Oscillatory Behavior of Three Dimensional α-Fractional Delay Differential Systems

Abstract

In the present work we study the oscillatory behavior of three dimensional $\alpha$-fractional nonlinear delay differential system. We establish some sufficient conditions that will ensure all solutions are either oscillatory or converges to zero, by using the inequality technique and generalized Riccati transformation. The newly derived criterion are also used to establish a new class of systems with delay which are not covered in the existing study of literature. Further, we constructed some suitable illustrations.

1. Introduction

In the literature there are many advanced strategies in the expansion of ordinary and partial differential equations of fractional order and they have been used as excellent sources and tools in order to model many phenomena in the different fields of engineering, science and technology. Further, these tools are also used in fields such as chemical processes, polymer rheology, mathematical biology, industrial robotics, viscoelasticity, and many more, see the monographs [1,2,3,4,5,6,7].

At the end of the nineteenth century, Henry Poincare initiated the method and used the qualitative analysis of nonlinear systems of integer order differential equations. Since then, there has been significant development in the theory of oscillation of integer order differential systems [8,9,10,11,12,13,14,15,16,17,18].

In a study [19], Vreeke et al. applied the differential systems in the application of physics in order to solve the problem of a nuclear reactor which involved two temperature feedback. In the current literature there are many established results in the oscillation theory of classical differential systems (see [20,21,22,23,24]). However, in the nonlinear fractional differential system development is relatively slow due to the occurrence of nonlocal behavior of fractional derivatives that possess weakly singular kernels.

In 2014, Khalil et al. introduced the idea of conformable fractional derivative as a kind of local derivative with no memory (see [25,26,27]). By following the idea of Khalil, an interesting application of the conformable fractional derivative in physics was discussed and the action principle for particles under the frictional forces were formulated, see [28].

The idea of conformable fractional derivatives was generalized by Katugampola, and today it is known as the Katugampola fractional derivative. Nowadays, many researchers have interest in this type of derivative for their useful properties (see [29,30,31]). In this respect, we list the contributions of Spanikova [32], Sadhasivam [33] and Chatzarakis [34] where the oscillation of $\alpha$-fractional nonlinear three dimensional delay differential systems were also studied.

Now we study oscillatory behavior of the following system having the form

$$\begin{array}{cc}\hfill D^{\alpha }\left(u\left(t\right)\right)& =p\left(t\right)g\left(v\left(\sigma \right(t\left)\right)\right),\hfill \\ \hfill D^{\alpha }\left(v\left(t\right)\right)& =-q\left(t\right)h\left(w\left(t\right))\right),\hfill \\ \hfill D^{\alpha }\left(w\left(t\right)\right)& =r\left(t\right)f\left(u\left(\delta \right(t\left)\right)\right),t\ge t_0,\hfill \end{array}$$

(1)

where $0<\alpha \le 1$, $D^{\alpha }$ denotes the $\alpha$-fractional derivative respect to t.

Based on the following assumptions:

$\left(A_1\right)$$p\left(t\right)\in C^{2\alpha }([t_0,\infty ),{\mathbb{R}}^{+})$, $q\left(t\right)\in C^{\alpha }([t_0,\infty ),{\mathbb{R}}^{+})$, $r\left(t\right)\in C([t_0,\infty ),{\mathbb{R}}^{+})$, $p\left(t\right)$, $q\left(t\right)$ and $r\left(t\right)$ are not identically zero on any interval of $[T_0,\infty )$, $T_0\ge t_0$, $r\left(t\right)$ and $q\left(t\right)$ are decreasing and positive;

$\left(A_2\right)$$g\in C^{\alpha }(\mathbb{R},\mathbb{R}),vg\left(v\right)>0,D^{\alpha }g\left(v\right)\ge l^{{}^{\prime }}>0$, $h\in C^{\alpha }(\mathbb{R},\mathbb{R}),wh\left(w\right)>0,D^{\alpha }h\left(w\right)\ge m^{{}^{\prime }}>0$, $f\in C(\mathbb{R},\mathbb{R}),yf\left(y\right)>0$ and $\frac{f\left(y\right)}{y}\ge k>0$ for $y\ne 0$;

$\left(A_3\right)$$\sigma \left(t\right)\le t$ with $D^{\alpha }\sigma \left(t\right)\ge l>0$, $\delta \left(t\right)\le t$ and satisfies ${lim}_{t\to \infty }\sigma \left(t\right)=\infty$, ${lim}_{t\to \infty }\delta \left(t\right)=\infty$;

$\left(A_4\right)$ The case will be considered as

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }{s}^{\alpha -1}\frac{1}{b\left(s\right)}ds=\infty ,{\int }_{t_0}^{\infty }{s}^{\alpha -1}\frac{1}{a\left(s\right)}ds=\infty ,\end{array}$$

where $b\left(t\right)=\frac{1}{q\left(t\right)},a\left(t\right)=\frac{1}{p\left(t\right)}$ and $c\left(t\right)=l^2{l}^{{}^{\prime }}{m}^{{}^{\prime }}r\left(t\right)$, $a\left(t\right),b\left(t\right)$ and $c\left(t\right)$ are positive real-valued continuous functions with $b\left(t\right)t^{1-\alpha }<1$.

The solution implies that, it is a vector-valued function such that $U\left(t\right)=\left(u\right(t),v(t),w(t\left)\right)$ with $T_1=min\left\{\delta \left(t_1\right),\sigma \left(t_1\right)\right\}$ for some $t_1\ge t_0$ which has the property such that $b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)\in C^{\alpha }([T_1,\infty ),\mathbb{R})$ and satisfies the system (1) on $[T_1,\infty )$. Denote by $\mathbb{P}$, the set of all solutions $U\left(t\right)$ of (1) which exist on some half line $[T_1,\infty ),T_1>t_0$. The researchers only focus to the nontrivial solutions of system (1) and satisfy $sup\left\{\left|u\right(\xi \left)\right|+\left|v\right(\xi \left)\right|+\left|w\right(\xi \left)\right|,t\le \xi <\infty \right\}>0$ for any $t\ge T_1$. We make a standing hypothesis that (1) has such a solution.

A proper solution $U\left(t\right)\in \mathbb{P}$ for the system (1) is called oscillatory if all the components are oscillatory, otherwise it is nonoscillatory. Further, the system (1) is said to be oscillatory if all proper solutions oscillate.

The main goal of this paper is to establish some new oscillation criteria for the system (1) by making use of the generalized Riccati transformation and inequality technique. The study is structured as follows. In Section 2, we recall some preliminary concepts relative to the $\alpha$- fractional derivative. In Section 3, some new conditions for the oscillatory behavior of the solutions of system (1) were presented. Illustrative examples are included in the final part of the paper in order to demonstrate the efficiency of new theorems.

2. Preliminaries

We begin this section with the following definition of the operator $D^{\alpha }$.

Definition 1.

[30] Let $y:[0,\infty )\to \mathbb{R}$, then α-fractional derivative of y is defined by

$$\begin{array}{c}\hfill D^{\alpha }\left(y\right)\left(t\right):=\underset{ϵ\to 0}{lim}{\displaystyle \frac{y\left(t{e}^{ϵt^{-\alpha }}\right)-y\left(t\right)}{ϵ}}fort>0\mathrm{and}\alpha \in (0,1].\end{array}$$

(2)

If y is differentiable α-times in some $(0,a)$ with $a>0$, ${\displaystyle \underset{t\to 0^{+}}{lim}{D}^{\alpha }\left(y\right)\left(t\right)}$ exists, then we have

$$\begin{array}{c}\hfill D^{\alpha }\left(y\right)\left(0\right):=\underset{t\to 0^{+}}{lim}{D}^{\alpha }\left(y\right)\left(t\right).\end{array}$$

$\alpha$-fractional derivative satisfies the following properties. [30]

Let $\alpha \in (0,1]$ and g, h be $\alpha$- differentiable for $t>0$. Then

$\left(p_1\right)$$D^{\alpha }\left(t^n\right)=n{t}^{n-\alpha }$ for all $n\in \mathbb{R}$.

$\left(p_2\right)$$D^{\alpha }\left(C\right)=0$ for all constant functions, $g\left(t\right)=C$.

$\left(p_3\right)$$D^{\alpha }\left(gh\right)=g{D}^{\alpha }\left(h\right)+h{D}^{\alpha }\left(g\right)$.

$\left(p_4\right)$$D^{\alpha }\left(\frac{g}{h}\right)=\frac{h{D}^{\alpha }\left(g\right)-g{D}^{\alpha }\left(h\right)}{h^2}$.

$\left(p_5\right)$$D^{\alpha }(g\circ h)\left(t\right)=g^{{}^{\prime }}\left(h\left(t\right)\right)D^{\alpha }h\left(t\right)$, for g is differentiable at $h\left(t\right)$.

$\left(p_6\right)$ If g is differentiable, then $D^{\alpha }\left(g\right)\left(t\right)=t^{1-\alpha }\frac{dg}{dt}\left(t\right)$.

Definition 2.

[30] Let $a\ge 0$, $t\ge a$ and a function y defined on $(a,t]$ with $\alpha \in \mathbb{R}$. Then, α-fractional integral as follows

$$\begin{array}{c}\hfill I_a^{\alpha }\left(y\right)\left(t\right):={\int }_a^t{\displaystyle \frac{y\left(x\right)}{x^{1-\alpha }}}dx\end{array}$$

(3)

provided improper integral exists.

3. Main Results

In this section, the oscillatory behavior of solutions for the system (1) under certain conditions are established. Next we give the following lemmas that will be used in our further discussion.

Lemma 1.

If $U\left(t\right)\in \mathbb{P}$ is a nonoscillatory solution for (1), then the component function $x\left(t\right)$ is always nonoscillatory.

Lemma 2.

Suppose that $\left(A_1\right)$ and $\left(A_4\right)$ holds. Then there exists a $t_1\ge t_0$ such that either

$\left(I\right)$$u\left(t\right)>0,D^{\alpha }u\left(t\right)>0,D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)>0$ for $t\ge t_1$.

or

$\left(II\right)$$u\left(t\right)>0,D^{\alpha }u\left(t\right)<0,D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)>0$ for $t\ge t_1$ holds.

Proof. 

Let $u\left(t\right)$ be an eventually positive solution for (1) on $[t_0,\infty )$. Now, system (1) will be reduced to the following inequality

$$\begin{array}{c}\hfill D^{\alpha }\left(\frac{1}{q\left(t\right)}{D}^{\alpha }\left(\frac{1}{p\left(t\right)}{D}^{\alpha }u\left(t\right)\right)\right)+l^2{l}^{{}^{\prime }}{m}^{{}^{\prime }}r\left(t\right)f\left(u\left(\delta \left(\sigma \left(t\right)\right)\right)\right)\le 0,t\ge t_1,\end{array}$$

(4)

which implies,

$$\begin{array}{c}\hfill D^{\alpha }\left(b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)\right)+c\left(t\right)f\left(u\left(\delta \left(\sigma \left(t\right)\right)\right)\right)\le 0,t\ge t_1.\end{array}$$

(5)

From $\left(5\right)$, we get $D^{\alpha }\left(b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)\right)\le 0$ for $t\ge t_0$. Then $b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)$ is decreasing on $(t_0,\infty )$. Thus the proof completes on using the Lemma 3.2 in [34]. ☐

The following notations are employed in the sequel.

$$\begin{array}{c}\hfill {\left(A_{\alpha }\right)}_{*}:=\underset{t\to \infty }{lim\; inf}t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}^{\alpha }\left(s\right)dsand{\left(B_{\alpha }\right)}_{*}:=\underset{t\to \infty }{lim\; inf}\frac{1}{t}\underset{t_0}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds,\end{array}$$

(6)

where $A_{\alpha }\left(t\right)=\frac{k}{2}\frac{c\left(t\right)}{a\left(t\right)}\frac{\delta \left(\sigma \right(t\left)\right)-T}{t}{\left(\delta \left(\sigma \left(t\right)\right)\right)}^{\alpha }$.

$$\begin{array}{c}\hfill d:=\underset{t\to \infty }{lim\; inf}tw\left(t\right)\mathrm{and}D:=\underset{t\to \infty }{lim\; sup}tw\left(t\right).\end{array}$$

(7)

Theorem 1.

Suppose that $\left(A_1\right)–\left(A_4\right)$ hold. Assume also that

$$\begin{array}{c}\hfill \underset{t_2}{\overset{\infty }{\int }}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)ds=\infty ,\end{array}$$

(8)

there exists a positive function $\rho \in C^{\alpha }([0,\infty );{\mathbb{R}}_{+})$ such that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\underset{t_0}{\overset{t}{\int }}\left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds=\infty .\end{array}$$

(9)

Then every solution of system $\left(1\right)$ is oscillatory.

Proof. 

Suppose that $\left(1\right)$ has a nonoscillatory solution $\left(u\right(t),v(t),w(t\left)\right)$ on $[t_0,\infty )$. From Lemma 1, $u\left(t\right)$ is always nonoscillatory. Without loss of generality, we shall assume that $u\left(t\right)>0$, $u\left(\delta \right(t\left)\right)>0$ and $u\left(\delta \right(\sigma \left(t\right)\left)\right)>0$ for $t\ge T\ge t_0$, since similar arguments can be made for $u\left(t\right)<0$ eventually. Suppose that Case (I) of Lemma 2 holds for $t\ge t_1$. Define

$$\begin{array}{c}\hfill w\left(t\right)=\rho \left(t\right){\displaystyle \frac{b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)}{a\left(t\right)D^{\alpha }u\left(t\right)}},t\ge t_1.\end{array}$$

(10)

Thus $w\left(t\right)>0$, differentiating $\alpha$ times with respect to t, using (5) and $\left(A_2\right)$, we have

$$\begin{array}{c}\hfill D^{\alpha }w\left(t\right)\le \frac{D^{\alpha }\rho \left(t\right)}{\rho \left(t\right)}w\left(t\right)-\frac{k\rho \left(t\right)c\left(t\right)}{a\left(t\right)}\frac{u\left(\delta \right(\sigma \left(t\right)\left)\right)}{D^{\alpha }u\left(t\right)}-\frac{1}{\rho \left(t\right)b\left(t\right)}{w}^2\left(t\right).\end{array}$$

(11)

Now, let $z_1(t,T)=(t-T),z_2(t,T)=\frac{{(t-T)}^2}{2}$ and define $U\left(t\right):=(t-T)t^{1-\alpha }u\left(t\right)-z_2(t,T)D^{\alpha }u\left(t\right)$.

Then $U\left(T\right)=0$ and differentiating the above, we get

$$\begin{array}{cc}\hfill D^{\alpha }U\left(t\right)& =t^{1-\alpha }(t^{1-\alpha }u\left(t\right)+(t-T)(1-\alpha )t^{-\alpha }u\left(t\right)+(t-T)t^{1-\alpha }{u}^{{}^{\prime }}\left(t\right)\hfill \\ \hfill & -z_2^{{}^{\prime }}(t,T)D^{\alpha }u\left(t\right)-z_2(t,T){\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}),\hfill \end{array}$$

which implies

$$\begin{array}{c}\hfill U^{{}^{\prime }}\left(t\right)\ge t^{1-\alpha }u\left(t\right)-z_2(t,T){\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}.\end{array}$$

(12)

By Taylor’s Theorem, we have

$$\begin{array}{c}\hfill {\int }_T^t{s}^{1-\alpha }{u}^{{}^{\prime }}\left(s\right)ds=z_1(t,T)D^{\alpha }u\left(T\right)+{\int }_T^t{z}_1(t,s){\left(D^{\alpha }u\left(s\right)\right)}^{{}^{\prime }}ds,\end{array}$$

since $D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)$ is decreasing, we get

$$\begin{array}{c}\hfill t^{1-\alpha }u\left(t\right)\ge t^{1-\alpha }u\left(T\right)+z_1(t,T)D^{\alpha }u\left(T\right)+{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}{\int }_T^t{z}_1(t,s)ds.\end{array}$$

Thus $U^{{}^{\prime }}\left(t\right)>0$ on $[T,\infty )$. From this we get $U\left(t\right)>0$ on $[T,\infty )$, which implies that

$$\begin{array}{c}\hfill \frac{u\left(t\right)}{D^{\alpha }u\left(t\right)}>\frac{z_2(t,T)}{(t-T)t^{1-\alpha }}=\frac{t-T}{2}{t}^{1-\alpha },t\in [T,\infty ).\end{array}$$

(13)

Next, define $V\left(t\right):=D^{\alpha }u\left(t\right)-t{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}$. In view of the fact that $D^{\alpha }V\left(t\right)=-t^{2-\alpha }{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{″}}$, which implies $V^{{}^{\prime }}\left(t\right)=-t^{2-\alpha }{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{″}}>0$ for $t\in [T,\infty )$, therefore $V\left(t\right)$ is strictly increasing on $[T,\infty )$.

We claim that there is a $t_1\in [T,\infty )$ such that $V\left(t\right)>0$ on $[t_1,\infty )$. Suppose not, $V\left(t\right)<0$ on $[t_1,\infty )$. Hence,

$$\begin{array}{c}\hfill D^{\alpha }\left(\frac{D^{\alpha }u\left(t\right)}{t}\right)=-\frac{t^{1-\alpha }}{t^2}(t{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}-D^{\alpha }u\left(t\right)),\end{array}$$

which gives

$$\begin{array}{c}\hfill {\left(\frac{D^{\alpha }u\left(t\right)}{t}\right)}^{{}^{\prime }}=-\frac{1}{t^2}V\left(t\right)>0,t\in [t_1,\infty ).\end{array}$$

Choose $t_2\in (t_1,\infty )$, for $t\ge t_2$, $\delta \left(\sigma \left(t\right)\right)\ge \delta \left(\sigma \left(t_2\right)\right)$. Since, $\frac{D^{\alpha }u\left(t\right)}{t}$ is strictly increasing,

$$\begin{array}{c}\hfill \frac{D^{\alpha }u\left(\delta \left(\sigma \left(t\right)\right)\right)}{\delta \left(\sigma \right(t\left)\right)}\ge \frac{D^{\alpha }u\left(\delta \left(\sigma \left(t_2\right)\right)\right)}{\delta \left(\sigma \left(t_2\right)\right)}:=m>0,\end{array}$$

the Equation (13) implies that

$$\begin{array}{c}\hfill u\left(\delta \left(\sigma \left(t\right)\right)\right)\ge \frac{t-T}{2}{t}^{1-\alpha }m\delta \left(\sigma \left(t\right)\right).\end{array}$$

(14)

Now, integrating (5) from $t_2$ to t, using $\left(A_2\right)$ and inequality in (14), we have

$$\begin{array}{c}\hfill \underset{t_2}{\overset{t}{\int }}\left({\left(b\left(s\right)D^{\alpha }\left(a\left(s\right)D^{\alpha }u\left(s\right)\right)\right)}^{{}^{\prime }}+\frac{km}{2}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)\right)ds\le 0.\end{array}$$

Then

$$\begin{array}{c}\hfill b\left(t_2\right)D^{\alpha }\left(a\left(t_2\right)D^{\alpha }u\left(t_2\right)\right)\ge \frac{km}{2}\underset{t_2}{\overset{t}{\int }}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)ds,\end{array}$$

which contradicts to (8). Hence $V\left(t\right)>0$ on $[t_1,\infty )$. Accordingly,

$$\begin{array}{c}\hfill t^{1-\alpha }{\left(\frac{D^{\alpha }u\left(t\right)}{t}\right)}^{{}^{\prime }}=-\frac{t^{1-\alpha }}{t^2}(t{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}-D^{\alpha }u\left(t\right))=-\frac{t^{1-\alpha }}{t^2}V\left(t\right)<0,t\in (t_1,\infty ),\end{array}$$

which gives $t{\left(D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}<D^{\alpha }u\left(t\right)$. Then $\delta \left(\sigma \right(t\left)\right)\le \delta \left(t\right)\le t$,

$$\begin{array}{c}\hfill \frac{D^{\alpha }u\left(\delta \left(\sigma \left(t\right)\right)\right)}{\delta \left(\sigma \right(t\left)\right)}\ge \frac{D^{\alpha }u\left(t\right)}{t},\end{array}$$

(15)

since $\frac{D^{\alpha }u\left(t\right)}{t}$ is strictly increasing. Using (13) and (15) in (11), we get

$$\begin{array}{c}\hfill D^{\alpha }w\left(t\right)\le \frac{D^{\alpha }\rho \left(t\right)}{\rho \left(t\right)}w\left(t\right)-\frac{k\rho \left(t\right)c\left(t\right)}{ta\left(t\right)}\frac{{\left(\delta \left(\sigma \left(t\right)\right)\right)}^{\alpha }(\delta \left(\sigma \left(t\right)\right)-T)}{2}-\frac{1}{\rho \left(t\right)b\left(t\right)}{w}^2\left(t\right).\end{array}$$

(16)

Therefore

$$\begin{array}{c}\hfill D^{\alpha }w\left(t\right)\le -\frac{k\rho \left(t\right)c\left(t\right)}{ta\left(t\right)}\frac{{\left(\delta \left(\sigma \left(t\right)\right)\right)}^{\alpha }(\delta \left(\sigma \left(t\right)\right)-T)}{2}+\frac{1}{4}b\left(t\right)\frac{{\left(D^{\alpha }\rho \left(t\right)\right)}^2}{\rho \left(t\right)},\end{array}$$

using (6) and $\left(p_6\right)$, we get

$$\begin{array}{c}\hfill w^{{}^{\prime }}\left(t\right)\le -t^{\alpha -1}\rho \left(t\right)A_{\alpha }\left(t\right)+\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(t\right)\right)}^2}{\rho \left(t\right)}{t}^{1-\alpha }b\left(t\right).\end{array}$$

(17)

Integrating,

$$\begin{array}{c}\hfill \underset{t_1}{\overset{t}{\int }}\left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\le w\left(t_1\right),\end{array}$$

which contradicts the hypothesis $\left(9\right)$. ☐

We now derive various oscillatory criteria on using the earlier results and we can generalize the Philos type kernel. Let us define a class of functions $\Omega$. Consider

$$\begin{array}{c}\hfill {\mathbb{D}}_0=\left\{(t,s):t>s\ge t_0\right\},\mathrm{and}\mathbb{D}=\left\{(t,s):t\ge s\ge t_0\right\}.\end{array}$$

The function $H\in C(\mathbb{D},\mathbb{R})$ belongs to the class $\Omega$, if

$\left(T_1\right)$$H(t,t)=0$ for $t\ge t_0$, and $H(t,s)>0$ for $(t,s)\in {\mathbb{D}}_0$.

$\left(T_2\right)$ The nonpositive partial derivative $\frac{\partial H}{\partial s}$ exist on ${\mathbb{D}}_0$ such that $h(t,s)=H(t,s)\frac{{\rho }^{{}^{\prime }}\left(s\right)}{\rho \left(s\right)}+\frac{\partial H}{\partial s}(t,s)$.

Theorem 2.

Assume that $\left(A_1\right)–\left(A_4\right)$ hold. Further there exists $\rho \in C^{\alpha }([0,\infty );{\mathbb{R}}_{+})$ such that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_1)}\underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{\rho \left(s\right)b\left(s\right)}{H(t,s)}{s}^{1-\alpha }{h}^2(t,s)\right)ds=\infty .\end{array}$$

(18)

Then each solution of system $\left(1\right)$ is oscillatory.

Proof. 

As we proceed in the proof of Theorem 1 and from (16), we have the inequality

$$\begin{array}{c}\hfill w^{{}^{\prime }}\left(t\right)\le \frac{{\rho }^{{}^{\prime }}\left(t\right)}{\rho \left(t\right)}w\left(t\right)-t^{\alpha -1}\rho \left(t\right)A_{\alpha }\left(t\right)-\frac{t^{\alpha -1}}{\rho \left(t\right)b\left(t\right)}{w}^2\left(t\right).\end{array}$$

(19)

Integrating,

$$\begin{array}{c}\underset{t_1}{\overset{t}{\int }}H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)ds\hfill \\ \le \underset{t_1}{\overset{t}{\int }}H(t,s)\frac{{\rho }^{{}^{\prime }}\left(s\right)}{\rho \left(s\right)}w\left(s\right)ds-\underset{t_1}{\overset{t}{\int }}H(t,s)w^{{}^{\prime }}\left(s\right)ds-\underset{t_1}{\overset{t}{\int }}H(t,s)\frac{s^{\alpha -1}}{\rho \left(s\right)b\left(s\right)}{w}^2\left(s\right)ds,\hfill \\ \le H(t,t_1)w\left(t_1\right)+\underset{t_1}{\overset{t}{\int }}\left(H(t,s)\frac{{\rho }^{{}^{\prime }}\left(s\right)}{\rho \left(s\right)}+\frac{\partial H}{\partial s}(t,s)\right)w\left(s\right)ds-\underset{t_1}{\overset{t}{\int }}H(t,s)\frac{s^{\alpha -1}}{b\left(s\right)\rho \left(s\right)}{w}^2\left(s\right)ds,\hfill \\ \le H(t,t_1)w\left(t_1\right)+\underset{t_1}{\overset{t}{\int }}\left(w\left(s\right)h(t,s)-H(t,s)\frac{s^{\alpha -1}}{\rho \left(s\right)b\left(s\right)}{w}^2\left(s\right)\right)ds,\hfill \\ \le H(t,t_1)w\left(t_1\right)+\underset{t_1}{\overset{t}{\int }}\frac{1}{4}\frac{\rho \left(s\right)b\left(s\right)}{H(t,s)}{s}^{1-\alpha }{h}^2(t,s)ds.\hfill \end{array}$$

From this we conclude that

$$\begin{array}{c}\hfill \underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{\rho \left(s\right)b\left(s\right)}{H(t,s)}{s}^{1-\alpha }{h}^2(t,s)\right)ds\le H(t,t_1)w\left(t_1\right).\end{array}$$

Since $0<H(t,s)\le H(t,t_1)$ for $t>s>t_1$, we have $0<\frac{H(t,s)}{H(t,t_1)}\le 1$, hence

$$\begin{array}{c}\hfill \frac{1}{H(t,t_1)}\underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{\rho \left(s\right)b\left(s\right)}{H(t,s)}{s}^{1-\alpha }{h}^2(t,s)\right)ds\le w\left(t_1\right).\end{array}$$

Letting $t\to \infty$,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_1)}\underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{\rho \left(s\right)b\left(s\right)}{H(t,s)}{s}^{1-\alpha }{h}^2(t,s)\right)ds\le w\left(t_1\right).\end{array}$$

Therefore assumption $\left(18\right)$ is contradicted. Thus every solution of (1) oscillates. ☐

We immediately obtain the following oscillation result for (1).

Theorem 3.

Assume that $\left(A_1\right)$–$\left(A_4\right)$ hold. Also assume that there exists a function $\rho \in C^{\alpha }([0,\infty );{\mathbb{R}}_{+})$ such that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_1)}\underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{H(t,s){\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds=\infty .\end{array}$$

(20)

Then every solution of system $\left(1\right)$ is oscillatory.

Proof. 

Proceeding as in the proof of Theorem 1, multiplying inequality (17) by $H(t,s)$ and integrating, we get

$$\begin{array}{c}\hfill \underset{t_1}{\overset{t}{\int }}\left(H(t,s)s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{H(t,s){\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\le -\underset{t_1}{\overset{t}{\int }}H(t,s)w^{{}^{\prime }}\left(s\right)ds\le H(t,t_1)w\left(t_1\right).\end{array}$$

Taking limsup as $t\to \infty$, and hence

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_1)}\underset{t_1}{\overset{t}{\int }}\left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)H(t,s)-\frac{1}{4}\frac{H(t,s){\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\le w\left(t_1\right),\end{array}$$

which contradicts the hypothesis in $\left(20\right)$. ☐

The following theorem is to be proved using the techniques employed in the previous theorems.

Theorem 4.

Suppose that the assumptions $\left(A_1\right)$–$\left(A_4\right)$ and (8) hold. Further assume also that Case (I) of Lemma 2 holds, then

$$\begin{array}{c}\hfill {\left(A_{\alpha }\right)}_{*}\le d-t^{\alpha -1}\frac{1}{b\left(s\right)}{d}^2,\end{array}$$

(21)

and

$$\begin{array}{c}\hfill {\left(B_{\alpha }\right)}_{*}\le D-D^2.\end{array}$$

(22)

Proof. 

Let $u\left(t\right)$ be a nonoscillatory solution of (5) such that $u\left(t\right)>0$, $u\left(\delta \right(t\left)\right)>0$ and $u\left(\delta \right(\sigma \left(t\right)\left)\right)>0$ for $t\ge T>t_0$, consider the case (I) of Lemma 2 holds, $u\left(t\right)$ satisfies the inequality $D^{\alpha }\left(b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)\right)\le 0,t\in [T,\infty )$. Define Riccati transformation

$$\begin{array}{c}\hfill w\left(t\right)=\frac{b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)}{a\left(t\right)D^{\alpha }u\left(t\right)}.\end{array}$$

Thus $w\left(t\right)>0$, differentiating $\alpha$ times with respect to t, using (5) and $\left(A_2\right)$, we have

$$\begin{array}{c}\hfill D^{\alpha }w\left(t\right)\le -\frac{kc\left(t\right)}{a\left(t\right)}\frac{u\left(\delta \right(\sigma \left(t\right)\left)\right)}{D^{\alpha }u\left(t\right)}-\frac{1}{b\left(t\right)}{w}^2\left(t\right).\end{array}$$

By using (15), (13) and (6), we obtain the above inequality

$$\begin{array}{c}\hfill w^{{}^{\prime }}\left(t\right)+t^{\alpha -1}{A}_{\alpha }\left(t\right)+t^{\alpha -1}\frac{1}{b\left(t\right)}{w}^2\left(t\right)\le 0.\end{array}$$

(23)

Given that $A_{\alpha }\left(t\right)>0$ and $w\left(t\right)>0$, which gives $w^{{}^{\prime }}\left(t\right)\le 0$ and

$$\begin{array}{c}\hfill -b\left(t\right)(w^{{}^{\prime }}\left(t\right)t^{1-\alpha }/w^2\left(t\right))>1.\end{array}$$

which yields that

$$\begin{array}{c}\hfill {\left(\frac{1}{w\left(t\right)}\right)}^{{}^{\prime }}>t^{\alpha -1}\frac{1}{b\left(t\right)}.\end{array}$$

Integrating the above inequality and denote $t_1^{\alpha -1}\frac{1}{b\left(t_1\right)}=M$, we have

$$\begin{array}{c}\hfill M(t-t_1)w\left(t\right)<1.\end{array}$$

(24)

which implies that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}w\left(t\right)=0,\underset{t\to \infty }{lim}tw\left(t\right)=0.\end{array}$$

(25)

From (9) and (24), $0<d<1$ and $0<D<1$. Even though if $d=0$ and $D=0$, there is nothing to prove. Now, to claim (21). Integrating (23) from t to ∞ and use (25), we get

$$\begin{array}{c}\hfill w\left(t\right)\ge \underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds+\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}\frac{1}{b\left(s\right)}{w}^2\left(s\right)ds.\end{array}$$

Multiplying by t and taking liminf as $t\to \infty$, by (25), $d\ge {\left(A_{\alpha }\right)}_{*}$. For given $ϵ>0$, there exists a $t_2\ge t_1$ as

$$\begin{array}{c}\hfill d-ϵ<tw\left(t\right)<d+ϵandt\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds\ge {\left(A_{\alpha }\right)}_{*}-ϵ,t\ge t_2.\end{array}$$

(26)

Again from (26),

$$\begin{array}{cc}tw\left(t\right)\hfill & \ge t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds+t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}\frac{1}{b\left(s\right)}{w}^2\left(s\right)ds\hfill \\ \hfill & \ge t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds+t^{\alpha }\frac{1}{b\left(t\right)}\underset{t}{\overset{\infty }{\int }}\frac{{\left(sw\left(s\right)\right)}^2}{s^2}ds\hfill \\ \hfill & \ge t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds+t^{\alpha }\frac{1}{b\left(t\right)}{(d-ϵ)}^2\underset{t}{\overset{\infty }{\int }}\frac{1}{s^2}ds\hfill \\ \hfill & =t\underset{t}{\overset{\infty }{\int }}{s}^{\alpha -1}{A}_{\alpha }\left(s\right)ds+t^{\alpha -1}\frac{1}{b\left(t\right)}{(d-ϵ)}^2.\hfill \end{array}$$

(27)

Therefore from (26) and (27), $d\ge {\left(A_{\alpha }\right)}_{*}-ϵ+{(d-ϵ)}^2$. Then

$$\begin{array}{c}\hfill d\ge {\left(A_{\alpha }\right)}_{*}+t^{\alpha -1}\frac{1}{b\left(t\right)}{d}^2,\end{array}$$

since $ϵ$ is arbitrarily small. Next to prove that (22). Multiply (23) by $s^2$, integrating from $t_1$ to t, and integration by parts follows that

$$\begin{array}{cc}\hfill \underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds& \le -\underset{t_1}{\overset{t}{\int }}{s}^2{w}^{{}^{\prime }}\left(s\right)ds-\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}\frac{1}{b\left(s\right)}{w}^2\left(s\right)ds\hfill \\ \hfill & \le -t^{2}w\left(t\right)+t_1^{2}w\left(t_1\right)+2\underset{t_1}{\overset{t}{\int }}sw\left(s\right)ds-\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}\frac{1}{b\left(s\right)}{w}^2\left(s\right)ds,\hfill \end{array}$$

implies

$$\begin{array}{c}\hfill t^{2}w\left(t\right)\le t_1^{2}w\left(t_1\right)-\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds+\underset{t_1}{\overset{t}{\int }}\left(2sw\left(s\right)-s^{\alpha +1}\frac{1}{b\left(s\right)}{w}^2\left(s\right)\right)ds.\end{array}$$

(28)

Thus, we obtain

$$\begin{array}{cc}tw\left(t\right)\hfill & \le \frac{t_1^{2}w\left(t_1\right)}{t}-\frac{1}{t}\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds+\frac{1}{t}\underset{t_1}{\overset{t}{\int }}{s}^{1-\alpha }b\left(s\right)ds,\hfill \\ \hfill & \le \frac{t_1^{2}w\left(t_1\right)}{t}-\frac{1}{t}\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds+\frac{1}{t}{t}^{1-\alpha }b\left(t\right)\underset{t_1}{\overset{t}{\int }}ds.\hfill \end{array}$$

(29)

By $\left(A_4\right)$, (29) imply that

$$\begin{array}{c}\hfill tw\left(t\right)\le \frac{t_1^{2}w\left(t_1\right)}{t}-\frac{1}{t}\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds+\frac{1}{t}(t-t_1).\end{array}$$

Thus

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}tw\left(t\right)\le 1-\underset{t\to \infty }{lim\; inf}\frac{1}{t}\underset{t_1}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds.\end{array}$$

Hence from (6), (7), $D\le 1-{\left(B_{\alpha }\right)}_{*}$. For any $ϵ>0$, there exists a $t_2\ge t_1$ such that

$$\begin{array}{c}\hfill D-ϵ<tw\left(t\right)<D+ϵand\frac{1}{t}\underset{t_0}{\overset{t}{\int }}{s}^{\alpha +1}{A}_{\alpha }\left(s\right)ds>{\left(B_{\alpha }\right)}_{*}-ϵ,t\ge t_2.\end{array}$$

(30)

Now, from (28) and (30) we get

$$\begin{array}{c}\hfill D\le -{\left(B_{\alpha }\right)}_{*}+ϵ(D+ϵ)(2-D+ϵ),t\ge t_2,\end{array}$$

since $ϵ$ is arbitrarily small, we have

$$\begin{array}{c}\hfill {\left(B_{\alpha }\right)}_{*}\le D-D^2,\end{array}$$

which proves (22). ☐

Lemma 3.

Suppose that $\left(A_1\right)$–$\left(A_4\right)$ and (8) hold. Also assume that Case (II) of Lemma 2 holds. If

$$\begin{array}{c}\hfill \underset{t_2}{\overset{\infty }{\int }}{\eta }^{\alpha -1}\frac{1}{a\left(\eta \right)}\left(\underset{\eta }{\overset{\infty }{\int }}\underset{\mu }{\overset{\infty }{\int }}{s}^{\alpha -1}c\left(s\right)dsd\mu \right)d\eta =\infty .\end{array}$$

(31)

Then $\underset{t\to \infty }{lim}u\left(t\right)=0$.

Proof. 

We consider the Case (II) of Lemma 2, $D^{\alpha }u\left(t\right)<0,D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)>0$ for $t\ge t_1$. Since $u\left(t\right)$ is positive and decreasing, we have $\underset{t\to \infty }{lim}u\left(t\right)=d^{{}^{\prime }}\ge 0$. Suppose not, $d^{{}^{\prime }}>0$. Given that $u\left(\delta \right(\sigma \left(t\right)\left)\right)\le \delta \left(t\right)\le t$, then $u\left(\delta \left(\sigma \left(t\right)\right)\right)\ge u\left(t\right)>d^{{}^{\prime }}$ for $t\ge t_2\ge t_1$ sufficiently large, $u\left(t\right)$ is decreasing. Integrating (5) from t to ∞ and using $u\left(\delta \left(\sigma \left(t\right)\right)\right)\ge d^{{}^{\prime }}$, we get

$$\begin{array}{c}\hfill {\int }_t^{\infty }{\left(b\left(s\right)D^{\alpha }\left(a\left(s\right)D^{\alpha }u\left(s\right)\right)\right)}^{{}^{\prime }}ds\le -{\int }_t^{\infty }k{s}^{\alpha -1}c\left(s\right)u\left(\delta \left(\sigma \left(s\right)\right)\right)ds\le -k{d}^{{}^{\prime }}{\int }_t^{\infty }{s}^{\alpha -1}c\left(s\right)ds,\end{array}$$

then,

$$\begin{array}{c}\hfill b\left(t\right)D^{\alpha }\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)\ge k{d}^{{}^{\prime }}{\int }_t^{\infty }{s}^{\alpha -1}c\left(s\right)ds.\end{array}$$

By $\left(A_4\right)$, we get

$$\begin{array}{c}\hfill {\left(a\left(t\right)D^{\alpha }u\left(t\right)\right)}^{{}^{\prime }}\ge k{d}^{{}^{\prime }}\frac{1}{b\left(t\right)t^{1-\alpha }}{\int }_t^{\infty }{s}^{\alpha -1}c\left(s\right)ds\ge k{d}^{{}^{\prime }}{\int }_t^{\infty }{s}^{\alpha -1}c\left(s\right)ds.\end{array}$$

Again integrating, we obtain

$$\begin{array}{cc}\hfill -a\left(t\right)D^{\alpha }u\left(t\right)& \ge k{d}^{{}^{\prime }}{\int }_t^{\infty }{\int }_{\mu }^{\infty }{s}^{\alpha -1}c\left(s\right)dsd\mu ,\hfill \end{array}$$

this implies that,

$$\begin{array}{cc}\hfill -u^{{}^{\prime }}\left(t\right)& \ge k{d}^{{}^{\prime }}{t}^{\alpha -1}\frac{1}{a\left(t\right)}{\int }_t^{\infty }{\int }_{\mu }^{\infty }{s}^{\alpha -1}c\left(s\right)dsd\mu .\hfill \end{array}$$

By integrating, once again it is get as

$$\begin{array}{cc}\hfill u\left(t_2\right)& \ge k{d}^{{}^{\prime }}{\int }_{t_2}^{\infty }\left({\eta }^{\alpha -1}\frac{1}{a\left(\eta \right)}{\int }_{\eta }^{\infty }{\int }_{\mu }^{\infty }{s}^{\alpha -1}c\left(s\right)dsd\mu \right)d\eta ,\hfill \end{array}$$

which contradicts to (31). Thus $d^{{}^{\prime }}=0$ and hence $\underset{t\to \infty }{lim}u\left(t\right)=0$. ☐

From Theorem 4, Nehari type oscillation criteria for (1).

Theorem 5.

Assume that $\left(A_1\right)-\left(A_4\right)$, (8) and (31) hold. If

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; inf}\frac{1}{t}\underset{t_0}{\overset{t}{\int }}\left(k{s}^{\alpha +1}\frac{c\left(s\right)}{a\left(s\right)}\frac{u\left(\delta \right(\sigma \left(s\right)\left)\right)-T}{s}{\left(u\left(\delta \left(\sigma \left(s\right)\right)\right)\right)}^{\alpha }\right)ds>\frac{1}{2},\end{array}$$

(32)

then u(t) is oscillatory or satisfies $u\left(t\right)=0$ as $t\to \infty$.

4. Examples

In this section, we provide some examples in order to see the effect of the main results.

Example 1.

Consider $\frac{1}{2}$-fractional delay differential system

$$\begin{array}{cc}\hfill D^{\frac{1}{2}}\left(u\left(t\right)\right)& =\frac{1}{\sqrt{t}}g\left(v\left(\frac{t}{2}\right)\right)\hfill \\ \hfill D^{\frac{1}{2}}\left(v\left(t\right)\right)& =-\frac{1}{\sqrt{t}}h\left(w\left(t\right)\right),\hfill \\ \hfill D^{\frac{1}{2}}\left(w\left(t\right)\right)& =\frac{1}{\sqrt{t}}f\left(u\left(\frac{t}{2}\right)\right),t\ge t_0,\hfill \end{array}$$

(33)

where $C_1=cos(ln2)$, $C_2=sin(ln2)$, $A_1=cos(ln4)$, $A_2=sin(ln4)$.

Here $\alpha =\frac{1}{2}$, $p\left(t\right)=\frac{1}{a\left(t\right)}=\frac{1}{\sqrt{t}}$, $q\left(t\right)=\frac{1}{b\left(t\right)}=\frac{1}{\sqrt{t}},r\left(t\right)=\frac{1}{\sqrt{t}}$, $f\left(u\right)=A_1\sqrt{(1-u^2)}-A_{2}u$, $g\left(v\right)=v$ and $h\left(w\right)=w$.

It is easy to see that

$$\begin{array}{ccc}\hfill D^{\alpha }g\left(v\right)& =& \frac{1}{\sqrt{t}}(C_1+C_2)\ge l^{{}^{\prime }}>0,\hfill \\ \hfill D^{\alpha }h\left(w\right)& =& \frac{1}{\sqrt{t}}(C_1-C_2)\ge m^{{}^{\prime }}>0,\hfill \\ \hfill f\left(u\right)/u& =& A_1\sqrt{\frac{1}{u^2}-1}-A_2\ge 0.2579=k>0,\hfill \end{array}$$

since $u^2<1$, $\sigma \left(t\right)=\delta \left(t\right)=\frac{t}{2}$ and $D^{\alpha }\sigma \left(t\right)=\frac{\sqrt{t}}{2}\ge l>0$, $c\left(t\right)=\frac{C_1^2-C_2^2}{4\sqrt{t}}$, $A_{\alpha }\left(t\right)=\frac{0.2579}{16}\frac{\frac{t}{4}-T}{\sqrt{t}}$. Now it is considered as,

$$\begin{array}{c}\hfill \underset{t_2}{\overset{\infty }{\int }}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)ds=\frac{C_1^2-C_2^2}{4}\underset{t_2}{\overset{\infty }{\int }}\frac{(s-T)}{\sqrt{s}}\frac{s}{4}ds\to \infty ast\to \infty .\end{array}$$

By taking $\rho \left(t\right)=16/k$ then ${\rho }^{{}^{\prime }}\left(t\right)=0$. Consider

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}& \left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\hfill \\ \hfill & =\underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}\left(s^{-\frac{1}{2}}\frac{16}{k}\frac{k(\frac{s}{4}-T)}{16}\frac{1}{\sqrt{s}}\right)ds\hfill \\ \hfill & =\underset{t\to \infty }{lim\; sup}\frac{1}{4}\underset{t_1}{\overset{t}{\int }}\left(\frac{s-4T}{s}\right)ds\to \infty ast\to \infty .\hfill \end{array}$$

Since, each of the conditions are verified in Theorem 1, all solutions of $\left(33\right)$ are oscillatory. Thus $(u\left(t\right),v\left(t\right),w\left(t\right))=(sin(lnt),C_{1}cos(lnt)-C_{2}sin(lnt),C_{1}sin(lnt)+C_{2}cos(lnt))$ is one such solution.

Note: The decreasing condition imposed on $q\left(t\right)$ and $r\left(t\right)$ is only a sufficient condition, however it is not a necessary one. The following example ensures the oscillatory behavior of the system (34) even though $q\left(t\right)$ and $r\left(t\right)$ is nondecreasing.

Example 2.

Consider $\frac{1}{3}$-fractional following differential system

$$\begin{array}{cc}\hfill D^{\frac{1}{3}}\left(u\left(t\right)\right)& =\frac{t^{\frac{2}{3}}}{1+\frac{3}{4}{cos}^{\frac{5}{3}}\left(t\right)}g\left(v(t-2\pi )\right),\hfill \\ \hfill D^{\frac{1}{3}}\left(v\left(t\right)\right)& =-t^{\frac{2}{3}}w\left(t\right),\hfill \\ \hfill D^{\frac{1}{3}}\left(w\left(t\right)\right)& =\frac{t^{\frac{2}{3}}}{1+{cos}^2\left(t\right)}f\left(u(t-\frac{3\pi }{2})\right),t\ge t_0.\hfill \end{array}$$

(34)

Here $\alpha =\frac{1}{3}$, $p\left(t\right)=\frac{1}{a\left(t\right)}=\frac{t^{\frac{2}{3}}}{1+\frac{3}{4}{cos}^{\frac{5}{3}}\left(t\right)},q\left(t\right)=\frac{1}{b\left(t\right)}=t^{\frac{2}{3}},r\left(t\right)=\frac{t^{\frac{2}{3}}}{1+{cos}^2\left(t\right)}$, $f\left(u\right)=u(1+u^2)$, $g\left(v\right)=v(1+\frac{3}{4}{v}^{\frac{5}{3}})$ and $h\left(w\right)=w$. It is easy to see that $D^{\alpha }g\left(v\right)=v^{\frac{1}{3}}+2v^2\ge 1=l^{{}^{\prime }}>0$ such that $y^2>1,y^{{}^{\prime }}>\frac{1}{3}$, $D^{\alpha }h\left(w\right)\ge 1=m^{{}^{\prime }}>0$, $f\left(u\right)/u=1+u^2\ge 1=k>0$, $\sigma \left(t\right)=t-2\pi$, $\delta \left(t\right)=t-\frac{3\pi }{2}$ and $D^{\alpha }\sigma \left(t\right)=t^{\frac{2}{3}}\ge l$ such that $t_1=l^{\frac{2}{3}}$, $t\ge t_1$, $c\left(t\right)=l^2\frac{t^{\frac{2}{3}}}{1+{cos}^2\left(t\right)}$, $A_{\alpha }\left(t\right)=\frac{l^2}{2}\frac{1+\frac{3}{4}{cos}^{\frac{5}{3}}\left(t\right)}{1+{cos}^2\left(t\right)}\frac{t-\frac{3\pi }{2}-T}{t}{(t-\frac{3\pi }{2})}^{\frac{1}{3}}$. Now consider,

$$\begin{array}{cc}\hfill \underset{t_2}{\overset{\infty }{\int }}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)ds& =\underset{t_2}{\overset{\infty }{\int }}{l}^2\frac{s^{\frac{2}{3}}}{1+{cos}^2\left(s\right)}(s-T)(s-\frac{3\pi }{2})ds\hfill \\ \hfill & \ge \frac{l^2}{2}\underset{t_2}{\overset{\infty }{\int }}{s}^{\frac{2}{3}}(s-T)(s-\frac{3\pi }{2})ds\to \infty ast\to \infty .\hfill \end{array}$$

If we take $\rho \left(t\right)=1$ then ${\rho }^{{}^{\prime }}\left(t\right)=0$. Consider

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}& \left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\hfill \\ \hfill & =\underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}\left(s^{-\frac{2}{3}}{l}^2\frac{s^{\frac{2}{3}}}{1+{cos}^2\left(s\right)}\frac{1}{2}\frac{1+\frac{3}{4}{cos}^{\frac{5}{3}}\left(s\right)}{1+{cos}^2\left(s\right)}\frac{s-\frac{3\pi }{2}-T}{s}{(s-\frac{3\pi }{2})}^{\frac{1}{3}}\right)ds\hfill \\ \hfill & \ge \underset{t\to \infty }{lim\; sup}\frac{7l^2}{16}\underset{t_1}{\overset{t}{\int }}\left((1-\frac{\frac{3\pi }{2}-T}{s}){(s-\frac{3\pi }{2})}^{\frac{1}{3}}\right)ds\to \infty ast\to \infty .\hfill \end{array}$$

Theorem 1 are satisfying the new conditions arriving at the solution for $\left(34\right)$ is oscillatory and it is given as $\left(u\right(t),v(t),w(t\left)\right)=(sint,cost,sint)$.

Example 3.

Consider the $\frac{1}{2}$-fractional differential system

$$\begin{array}{cc}\hfill D^{\frac{1}{2}}\left(u\left(t\right)\right)& =e^{2t}{t}^{\frac{1}{2}}g\left(v(t-1)\right),\hfill \\ \hfill D^{\frac{1}{2}}\left(v\left(t\right)\right)& =-e^{-2t}{t}^{\frac{1}{2}}w\left(t\right),\hfill \\ \hfill D^{\frac{1}{2}}\left(w\left(t\right)\right)& ={\left(et\right)}^{\frac{1}{2}}f\left(u(t-\frac{1}{2})\right),t\ge t_0.\hfill \end{array}$$

(35)

Here $\alpha =\frac{1}{2}$, $\frac{1}{a\left(t\right)}=p\left(t\right)=e^{2t}{t}^{\frac{1}{2}},\frac{1}{b\left(t\right)}=q\left(t\right)=e^{-2t}{t}^{\frac{1}{2}},r\left(t\right)={\left(et\right)}^{\frac{1}{2}}$, $g\left(v\right)=v$, $h\left(w\right)=w$ and $f\left(u\right)=u$. Now it is easy to check that $D^{\alpha }g\left(v\right)=v^{\frac{1}{2}}=e^{-\frac{t}{2}}=l^{{}^{\prime }}>0$, $D^{\alpha }h\left(w\right)=w^{\frac{1}{2}}=e^{\frac{t}{2}}=m^{{}^{\prime }}>0$, $f\left(u\right)/u=1=k>0$, $\sigma \left(t\right)=t-1$, $\delta \left(t\right)=t-\frac{1}{2}$ and $D^{\alpha }\sigma \left(t\right)=t^{\frac{1}{2}}\ge l$ such that $t_1=l^{\frac{1}{2}}$ for $t\ge t_1$, $c\left(t\right)=l^2{\left(et\right)}^{\frac{1}{2}}$, $A_{\alpha }\left(t\right)=\frac{l^2}{2}{e}^{\frac{1}{2}}{e}^{2t}(t-\frac{1}{2}-T){(t-\frac{1}{2})}^{\frac{1}{2}}$. Now,

$$\begin{array}{c}\hfill \underset{t_2}{\overset{\infty }{\int }}c\left(s\right)(s-T)\delta \left(\sigma \left(s\right)\right)ds=\underset{t_2}{\overset{\infty }{\int }}{l}^2{\left(es\right)}^{\frac{1}{2}}(s-T)(s-\frac{1}{2})ds=l^2{e}^{\frac{1}{2}}\underset{t_2}{\overset{\infty }{\int }}{s}^{\frac{1}{2}}(s-T)(s-\frac{1}{2})ds\to \infty .\end{array}$$

Taking $\rho \left(t\right)=\frac{1}{t^{\frac{7}{2}}{e}^{2t}}$ then ${\rho }^{{}^{\prime }}\left(t\right)=-\frac{t^{\frac{7}{2}}}{2t^7{e}^{2t}}(4t+7)$. Consider

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}& \left(s^{\alpha -1}\rho \left(s\right)A_{\alpha }\left(s\right)-\frac{1}{4}\frac{{\left({\rho }^{{}^{\prime }}\left(s\right)\right)}^2}{\rho \left(s\right)}{s}^{1-\alpha }b\left(s\right)\right)ds\hfill \\ \hfill & =\underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}\left(s^{-\frac{1}{2}}\frac{l^2{e}^{\frac{1}{2}}}{s^{\frac{7}{2}}{e}^{2s}}\frac{1}{2}{e}^{2s}(s-\frac{1}{2}-T){(s-\frac{1}{2})}^{\frac{1}{2}}-\frac{{(4s+7)}^2}{4s^9{e}^{4s}}{e}^{4s}{s}^{\frac{7}{2}}\right)ds\hfill \\ \hfill & \le \underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}\left(\frac{l^2{e}^{\frac{1}{2}}}{2s^{\frac{7}{2}}}s-\frac{s^2}{4s^{\frac{11}{2}}}\right)ds\le \underset{t\to \infty }{lim\; sup}\underset{t_1}{\overset{t}{\int }}\left(\frac{l^2{e}^{\frac{1}{2}}}{2s^{\frac{5}{2}}}-\frac{1}{4s^{\frac{7}{2}}}\right)ds<\infty .\hfill \end{array}$$

Here further the condition (9) of the above Theorem 1 seems to be not satisfied, in view of the fact that $\left(A_4\right)$ fails to hold, and hence the system (35) is not oscillatory. In fact, $(u\left(t\right),v\left(t\right),w\left(t\right))=(e^t,e^{-t},e^t)$ it is a solution for (35), and nonoscillatory.

Remark 1.

The results obtained in this article further can be extended to a neutral system with forced term

$$\begin{array}{cc}\hfill D^{\alpha }\left(u\left(t\right)+p\left(t\right)u\left(\delta \right(t\left)\right)\right)& =a\left(t\right)h_1\left(v\left(\tau \right(t\left)\right)\right),\hfill \\ \hfill D^{\alpha }\left(v\left(t\right)\right)& =-b\left(t\right)h_2\left(w\left(t\right))\right),\hfill \\ \hfill D^{\alpha }\left(w\left(t\right)\right)& =c\left(t\right)h_3\left(u\left(\sigma \right(t\left)\right)\right)+e\left(t\right),t\ge t_0,\hfill \end{array}$$

for the cases

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }a\left(s\right)d_{\alpha }s<\infty ,{\int }_{t_0}^{\infty }b\left(s\right)d_{\alpha }s=\infty ,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }a\left(s\right)d_{\alpha }s<\infty ,{\int }_{t_0}^{\infty }b\left(s\right)d_{\alpha }s<\infty .\end{array}$$

5. Conclusions

Through this article, we have derived some new oscillation results for a certain class of nonlinear three-dimensional $\alpha$-fractional differential systems by using the Riccati transformation and inequality technique. This work extends and also improves some classical results in the literature [16,18,32] to the $\alpha$-fractional systems and studied the oscillation criteria. Further, the present results are essentially new and, in order to illustrate the validity of the obtained results, we have provided three examples.