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Nonoscillatory solutions for firstorder neutral dynamic equations with continuously distributed delay on time scales
Advances in Difference Equations volume 2019, Article number: 65 (2019)
Abstract
In this paper, we establish the existence of nonoscillatory solutions to the neutral dynamic equation
on a time scale \(\mathbb{T}\). Some examples are given to illustrate the main results.
1 Introduction
In this paper, we will consider the existence of nonoscillatory solutions to the firstorder neutral dynamic equation of the form
on a time scale \(\mathbb{T}\), where \(a, b, c, d\in\mathbb{T}\) with \(a\leq b\) and \(c\leq d\) and \(p(t,\eta), g(t,\eta), \omega(t,\eta), h(t,\eta) \in C(\mathbb{T},\mathbb{T})\). Throughout this paper, we assume that \(\mathbb{T}=[t_{0},\infty)_{\mathbb{T}}:=\{t\in\mathbb{T}: t\geq t_{0}\}\).
In recent years, there has been much research activity concerning the nonoscillation of solutions of various equations on time scales, and we refer the reader to [1,2,3,4]. Mathsen, Wang and Wu [5] established some sufficient conditions for the existence of positive solutions of the delay equation
Zhu and Wang [6] established the existence of nonoscillatory solutions to the neutral equation
Gao, Chen and Shi [7] established the existence of oscillatory solutions to the neutral equation
Up to now there have been few studies on equations with continuously distributed delay on time scales. For some related works, the reader is referred to the papers [8,9,10]. In this paper, we obtain some new sufficient conditions for the existence of nonoscillatory solutions of equation (1.1). The method used in this paper is motivated by the work of Zhu and Wang in [6]. And our work enriches the research with the case of equations with continuously distributed delay on time scales.
This paper is organized as follows. In Sect. 2, we recall some basic concepts and some preliminaries briefly. In Sect. 3, we focus on the study of sufficient conditions for the existence of nonoscillatory solutions to the firstorder neutral dynamic equation (1.1). In Sect. 4, some examples are presented to illustrate our main results for equation (1.1).
2 Preliminaries
For convenience, we recall some concepts related to time scales. More details can be found in [11, 12].
Definition 2.1
(see [12, Definition 1.10])
Assume \(f:\mathbb {T}\rightarrow\mathbb{R}\) is a function and \(t\in\mathbb{T^{\kappa}}\). Then we define \(f^{\Delta}(t)\) to be the number (provided it exists) with the property that given any \(\varepsilon> 0\), there is a neighborhood U of t (i.e., \(U=(t\delta,t+\delta )\cap\mathbb{T}\) for some \(\delta>0\)) such that
We call \(f^{\Delta}(t)\) the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable (or, in short, differentiable) on \(\mathbb {T^{\kappa}}\) provided \(f^{\Delta}(t)\) exists for all \(t\in\mathbb {T^{\kappa}}\). The function \(f^{\Delta}:\mathbb{T}^{\kappa}\to\mathbb{R}\) is then called the (delta) derivative of f on \(\mathbb {T}^{\kappa}\).
Using the definition of delta derivative, it’s easy to see the following results (see [12, Theorem 1.16]). If f is continuous at \(t\in\mathbb{T}\) and t is rightscattered, then f is differentiable at t with
Moreover, if t is rightdense then f is differentiable at t iff the limit
exists as a finite number. In this case
In the sequel, as in [6], we denote by \(C([t_{0},\infty)_{\mathbb{T}},\mathbb{R})\) all continuous functions mapping \([t_{0},\infty)_{\mathbb{T}}\) into \(\mathbb{R}\), and let
Endowing \(BC[t_{0},\infty)_{\mathbb{T}}\) with the norm \(\ x\=\sup_{t\in [t_{0},\infty)_{\mathbb{T}}}x(t)\) makes \((BC[t_{0},\infty)_{\mathbb{T}}, \ \cdot\)\) a Banach space. To prove our main results, we need the following lemmas.
Lemma 2.1
(see [6, Lemma 4])
Suppose that \(X\subseteq BC[t_{0},\infty)_{\mathbb{T}} \) is bounded and uniformly Cauchy. Further, suppose that X is equicontinuous on \([t_{0},t_{1}]_{\mathbb{T}}\) for any \(t_{1}\in[t_{0},\infty)_{\mathbb {T}}\). Then X is relatively compact.
Lemma 2.2
(see [13, Krasnoselskii’s Fixed Point Theorem])
Suppose that Ω is a Banach space and X is a bounded, convex and closed subset of Ω. Suppose further that there exist two operators U, \(S:X\rightarrow\varOmega\) such that

(i)
\(Ux+Sy\in X\) for all x, \(y\in X\);

(ii)
U is a contraction mapping;

(iii)
S is completely continuous.
Then \(U+S\) has a fixed point in X.
3 Main results
Theorem 3.1
Assume that \(p(t,\eta)>0\), \(\omega(s,\nu )>0\) for any \(t\in\mathbb{T}, \eta\in[a,b]_{\mathbb{T}}\) and \(\nu\in [c,d]_{\mathbb{T}}\). And suppose further that there exists a constant α such that
and
Then (1.1) has a bounded nonoscillatory solution.
Proof
We define the Banach space \(BC[t_{0},\infty)_{\mathbb {T}}\) as in (2.1) and let
It is easy to check that X is a bounded, convex and closed subset of \(BC[t_{0},\infty)_{\mathbb{T}}\). We will apply the Krasnoselskii’s Fixed Point Theorem to obtain the desired result. To this end, first of all, we define two operators U, \(S:X\rightarrow BC[t_{0},\infty)_{\mathbb{T}}\) as follows:
and
And then, we will check that U and S satisfy the conditions in Lemma 2.2.
(1) For \(x, y\in X\), from (3.1) and (3.2), we have
So \(Ux+Sy\in X \) for any x, \(y\in X\).
(2) For \(x, y\in X\) and \(t\in\mathbb{T}\), we have
which implies that U is a contraction mapping on X.
(3) We now show that S is completely continuous. Clearly, S maps X into itself.
Let \(x_{n},x\in X\) and \(\x_{n}x\\rightarrow0\) as \(n\rightarrow \infty\). Then, from (3.2), we get that
which implies that S is continuous on X.
For any \(\varepsilon>0\), by (3.2), we can choice \(t_{1} \in[t_{0},\infty)_{\mathbb{T}}\) such that
Then, for any \(x \in X\) and u, \(v \in[t_{1},\infty)_{\mathbb{T}}\), we have
Hence, SX is uniformly Cauchy.
For any \(t_{2}\in[t_{0},\infty)_{\mathbb{T}}\), by (3.2), for any \(s\in[t_{0},\infty)_{\mathbb{T}}\), we have that \(\int _{c}^{d}\omega(s,\nu)\Delta\nu\leq N\) for some positive constant N. And so, for any \(\varepsilon>0\), take \(\delta=\varepsilon/N\), then for any \(x\in X\), when \(u,v\in[t_{0},t_{2}]\) with \(uv<\delta\), we have
Thus SX is equicontinuous on \([t_{0},t_{2}]_{\mathbb{T}}\).
By Lemma 2.1, SX is relatively compact. And so S is a completely continuous mapping.
Finally, by Lemma 2.2, there exists \(x \in X\) such that \((U+S)x=x\), which implies that \(x(t)\) is a bounded nonoccillatory solution of (1.1). The proof is complete. □
Theorem 3.2
Assume that \(p(t,\eta)<0\), \(\omega (s,\nu)>0\) for any \(t\in\mathbb{T}\), \(\eta\in[a,b]_{\mathbb{T}}\) and \(\nu\in [c,d]_{\mathbb{T}}\). And suppose further that there exists a constant α such that (3.2) and
hold. Then (1.1) has a bounded nonoscillatory solution.
Proof
We define the Banach space \(BC[t_{0},\infty)_{\mathbb {T}}\) as in (2.1) and let
It is easy to check that X is a bounded, convex and closed subset of \(BC[t_{0},\infty)_{\mathbb{T}}\).
Now we define two operators U and \(S:X\rightarrow BC[t_{0},\infty )_{\mathbb{T}}\) as follows:
and
Similar to the proof of Theorem 3.1, we can show that U and S satisfy the conditions in Lemma 2.2, which implies the desired result. The proof is complete. □
Theorem 3.3
Assume that \(p(t,\eta)>0\), \(\omega (s,\nu)<0\) for any \(t\in\mathbb{T}\), \(\eta\in[a,b]_{\mathbb{T}}\) and \(\nu\in [c,d]_{\mathbb{T}}\). And suppose further that there exists a constant α such that (3.1) and
hold. Then (1.1) has a bounded nonoscillatory solution.
Proof
We define the Banach space \(BC[t_{0},\infty)_{\mathbb {T}}\) as in (2.1) and let
It is easy to check that X is a bounded, convex and closed subset of \(BC[t_{0},\infty)_{\mathbb{T}}\).
Now we define two operators U and \(S:X\rightarrow BC[t_{0},\infty )_{\mathbb{T}}\) as follows:
and
Similar to the proof of Theorem 3.1, we can show that U and S satisfy the conditions in Lemma 2.2, which implies the desired result. The proof is complete. □
Theorem 3.4
Assume that \(p(t,\eta)<0\), \(\omega (s,\nu)<0\) for any \(t\in\mathbb{T}\), \(\eta\in[a,b]_{\mathbb{T}}\) and \(\nu\in [c,d]_{\mathbb{T}}\). And suppose further that there exists a constant α such that (3.3) and (3.4) hold, then (1.1) has a bounded nonoscillatory solution.
Proof
We define the Banach space \(BC[t_{0},\infty)_{\mathbb {T}}\) as in (2.1) and let
It is easy to check that X is a bounded, convex and closed subset of \(BC[t_{0},\infty)_{\mathbb{T}}\).
Now we define two operators U and \(S:X\rightarrow BC[t_{0},\infty )_{\mathbb{T}}\) as follows:
and
Similar to the proof of Theorem 3.1, we can show that U and S satisfy the conditions in Lemma 2.2, which implies the desired result. The proof is complete. □
4 Examples
Example 4.1
Let \(q>1\) and \(\mathbb{T}=q^{\mathbb{N}}\). For some fixed natural number k, we consider the following equation:
i.e., \(p(t,\eta)=\frac{1}{(t+2\eta)\eta}\), \(\omega(t,\nu)=\frac {1}{t^{2}\nu(t+4\nu)}\), \(a=q=c\), \(b=q^{k}=d\). Then we have
and
Therefore, by Theorem 3.1 with \(\alpha=\frac{1}{2}\), (4.1) has a bounded nonoscillatory solution.
Example 4.2
Let \(\mathbb{T}=\mathbb{N}^{2}\). Consider the following equation:
i.e., \(p(t,\eta)=\frac{1}{5(t+2\eta)\eta}\), \(\omega(t,\nu)=\frac {3}{10t(1+\sqrt{t})^{2}(t+2\nu)\nu}\), \(a=1=c\), \(b=9=d\). Then we have
and
Therefore, by Theorem 3.2 with \(\alpha=\frac{1}{4}\), (4.2) has a bounded nonoscillatory solution.
Example 4.3
Let \(\mathbb{T}=\mathbb{N}^{\frac{1}{2}}\). Consider the following equation
i.e., \(p(t,\eta)=\frac{\eta}{t+\eta}\), \(\omega(t,\nu)=\frac{\nu }{4t\sqrt{1+t^{2}}(t+\nu)}\), \(a=1=c\), \(b=\sqrt{3}=d\). Then we have
and
Therefore, by Theorem 3.3 with \(\alpha=\frac{1}{2}\), (4.3) has a bounded nonoscillatory solution.
Example 4.4
Fixed \(h>0\) and let \(\mathbb{T}=h\mathbb{N}\). Consider the following equation:
i.e., \(p(t,\eta)=\frac{1}{t+3\eta}\), \(\omega(t,\nu)=\frac {h}{t(t+h)(t+12\nu)}\), \(a=h=c\), \(b=3h=d\). Then we have
and
Therefore, by Theorem 3.4 with \(\alpha=\frac{1}{2}\), (4.4) has a bounded nonoscillatory solution.
5 Conclusion
In this paper, using Krasnoselskii’s Fixed Point Theorem, we obtain some sufficient conditions for the existence of nonoscillatory solutions to the firstorder neutral dynamic equation (1.1) in four cases. And we give some examples to illustrate the main results. A similar method can be used to prove the existence of nonoscillatory solutions for other dynamic equations.
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Funding
This work is partially supported by National Natural Science Foundation of China (No. 11401116, No. 11761011, No. 11461003) and Natural Science Foundation of Guangxi (No. 2014GXNSFBA118003).
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Chen, Z., Lv, J., He, X. et al. Nonoscillatory solutions for firstorder neutral dynamic equations with continuously distributed delay on time scales. Adv Differ Equ 2019, 65 (2019). https://doi.org/10.1186/s1366201920156
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DOI: https://doi.org/10.1186/s1366201920156
MSC
 34K11
 39A10
 39A99
Keywords
 Neutral dynamic equation
 Nonoscillatory solution
 Time scale