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Oscillation theorems for three classes of conformable fractional differential equations
Advances in Difference Equations volume 2019, Article number: 313 (2019)
Abstract
In this paper, we consider the oscillation theory for fractional differential equations. We obtain oscillation criteria for three classes of fractional differential equations of the forms
and
where \(T_{\alpha}\) denotes the conformable differential operator of order α, \(0<\alpha\leqslant1\).
1 Introduction
Fractional differential equations have been of great interest recently. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self-similar and porous structures, fluid flows, electrical networks, chemical physics, and many other branches of science.
The oscillation of fractional differential equations as a new research field has received significant attention, and some interesting results have already been obtained. We refer to [1,2,3,4,5,6,7,8,9,10,11] and the references therein. The definition of the fractional-order derivative used is either the Caputo or the Riemann–Liouville fractional-order derivative involving an integral expression and the gamma function. Because of the definition, the oscillation of these types of fractional equations cannot be studied by regular methods, for example, by the Riccati transformation. It can only be studied by transforming it into an integer-order equation. In 2012, Chen et al. [4] studied the oscillation behavior of the following fractional differential equation:
where \(D_{-}^{\alpha}y\) denotes the Liouville right-sided fractional derivative of order α,
By the Riccati transformation the authors obtained some sufficient conditions.
Recently, Khalil et al. [12] introduced a new well-behaved definition of local fractional derivative, called the conformable fractional derivative, depending just on the basic limit definition of the derivative. This new theory is improved by Abdeljawad [13]. For recent results on conformable fractional derivatives, we refer the reader to [14,15,16,17,18,19,20,21,22,23]. This new definition satisfies formulas of the derivatives of the product and quotient of two functions and has a simpler chain rule. In addition to the definition of conformable fractional derivative, a definition of conformable fractional integral, the Rolle theorem, and the mean value theorem for conformable fractional differentiable functions were given. These properties are more conducive to the study of the oscillation of fractional-order equations.
In fact, some works in this field have shown the significance of conformable fractional derivative. For example, [24] discusses the potential conformable quantum mechanics, [25] discusses the conformable Maxwell equations, and [26, 27] show that the conformable fractional derivative models present good agreements with experimental data, but there are less oscillation results.
In the paper, we study oscillation criteria of conformable fractional differential equations. Our main goal is to generalize the oscillatory criteria in [28,29,30,31,32,33,34,35,36,37] to the conformable fractional derivative. The three equations represent three classes of equations of different orders. For example, in 2016, Akca et al. [33] studied the equation
and obtained the following:
Theorem 1.1
Assume that \(0<\varsigma:=\liminf_{t\rightarrow\infty}\int_{\tau(t)}^{t}\sum_{i=1}^{m}p_{i}(s)\,ds\leqslant\frac{1}{e}\) and for some \(r\in\mathbb {N}\), we have
where \(h(t)=\max_{1\leqslant i\leqslant m}h_{i}(t)\), \(h_{i}(t)=\sup_{0\leqslant s\leqslant t}\tau_{i}(s)\), \(a_{1}(t,s):=\exp \{\int_{s}^{t}\sum_{i=1}^{m}p_{i}(\zeta )\,d\zeta \}\), \(a_{r+1} (t,s):=\exp \{\int_{s}^{t}\sum_{i=1}^{m}p_{i}(\zeta )a_{r}(\zeta,\tau_{i}(\zeta))\,d\zeta \}\), and \(\lambda_{0}\) is the smaller root of the equation \(e^{\varsigma\lambda}=\lambda\). Then the above equation oscillates.
From this we can unify the oscillation theory of integral-order and fractional-order differential equations. Through the inequality principle, iterative sequences, and the Riccati transformation method this can be extended to the conformable fractional derivatives by Lemma 2.2.
A solution x is called oscillatory if it is eventually neither positive nor negative. Otherwise, the solution is said to be nonoscillatory. An equation is oscillatory if all its solutions oscillate. In this paper, x is differentiable on \([t_{0},\infty)\). This paper is organized as follows. In Sect. 2, we introduce some notation and definitions on conformable fractional integrals. In Sect. 3, we present the main theorems on α-order equations. Section 4 is devoted to the oscillatory results on 2α-order equation. In Sect. 5, we demonstrate the oscillatory results for 3α-order equations. In each section, we give examples to illustrate the significance of the results.
2 Conformable fractional calculus
For the convenience of the reader, we give some background from fractional calculus theory. These materials can be found in the recent literature, see [12, 13, 23].
Definition 2.1
([13]) The (left) fractional derivative of a function \(f : [ a ,\infty)\rightarrow R\) of order \(\alpha\in(0,1]\) starting from a is defined by
When \(a=0\), we write \(T_{\alpha}\).
Note that if f is differentiable, then \((T_{\alpha}^{a}f)(t)=(t-a)^{1-\alpha}f'(t)\).
Definition 2.2
([13]) The left fractional integral of order \(\alpha\in(0,1]\) starting at a is defined by
Definition 2.3
([13]) Let \(f:[a,\infty)\rightarrow\mathbb{R}\) be a continuous function, and let \(\alpha\in(0,1]\). Then, for all \(t>a\), we have
Definition 2.4
([13]) Let \(f:(a,b)\rightarrow\mathbb{R}\) be let a differentiable function, and let \(\alpha\in(0,1]\). Then, for all \(t>a\), we have
Proposition 2.1
([13]) Let \(f:(a,\infty)\rightarrow\infty\rightarrow\mathbb{R}\) be a twice differentiable function, and let \(0 < \alpha, \beta\leqslant 1\) be such that \(1<\alpha+\beta\leqslant2\). Then
Proposition 2.2
([23]) Let \(\alpha\in(0,1]\), and let f and g be α-differentiable at a point \(t>0\) on \([a,\infty)\). Then
-
(1)
\(T_{\alpha}^{a}(af+bg)=aT_{\alpha}^{a}(f)+bT_{\alpha}^{a}(g)\) for all \(a, b\in\mathbb{R}\),
-
(2)
\(T_{\alpha}^{a}(\lambda)=0\) for all constant functions \(f(t)=\lambda\),
-
(3)
\(T_{\alpha}^{a}(fg)=fT_{\alpha}^{a}(g)+gT_{\alpha}^{a}(f)\),
-
(4)
\(T_{\alpha}^{a}(\frac{f}{g})=\frac{gT_{\alpha}^{a}(f)-fT_{\alpha}^{a}(g)}{g^{2}}\),
-
(5)
\(T_{\alpha}^{a}(t^{n})=nt^{n-\alpha}\) for all \(n\in\mathbb{R}\), and
-
(6)
\(T_{\alpha}^{a}(f\circ g)(t)=f'(g(t))T_{\alpha}^{a}(g)(t)\) for f differentiable at \(g(t)\).
Lemma 2.1
([13]) Let \(f, g:[a,b]\rightarrow\mathbb{R}\) be two functions such that fg is differentiable, and let \(\alpha\in(0,1]\). Then
Lemma 2.2
Let \(f : (t_{0} , \infty) \rightarrow\mathbb{R}\) be differentiable, and let \(\alpha\in(0,1]\). If \(T_{\alpha}^{t_{0}}f(t)=M(t)\), then for all \(t > s>t_{0}\), we have
Proof
We can conclude from \(T_{\alpha}^{t_{0}}f(t)=M(t)\) that
that is,
Then applying \(I_{\alpha}\) to the latter from s to t, we have
that is,
The proof of Lemma 2.2 is complete. □
3 α-Order conformable fractional differential equations with finite nonmonotone delay arguments
In this section, we deal with the differential equations of the form
where \(T_{\alpha}\) denotes the conformable differential operator of order \(\alpha\in(0,1]\), \(p_{i}(t)\), \(1\leqslant i\leqslant m\), are nonnegative functions, \(\tau_{i}(t)\), \(1\leqslant i\leqslant m\), are nonmonotone functions of positive real numbers such that
To prove our main results, we establish some fundamental results in this section.
Lemma 3.1
Assume that \(x(t)\) is an eventually positive solution of (3.1) and \(a_{r}(t,s)\), \(r\in\mathbb{N}^{+}\), is defined as
Then
Proof
Let \(x(t)\) be an eventually positive solution of equation (3.1). Then there exists \(t_{1}>t_{0}\) such that \(x(t)>0\) and \(x(\tau_{i}(t))>0\), \(1\leqslant i\leqslant m\), for all \(t\geqslant t_{1}\), so
This means that \(x(t)\) is monotonically decreasing, that is, \(x(\tau_{i}(t))\geqslant x(t)\), \(1\leqslant i\leqslant m\), and it is easy to put it into the original equation:
Dividing this equation by \(x(t)\), we get
that is,
Integrating the last inequality from s to t, \(0\leqslant s\leqslant t\), we get
that is,
So
that is, estimate (3.3) is valid for \(r = 1\). Supposing that (3.3) is established for \(r=n\), we obtain
so
Repeating these steps can, we obtain
that is, \(x(t)a_{n+1}(t,s)\leqslant x(s)\). So Lemma 3.1 is proved by mathematical induction. □
Lemma 3.2
Assume that \(x(t)\) is an eventually positive solution of (3.1) and
where
Then
where \(\lambda_{0}\) is the smaller root of the equation \(\lambda=e^{\beta\lambda}\).
Proof
Let \(x(t)\) be an eventually positive solution of equation (3.1). Then there exists \(t_{1}>t_{0}\) such that \(x(t)>0\) and \(x(\tau_{i}(t))>0\), \(1\leqslant i \leqslant m\), for all \(t\geqslant t_{1}\). Thus we can conclude from (3.1) that
This means that \(x(t)\) is monotonically decreasing and positive.
By (3.4), for any \(\varepsilon\in(0,\beta)\), there is \(t_{\varepsilon}\) such that
We will show that
where \(\lambda_{1}\) is the smaller root of the equation
For contradiction, we assume that
Therefore
Then for any \(\delta\in(0,\gamma)\), there exists \(t_{\delta}\) such that \(\frac{x(h(t))}{x(t)}\geqslant\gamma-\delta\) for \(t\geqslant t_{\delta}\). Dividing both sides of (3.1) by \(x(t)\), we have
Integrating the latter from \(h(t)\) to t, we obtain
or
so
Therefore
which implies
which is a contradiction to hypothesis (3.8). So (3.7) is true. Since (3.7) implies (3.6), the proof of Lemma 3.2 is complete. □
Theorem 3.1
Assume that (3.4) holds and for some r, we have
where \(h(t)\) is defined by (3.5), \(a_{r} (t,s)\) is defined by (3.2), and \(\lambda_{0}\) is the smaller root of the equation \(e^{\beta\lambda}=\lambda\). Then equation (3.1) oscillates.
Proof
If equation (3.1) has a solution \(x(t)\), then \(-x(t)\) is also a solution of equation (3.1), so we only consider the situation where a solution of (3.1) is eventually positive, that is, there is an integer \(t_{1}\geqslant t_{0}\) such that \(x(t)>0\) and \(x(\tau_{i}(t))>0\), \(1\leqslant i\leqslant m\), for all \(t\geqslant t_{1}\). By (3.1) we have
It is shown that \(x(t)\) is an eventually decreasing function.
By Lemma 3.2 inequality (3.6) holds. It can be easily seen that \(\lambda_{0}>1\), so for any real number \(0<\varepsilon\leqslant\lambda_{0}-1\), we have
Then there is \(t^{\ast}\in(h(t),t)\) satisfying
Then integrating from \(t^{\ast}\) to t equation (3.1) and substituting into (3.3), we have
Combining this with (3.10), we have
Dividing (3.1) by \(x(t)\), substituting into (3.3), and then integrating from \(h(t)\) to \(t^{\ast}\), we have
and because of \(T_{\alpha}^{t_{0}} x(t)<0\), we have
that is,
Adding (3.12) to (3.11), we get
This inequality holds for all \(0<\varepsilon\leqslant\lambda_{0}-1\), so as \(\varepsilon\rightarrow0\), we obtain
This is a contradiction to (3.9). The proof of Theorem 3.1 is complete. □
Lemma 3.3
Assume that \(x(t)\) is an eventually positive solution of (3.1) and that β and \(h(t)\) are defined by (3.4) and (3.5). Then
Proof
Assume that \(x(t)>0\) for \(t>T_{1}\geqslant t_{0}\). Then there exists \(T_{2}\geqslant T_{1}\) such that \(x(\tau_{i}(t))>0\), \(i=1,2,\dots,m\). In view of (3.1), \(T_{\alpha}^{t_{0}}x(t)\leqslant0\) on \([T_{2},\infty)\). Clearly, (3.13) holds for \(\beta=0\). If \(0<\beta\leqslant\frac{1}{e}\), then for any \(\varepsilon\in(0,\beta)\), there exists \(N_{\varepsilon}\) such that
For fixed ε, we will show that for each \(t>N_{\varepsilon}\), there exists \(\lambda_{t}\) such that \(h(\lambda_{t})< t<\lambda_{t}\) and
In fact, for a given \(t>N_{\varepsilon}\), \(f(\lambda):=\int^{\lambda}_{t}(\zeta-t_{0})^{\alpha-1}\sum_{i=1}^{m}p_{i}(\zeta)\,d\zeta\) is continuous. Because of \(\lim_{t\rightarrow\infty}h(t)=\infty\) and (3.14), we have \(\lim_{\lambda\rightarrow\infty}f(\lambda)>\beta-\varepsilon >0\). Hence there exists \(\lambda_{t}>t\) such that \(f(\lambda)=\beta-\varepsilon\), that is, (3.15) holds. From (3.14) we have
and therefore \(h(\lambda_{t})< t\).
Integrating (3.1) from t (\(>T_{3}=\max\{T_{2},N_{\varepsilon}\}\)) to \(\lambda_{t}\), we have
We see that \(h(t)\leqslant h(\zeta)\leqslant h(\lambda_{t})< t\) for \(t\leqslant y\leqslant\lambda_{t}\). Integrating (3.1) from \(\tau_{i}(\zeta)\) to t, we have that for \(t\leqslant\zeta\leqslant \lambda_{t}\),
From (3.16) and (3.17) we have
Noting the known formula
or
we have
Substituting this into (3.18), we have
Hence
and then
Substituting this into (3.19), we obtain
and hence
In general, we have
It is not difficult to see that if ε is small enough, then \(1\geqslant d_{n}>d_{n-1}\), \(n=2,3,\dots\) . Hence \(\lim_{n\rightarrow\infty}d_{n}=d\) exists and satisfies
that is,
Because of \(T_{\alpha}^{t_{0}}\leqslant0\), we have \(d<1\). Therefore, for all large t,
Letting \(\varepsilon\rightarrow0\), we obtain that
This shows that (3.13) holds. □
Theorem 3.2
Assume (3.4) holds and that for some r, we have
where \(h(t)\) is defined by (3.5), \(a_{r} (t,s)\) is defined by (3.2), and \(\lambda_{0}\) is the smaller root of the equation \(e^{\beta\lambda}=\lambda\). Then equation (3.1) oscillates.
Proof
If equation (3.1) has a solution \(x(t)\), then \(-x(t)\) is also a solution of equation (3.1), so we only consider the situation where a solution of (3.1) is eventually positive, that is, \(x(t)>0\) and \(x(\tau_{i}(t))>0\), \(1\leqslant i \leqslant m\), for all \(t\geqslant T_{3}\). By (3.1) we have
Integrating from \(h(t)\) to t the latter and substituting into (3.3), we have
Consequently,
which gives
and by (3.13) the last inequality leads to
which contradicts (3.20). The proof of the theorem is complete. □
Example 3.1
We consider the delay differential equation
where
By (3.5) we obtain
So \(h(t)=\max_{1\leqslant i\leqslant2}\{h_{i}(t)\}=h_{1}(t)\).
The functions \(F_{r}:\mathbb{N}\rightarrow\mathbb{R}^{+}\) are defined as \(F_{r}(t)=\int^{t}_{h(t)}\sum_{i=1}^{m}p_{i}(\zeta)a_{r}(h(t),\tau _{i}(\zeta))\,d\zeta\). When \(t=3k+2.6\), \(t\in\mathbb{N}\), for any \(r\in\mathbb{N}^{+}\), the function \(F_{r}(t)\) attains its maximum. In particular,
where
so
and therefore
Now we see that
The solution of \(\lambda=e^{\beta\lambda}\) is \(\lambda_{0}=1.435\), so we get
Therefore equation (3.21) satisfies the conditions of Theorems 3.1 and 3.2, and thus equation (3.21) oscillates.
4 Oscillation of 2α-order neutral conformable fractional differential equation
In this section, we deal with differential equations of the form
where \(T_{\alpha}\) denotes the conformable differential operator of order \(\alpha\in(0,1]\), \(\beta\geqslant1\) is a quotient of odd positive integers, and the functions r, p, q, τ, σ are such that \(r, p, q, \tau, \sigma\in C^{1}([t_{0},\infty),(0,\infty))\). We also assume that, for all \(t\geqslant t_{0}\), \(\tau(t)\leqslant t\), \(\sigma(t)\leqslant t\), \(T_{\alpha}^{t_{0}}\sigma(t)>0\), \(\lim_{t\rightarrow\infty}\tau(t)=\lim_{t\rightarrow\infty }\sigma(t)=\infty\), \(0\leqslant p(t)<1\), \(q(t)\geqslant0\), and q does not vanish eventually.
We further use the following notation:
Lemma 4.1
Let \(\beta\geqslant1\) be a ratio of two odd numbers. Then
Theorem 4.1
Assume that \(\pi(t)=\int_{t}^{\infty}(s-t_{0})^{\alpha-1}r(s)^{-1/\beta }\,ds<\infty\) and there exists a function \(\rho\in C^{1}([t_{0},\infty),(0,\infty))\) such that
Suppose that there exists a function \(\delta\in C^{1}([t_{0},\infty),(0,\infty))\) such that
where
and \((\varphi(t))_{+}:=\max\{0,\varphi(t)\}\). Then equation (4.1) oscillates.
Proof
Let \(x(t)\) be a nonoscillating solution of (4.1) on \([t_{0},\infty)\). Without loss of generality, we may assume that there exists \(t_{1}\geqslant t_{0}\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\sigma(t))>0\) for all \(t\geqslant t_{1}\). Then \(z(t)\geqslant x(t)>0\), and since
the function \([r(t)T_{\alpha}^{t_{0}}z(t)]^{\beta}\) is nonincreasing for all \(t\geqslant t_{1}\). Therefore \(T_{\alpha}^{t_{0}} z(t)\) does not change sign eventually, that is, there exists \(t_{2}\geqslant t_{1}\) such that either \(T_{\alpha}^{t_{0}} z(t)>0\) or \(T_{\alpha}^{t_{0}} z(t)<0\) for all \(t\geqslant t_{2}\).
Case I. Assume first that \(T_{\alpha}^{t_{0}} z(t)>0\) for all \(t\geqslant t_{2}\). Note that \(T_{\alpha}^{t_{0}}z(t)|_{t=\sigma(t)}=T_{\alpha}^{t_{0}}(z(\sigma(t)))\). Then
from which it follows that
Since \(x(t)\leqslant z(t)\), we see that
Put
Clearly, \(w(t)>0\). Applying \(T_{\alpha}^{t_{0}}\) to (4.9) and using (4.6) and (4.8), we obtain
where \(T_{\alpha}^{t_{0}}\rho_{+}(t)=\max\{T_{\alpha}^{t_{0}}\rho (t),0\}\). Set
By calculation letting \(v_{0}=(\sigma(t)-t_{0})^{(1-\alpha)\beta}\frac{1}{(\beta+1)^{\beta }}\frac{(T_{\alpha}^{t_{0}}\rho_{+}(t))^{\beta}}{ \rho^{\beta-1}(t)}\frac{r(\sigma(t))}{(T_{\alpha}^{t_{0}}\sigma (t))^{\beta}}\), we have that when
the function \(F(v)\) attains its maximum \(F(v_{0})\). So
Therefore
Applying \(I_{\alpha}\) to the last inequality from \(t_{0}\) to t, we have
Letting \(t\rightarrow\infty\) in this inequality, we get a contradiction to (4.3).
Case II. Assume now that \(T_{\alpha}^{t_{0}} z(t)<0\) for all \(t\geqslant t_{0}\). It follows from (4.1) that \(T_{\alpha}^{t_{0}}(r(T_{\alpha}^{t_{0}} z)^{\beta})<0\) for all \(s\geqslant t \geqslant t_{2}\), and thus
Dividing (4.10) by \((s-t_{0})^{1-\alpha}\) and then integrating from t to l, \(l\geqslant t\geqslant t_{2}\), we have
Letting \(l\rightarrow\infty\), we get
which implies that
Hence we conclude that
Using (4.12) in (4.5), we have
Define a generalized Riccati substitution by
By (4.11), \(w(t)\geqslant0\) for all \(t\geqslant t_{2}\). Applying \(T_{\alpha}^{t_{0}}\) to (4.14), we have
Let \(A:=w(t)/(\delta(t)r(t))\) and \(B=1/(r(t)\pi^{\beta}(t))\). Using Lemma 4.1, we conclude that
On the other hand, we get by (4.13) that \(T_{\alpha}^{t_{0}}z<0\) and from \(\sigma(t)\leqslant t\) that
Thus (4.15) yields
that is,
Denote \(C:=\beta/(\delta(t)r(t))^{1/\beta}\), \(D:=(\varphi(t))_{+}\), and \(v:=w(t)\). Applying inequality (4.2), we obtain
Applying \(I_{\alpha}\) to the latter inequality from \(t_{0}\) to t, we have
which contradicts (4.4). Therefore (4.1) oscillates. □
Example 4.1
We consider the equation
where \(p(t)=\frac{1}{5}\) and \(q(t)=(2+\frac{4\sqrt{2}}{5})t\). Let \(\rho(t)=1\) and \(\delta(t)=1/t\). Then we have
and it is obvious that (4.3) holds. Because of \(\varphi(t)=2/\sqrt{t}\), \(\psi(t)=(q_{0}(1-2\sqrt{2}p_{0}))/t=\frac{34}{25}\). So
and we can conclude that condition (4.4) is satisfied. Hence by Theorem 4.1 we deduce that (4.18) oscillates.
5 Oscillation of 3α-order damped conformable fractional differential equation
This section deals with oscillatory behavior of all solutions of the 3α-order nonlinear delay damped equation of the form
where \(0<\alpha\leqslant1\), and \(\beta\geqslant1\) is the ratio of positive odd integers. We further assume that the following conditions are satisfied:
- (H1):
-
\(r_{1},r_{2},p,q\in C(I,\mathbb{R}^{+})\), where \(I=[t_{0},\infty)\), \(\mathbb{R}^{+}=(0,\infty)\);
- (H2):
-
\(g\in C^{1}(I,\mathbb{R})\), \(T_{\alpha}^{t_{0}} g(t)\geqslant0\) and \(g(t)\rightarrow\infty\) as \(t\rightarrow\infty\);
- (H3):
-
\(f \in C(\mathbb{R},\mathbb{R})\) is such that \(xf(x)>0\) for \(x\neq0\), and \(f(x)/x^{\gamma}\geqslant k>0\), where γ is the ratio of positive odd integers.
We define
for \(t_{0}\leqslant t_{1}\leqslant t\leqslant\infty\) and assume that
and
A function y is called a solution of (5.1) if \(y, r_{1}(T_{\alpha}^{t_{0}} y)^{\beta}, r_{2}(r_{1}(T_{\alpha}^{t_{0}} y)^{\beta})\in C^{1}([t_{y},\infty),\mathbb{R})\) and y satisfies (5.1) for \([t_{y},\infty)\) for some \(t_{y}\geqslant t_{0}\).
For brevity, we define
on I. Then (5.1) can be written as
The purpose of this section is to ensure that any solution of (5.1) oscillates when the related second-order linear ordinary fractional differential equation without delay
is nonoscillatory.
Next, we state and prove the following lemmas.
Lemma 5.1
Let y be a nonoscillatory solution of (5.1) on I. Suppose (5.4) is nonoscillatory. Then there exists \(t_{2}\in[t_{1},\infty)\) such that \(y(t)L_{1}y(t)>0\) or \(y(t)L_{1}y(t)<0\), \(t\geqslant t_{2}\).
Proof
Let y be a nonoscillatory solution of (5.1) on \([t_{1},\infty)\), say \(y(t)>0\) and \(y(g(t))>0\) for \(t\geqslant t_{1}\geqslant t_{0}\). Let \(x=-L_{1}y(t)\). By (5.1) we have
Let \(u(t)\) be a positive solution of (5.4), say \(u(t)>0\) for \(t\geqslant t_{1}\geqslant t_{0}\). If x is oscillatory, then x has consecutive zeros at a and b (\(t_{1}< a< b\)) such that \(T_{\alpha}^{t_{0}} x(a)\geqslant0\), \(T_{\alpha}^{t_{0}} x(b)\leqslant0\), and \(x(t)>0\) for \(t\in(a,b)\). Then we obtain
which yields a contradiction. This completes the proof. □
Lemma 5.2
If y is a nonoscillatory solution of (5.1) and \(y(t)L_{1}y(t)>0\), \(t\geqslant t_{1}\geqslant t_{0}\), then
and
Proof
If y is a nonoscillatory solution of (5.1), then \(y(t)>0\), \(y(g(t))>0\), and \(L_{1}y(t)>0\) for \(t\geqslant t_{1}\geqslant t_{0}\). It is easy to see that
which implies that \(L_{2}y(t)\) is nonincreasing on \([t_{1},\infty)\). Applying \(I_{\alpha}\) to \(T_{\alpha}^{t_{0}}L_{1}y(t)=\frac{L_{2}y(t)}{r_{2}(t)}\) from \(t_{1}\) to t and Lemma 2.2, we get
Then
Now, applying \(I_{\alpha}\) to the last inequality from \(t_{1}\) to t, we can obtain from Lemma 2.2 that
This completes the proof. □
In the following two lemmas, we consider the second-order delay differential inequality
where the function \(r_{2}\) is as in (5.1), \(Q(t)\in C(I,\mathbb{R}^{+})\), and \(h(t)\in C^{1}(I,\mathbb{R})\) is such that \(h(t)\leqslant t\), \(T_{\alpha}^{t_{0}} h(t)\geqslant0\) for \(t\geqslant t_{0}\), and \(h(t)\rightarrow\infty\) as \(t\rightarrow\infty\).
Lemma 5.3
If
then all bounded solutions of (5.7) are oscillatory.
Proof
Let \(x(t)\) be a bounded nonoscillatory solution of (5.7), say \(x(t)>0\) and \(x(h(t))>0\) for \(t\geqslant t_{1}\) for some \(t_{1}\geqslant t_{0}\). By (5.7), \(r_{2}T_{\alpha}^{t_{0}} x(t)\) is strictly increasing on \([t_{1},\infty)\). Hence, for any \(t_{2}\geqslant t_{1}\), applying \(I_{\alpha}\) from \(t_{2}\) to t in \(T_{\alpha}^{t_{0}} x(t)=\frac{r_{2}(t)T_{\alpha}^{t_{0}} x(t)}{r_{2}(t)}\) and Lemma 2.2 yield
so \(T_{\alpha}^{t_{0}} x(t_{2})<0\), as otherwise (5.3) would imply \(x(t)\rightarrow\infty\) as \(t\rightarrow\infty\), a contradiction to the boundedness of x. Altogether,
Now, for \(v\geqslant u\geqslant t_{1}\), repeating the previous steps, we have
For \(t\geqslant s\geqslant t_{1}\), setting \(u=h(s)\) and \(v=h(t)\) in (5.9), we get
Applying \(I_{\alpha}\) to (5.7) from \(h(t)\geqslant t_{1}\) to t, we obtain from Lemma 2.2 that
that is,
Taking lim sup as \(t\rightarrow\infty\) on both sides of this inequality yields a contradiction to (5.8). This completes the proof. □
Lemma 5.4
If
then all bounded solutions of (5.7) are oscillatory.
Proof
Let x be a bounded nonoscillatory solution of (5.7), say \(x(t)>0\) and \(x(h(t))>0\) for \(t\geqslant t_{1}\) for some \(t_{1}\geqslant t_{0}\). As in Lemma 5.1, we obtain
Applying \(I_{\alpha}\) to(5.7) from \(u\geqslant t_{1}\) to t, we obtain from the previous forms that
so
We obtain from (5.11) that
that is,
Taking lim sup as \(u, t\rightarrow\infty\) on both sides of this inequality yields a contradiction to (5.10). This completes the proof. □
Theorem 5.1
Assume that (5.2) and (5.3) hold and \(\beta\geqslant \gamma\). Suppose that there exist two functions \(m,h\in C^{1}(I, \mathbb{R})\) such that
satisfying
and for \(t\geqslant t_{1}\),
and that (5.8) or (5.10) holds with
with \(c,c^{*}> 0\). Then every solution y of (5.1) and \(L_{2}y(t)\) are oscillatory.
Proof
Let y be a nonoscillatory solution of (5.1) on \([t_{1},\infty)\), \(t_{1}\geqslant t_{0}\). We assume that \(y(t)>0\) and \(y(g(t))>0\) for \(t\geqslant t_{1}\). From Lemma 5.1 we have \(L_{1}y(t)<0\) or \(L_{1}y(t)>0\) for \(t\geqslant t_{1}\).
Step 1. We assume that \(L_{1}y(t) > 0\) on \([t_{1},\infty)\). By (5.1) \(L_{2}y\) is strictly decreasing. Hence, for any \(t_{2}\geqslant t_{1}\), we have from Lemma 2.2 that
So \(L_{2}y(t_{2})>0\) as otherwise (5.3) would imply \(L_{1}y(t)\rightarrow-\infty\) as \(t\rightarrow\infty\), a contradiction to the positivity of \(L_{1}y\). Altogether, \(L_{2}y>0\) on \([t_{1},\infty)\).
Define the following generalized Riccati transformation:
By the product and quotient rules, α-differentiating w, we obtain
Using (5.1), (5.5), and assumption (H3) on f, we obtain
By the definition of \(L_{1}y(t)\) and (5.5) we obtain
Then
and we obtain
Since \(L_{3}y(t)<0\), we have \(0 < L_{2}y(t)\leqslant L_{2}y(t_{1})\), \(L_{2}y(t_{1})=c_{1}\) for \(t\geqslant t_{1}\). Then
and thus we get from Lemma 2.2 that
(note that \(L_{1}y(t_{1})>0\)), where
Therefore, we get for all \(t\geqslant t_{2}\) that
(note that \(y(t_{2})>0\)), where
Then we get
Using (5.16) in (5.17), we get
Then
Using (5.16) and (5.18) in (5.15), we get
where \(c^{*}=\gamma c_{2}^{\gamma-\beta}\), and A and B are as in (5.13). Applying \(I_{\alpha}\) to (5.19) from \(t_{0}\) to t, we get
which contradicts (5.12).
Step 2. Let \(L_{1}y(t) < 0\) on \([t_{1},\infty)\). We consider the function \(L_{2}y(t)\). The case \(L_{2}y(t)\leqslant0\) cannot hold for all large t, say \(t\geqslant t_{2}\geqslant t_{1}\), since by double integration of
we get from (5.2) that \(y(t) \leqslant0\) for all large t, which is a contradiction. Thus we assume that \(y(t)>0\), \(L_{1}y(t)<0\), and \(L_{2}y(t)\geqslant0\) for all large t, say \(t\geqslant t_{3}\geqslant t_{2}\). Now, for \(v\geqslant u\geqslant t_{3}\), we have
Letting \(u=g(t)\) and \(v=h(t)\), we obtain
where \(x(t)=(-L_{1}y(t))^{1/\beta}>0\) for \(t\geqslant t_{3}\). By (5.1), since that \(x(t)\) is decreasing and \(g(t)\leqslant h(t)\leqslant t\), we get
where \(z(t)= x^{\beta}(t)\). Because \(z(t)\) is decreasing and \(\beta\geqslant\gamma\), there exists a constant \(c_{4}>0\) such that \(z^{\gamma/\beta-1}(t)\geqslant c_{4}\) for \(t\geqslant t_{2}\). Then we have
Proceeding exactly as in the proofs of Lemmas 5.3 and 5.4, we arrive at the desired conclusion, thus completing the proof. □
Example 5.1
where \(r_{1}(t)=t^{-\frac{3}{2}}\), \(r_{2}(t)=1\), \(q(t)=\frac{1}{2}(t-2)^{-2}t^{-\frac{1}{2}}+2(t-2)^{-2}t^{-1}+1\), \(p(t)=t^{-\frac{5}{2}}\), \(g(t)=t-2\), \(h(t)=t-2\), \(\alpha=\frac{1}{2}\), \(\beta=1\), \(\gamma=1\), \(c^{\ast}=1\). By taking \(m(t)=1\) we get
so
and we obtain that
Hence
So
Then we see that (5.8) and (5.10) are clearly satisfied, and it is easy to verify that the equation
is nonoscillatory, and one nonoscillatory solution of (5.21) is \(z(t)=18t^{\frac{1}{3}}\). Then we get that equation (5.20) is oscillatory.
Example 5.2
where \(r_{1}(t)=r_{2}(t)=t^{-\frac{1}{2}}\), \(p(t)=2t^{-\frac{1}{2}}\), \(q(t)=3\), \(k=1\), \(g(t)=t\), \(\alpha=\frac{1}{2}\), \(\beta=\gamma=1\), \(c^{\ast}=c=1\). Letting \(m(t)=1\) and \(h(t)=t\), we can obtain
so all conditions except (5.12) are satisfied.
Equation (5.22) can be rewritten as
It is obvious that the equation is nonoscillatory. It has a nonoscillatory solution \(x=e^{\frac{1}{2}t}\cos\frac{\sqrt{2}}{2}t\). We can obtain that condition (5.12) indispensable.
6 Conclusion
In this paper, we study three kinds of different order conformable fractional equations and obtain oscillatory results of three equations. Those results unify the oscillation theory of the integral-order and fractional-order differential equations.
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Funding
This research is supported by the Natural Science Foundation of China (61703180, 61803176), and supported by Shandong Provincial Natural Science Foundation (ZR2016AM17, ZR2017MA043).
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Feng, L., Sun, S. Oscillation theorems for three classes of conformable fractional differential equations. Adv Differ Equ 2019, 313 (2019). https://doi.org/10.1186/s13662-019-2247-5
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DOI: https://doi.org/10.1186/s13662-019-2247-5
MSC
- 34C10
- 26A33
- 65Q10
Keywords
- Oscillation
- Conformable fractional calculus
- Differential equation