \), for \(t\in [ \tau^{-3} ( t_{0} ),\tau^{- ( N-1 ) } ( t_{0} ) ] _{\mathbb{T}}\), we have
Fix \(t\in [ \tau^{-3} ( t_{0} ) , \tau ^{-(N-1)} ( t_{0} ) ] _{\mathbb{T}}\). Let \(s\in \mathbb{T}\) be such that \(t\leq s\leq\sigma ( t^{\ast} ) \leq\tau^{-1} ( t )\) (here \(t^{\ast}\) is as described above, and note \(\tau ( t ) \leq\tau ( s ) \leq t\)) and integrating (1.5) from \(\tau ( s ) \) to t yields
and this together with x being nonincreasing on \([ \tau ^{-1} ( t_{0} ),\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\) and (3.14) will give
so from (3.15), (3.16) and (3.17), we obtain
Let \(F ( s )=\int_{t}^{s}p ( u ) \Delta u\), and note
Hence,
and so we obtain
Note \(\sigma ( t^{\ast} ) \in [ t,\tau^{-1} ( t ) ] _{\mathbb{T}}\), \(t\in [ \tau^{-3} ( t_{0} ),\tau^{- ( N-1 ) } ( t_{0} ) ] _{\mathbb{T}}\), and x is positive on \([ t_{0},\tau ^{-N} ( t_{0} ) ] _{\mathbb{T}}\) (so \(x ( \sigma ( t^{\ast} ) ) >0\)). Thus from (3.18) and (3.20), we obtain
and so we have
Fix \(t\in [ \tau^{-3} ( t_{0} ),\tau^{- ( N-2 ) } ( t_{0} ) ] _{\mathbb{T}}\) and with \(t^{\ast}\) as described above we have \(t\leq\sigma ( t^{\ast } ) \leq\tau^{-1} ( t ) \leq\tau^{- ( N-1 ) } ( t_{0} ) \), so from (3.22) we have
and since x is nonincreasing on \([ \tau^{-1} ( t_{0} ),\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\) and \(\tau ( \sigma ( t^{\ast} ) ) \leq t\leq \tau^{- ( N-1 ) } ( t_{0} ) \) we have
Substituting (3.23) into (3.21), we obtain for \(t\in [ \tau^{-3} ( t_{0} ),\tau^{- ( N-2 ) } ( t_{0} ) ] _{\mathbb{T}}\) that
and so we have
Repeating the above procedure, we obtain for \(t\in [ \tau ^{-3} ( t_{0} ),\tau ^{- ( N-m ) } ( t_{0} ) ] _{\mathbb{T}}\)
The proof is complete. □
4 Distributions of zeros of solutions
In this section, we study the distribution of zeros of solutions of (1.5) using the lower and upper bounds for \(x ( \tau ( t ) ) /x ( t ) \) in Section 3.
Theorem 4.1
Assume that \(\mathbb{T}\) is a time scale and \(t^{\prime }, t_{0}\in\mathbb{T}\), \(t_{0}\geq t^{\prime}\), \(x(t)\) is a solution of (1.5) on \([t^{\prime},\infty)_{\mathbb{T}}\), and there exist \(\rho\in(0,1)\) and \(n_{0}, m_{0} \in \{1,2,\ldots\}\) with \(f_{n_{0}} (\rho ) \geq g_{m_{0}} ( \rho )\), and with
assume \(\infty>f_{k} ( \rho )> 0\), \(g_{k} ( \rho )> 0\) for \(n\in\{2,3,\ldots, N-3\}\) and
where \(E= \{ \lambda:\lambda >0, 1-\lambda p ( t ) \mu ( t ) >0\textit{ for }t\in [ \tau^{-2} ( t_{0} ) , \tau ^{-N} ( t_{0} ) ] _{\mathbb{T}} \}\) and
Then every solution of (1.5) cannot be totally positive or totally negative on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\).
Proof
Note
Without loss of generality assume x is positive on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\). From Theorem 3.1 we have
and from Theorem 3.2 we have (note \(m^{\star}=N-(2+n^{\star}) \leq N-3\))
Note since \(N=2+n^{\star}+m^{\star}\) we have (take \(t=\tau^{- ( N-m^{\star} ) } ( t_{0} )\))
which contradicts (4.1). The proof is complete. □
Theorem 4.2
Assume that \(\mathbb{T}\) is a time scale and \(t^{\prime }, t_{0}\in\mathbb{T}\), \(t_{0}\geq t^{\prime}\), \(x(t)\) is a solution of (1.5) on \([t^{\prime},\infty)_{\mathbb{T}}\), and there exist \(\rho\in(0,1)\) and a positive integer \(N\geq4\) and \(m_{0} \in\{1,2,\ldots,N-3\}\) with
and with
assume \(\infty>f_{k} ( \rho )> 0\), \(g_{k} ( \rho )> 0\) for \(n\in\{2,3,\ldots, N-3\}\) and
where \(E= \{ \lambda:\lambda >0, 1-\lambda p ( t ) \mu ( t ) >0\textit{ for }t\in [ \tau^{-2} ( t_{0} ) , \tau ^{-N} ( t_{0} ) ] _{\mathbb{T}} \}\) and
Then every solution of (1.5) cannot be totally positive or totally negative on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\).
Proof
Note
Without loss of generality assume x is positive on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\). From Theorem 3.2, we have
so in particular
Integrating (1.5) from \(\tau ( t_{m^{\star}} ) \) to \(t_{m^{\star}}\), we obtain
and this together with (4.3) gives
which contradicts (4.2). The proof is complete. □
Theorem 4.3
Assume that \(\mathbb{T}\) is a time scale and \(t^{\prime }, t_{0}\in\mathbb{T}\), \(t_{0}\geq t^{\prime}\), \(x(t)\) is a solution of (1.5) on \([t^{\prime},\infty)_{\mathbb{T}}\), and there exist \(\rho\in(0,1)\), a constant L and \(n_{0}, m_{0} \in\{1,2,\ldots\}\) with
and with
assume \(\infty>f_{k} ( \rho )> 0\), \(g_{k} ( \rho )> 0\) for \(n\in\{2,3,\ldots, N-3\}\) and
where \(E= \{ \lambda:\lambda >0, 1-\lambda p ( t ) \mu ( t ) >0\textit{ for } t\in [ \tau^{-2} ( t_{0} ) , \tau ^{-N} ( t_{0} ) ] _{\mathbb{T}} \}\) and
Suppose \(f_{n^{\star}-1} ( \rho ) \geq1\), \(f_{n^{\ast }} ( \rho )> f_{n^{\ast}-1} ( \rho )\) and for \(t^{\ast}\in [ \tau ( t_{1} ) ,t_{1} ] _{\mathbb{T}}\) (here \(t_{1}=\tau^{- ( N-m^{\star} ) } ( t_{0} )\)) that
Then every solution of (1.5) cannot be totally positive or totally negative on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\).
Proof
Note
Without loss of generality assume x is positive on \([ t_{0},\tau^{-N} ( t_{0} ) ] _{\mathbb{T}}\). From Theorem 3.1, we have
and from Theorem 3.2, we have
so in particular (with \(t_{1}=\tau^{- ( N-m^{\star} ) } ( t_{0} )=\tau^{- ( 2+n^{\star} ) } ( t_{0} ) \)) we have
From (4.6) and \(f_{n^{\ast}} ( \rho )> f_{n^{\ast }-1} ( \rho )\) we have
Now since x is nonincreasing on \([ \tau^{-1}( t_{0}),\tau ^{-N } ( t_{0} ) ] _{\mathbb{T}}\) and \(f_{n^{\star}-1} ( \rho ) \geq1\) (and trivially note \(\frac{x ( \tau ( t_{1} ) ) }{ x (\tau ( t_{1} ) ) }=1\)) there exists a \(t^{\ast}\in [ \tau ( t_{1} ) ,t_{1} ] _{\mathbb{T}}\) with
Integrating (1.5) from \(\sigma ( t^{\ast} ) \) to \(t_{1}\), we obtain
which implies
From (4.8), (4.9) and (4.10), we obtain
Divide (1.5) by x and integrate from \(\tau ( t_{1} ) \) to \(t^{\ast}\), and we get
which implies
From (4.9), (4.12) and Lemma 2.2, we obtain
and from (4.5), (4.11) and (4.13) we have
which contradicts (4.4). The proof is complete. □
Remark 4.1
When \(\mathbb{T=R}\) equation (1.5) is the delay differential equation
Theorem 3.1 and Theorem 3.2 are related to the results in [9], Lemma 2.1 and Lemma 2.2, and Theorem 4.3 is motivated from results in [13], Theorem 3.
References
Agarwal, RP, Bohner, M: An oscillation criterion for first order delay dynamic equations. Funct. Differ. Equ. 16, 11-17 (2009)
Bohner, M: Some oscillation criteria for first order delay dynamic equations. Far East J. Appl. Math. 18, 289-304 (2005)
Bohner, M, Karpuz, B, Ocalan, O: Iterated oscillation criteria for delay dynamic equations of first order. Adv. Differ. Equ. 2008, Article ID 458687 (2008)
El-Morshedy, HA: On the distribution of zeros of solutions of first order delay differential equations. Nonlinear Anal. 74, 3353-3362 (2011)
Karpuz, B, Ocalan, O: New oscillation tests and some refinements for first-order delay dynamic equations. Turk. J. Math., 1-14 (2015)
Liang, FX: The distribution of zeros of solutions of first-order delay differential equations. J. Math. Anal. Appl. 186, 383-392 (1994)
Sahiner, Y, Stavroulakis, IP: Oscillations of first order delay dynamic equations. Dyn. Syst. Appl. 15, 645-655 (2006)
Wu, H: The distribution of zeros of solutions of advanced dynamic equations on time scales. Bull. Malays. Math. Soc. 38, 1-17 (2015)
Wu, HW, Xu, YT: The distribution of zeros of solutions of neutral differential equations. Appl. Math. Comput. 156, 665-677 (2004)
Xianhua, T, Jianshe, Y: Distribution of zeros of solutions of first order delay differential equations. Appl. Math. J. Chin. Univ. Ser. B 14, 375-380 (1999)
Zhang, BG, Xinghua, D: Oscillation of delay differential equations on time scales. Math. Comput. Model. 36, 1307-1318 (2002)
Zhou, Y: The distribution of zeros of solutions of first order functional differential equations. Bull. Aust. Math. Soc. 59, 305-314 (1999)
Zhang, BG, Zhou, Y: The distribution of zeros of solutions of differential equations with a variable delay. J. Math. Anal. Appl. 256, 216-228 (2001)
Zhou, Y, Zhang, BG: An estimate of numbers of terms of semicycles of delay difference equations. Comput. Math. Appl. 41, 571-578 (2001)
Wu, HW, Cheng, SS, Wang, QR: The distribution of zeros of solutions of functional differential equations. Appl. Math. Comput. 193, 154-161 (2007)
Tang, XH, Yu, JS: The maximum existence interval of positive solutions of first order delay differential inequalities with applications. Math. Pract. Theory 30, 447-452 (2000)
Zhang, BG, Zhou, Y: The semicycles of solutions of delay difference equations. Comput. Math. Appl. 38, 31-38 (1999)
Yu, JS, Zhang, BG, Wang, ZC: Oscillation of delay difference equations. Appl. Anal. 53, 117-124 (1994)
Zhang, BG, Lian, F: The distribution of generalized zeros of solutions of delay differential equations on time scales. J. Differ. Equ. Appl. 10, 759-771 (2004)
Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 18, 18-56 (1990)
Bohner, M, Peterson, A: Dynamic Equations on Time Scales - An Introduction with Applications. Birkhäuser, Boston (2001)
Acknowledgements
The authors are grateful to the anonymous referees and the editor for their careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have contributed equally to this manuscript. They read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
O’Regan, D., Saker, S., Elshenhab, A. et al. Distributions of zeros of solutions to first order delay dynamic equations. Adv Differ Equ 2017, 205 (2017). https://doi.org/10.1186/s13662-017-1259-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-017-1259-2