Abstract
In this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.
1 Introduction
Fractional differential equations are generalizations of classical differential equations of integer order and have recently proved to be valuable tools in the modelling of many phenomena in various fields of science and engineering. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, viscoelasticity, chemical physics, electrical networks, fluid flows, control, dynamical processes in self-similar and porous structures, etc.; see, for example, [1–6]. Fractional derivatives have appeared in lots of work where they are used for better descriptions of material properties. Mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.This growth in use has been caused by the intensive development of the theory of fractional calculus itself and its applications. The books on the subject of fractional integrals and fractional derivatives by Diethelm [7], Miller and Ross [8], Podlubny [9] and Kilbaset al. [10] summarise and organise much of the field of fractional calculus including many of the theories and applications of fractional differential equations. Many papers have studied some aspects of fractional differential equations. Most have focused on the existence of, methods for defining or stability of the solutions (or positive solutions) to nonlinear initial (or boundary) value problems for fractional differential equations (or systems) using nonlinear analysis techniques (fixed-point theorems, Leray-Schauder theory). We refer to [11–21] and the references cited therein.
Recently, research on the oscillation of various equations including differential equations, difference equations and dynamic equations on time scales, has been a hot topic the literature. A lot of effort has been committed to establishing new oscillation criteria for these equations; see the monographs [22, 23]. In these investigations, we notice that very little attention has been paid to the oscillation of fractional differential equations.
In 2006, a definition for a fractional derivative called the modified Riemann-Liouville derivative, was suggested by Jumarie [24] and its application have subsequently been studied by many researchers [25–28].
Recently, Qin and Zheng [29] established oscillation criteria for linear fractional differential equations with damping term of the form:
where
Now, in this paper, we are concerned with the oscillation of fractional differential equations with damping term in the form of:
where:
Some of the key properties of Jumarie’s modified Riemann-Liouville derivative of order are listed as follows:
where f (t, ξ, α) = (t − ξ)−α (f (ξ) − f(0)).
As usual, a solution x (t) of (1) is called oscillatory if it has arbitrarily large zeros, otherwise it is called non-oscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.
In the rest of this paper, we denote for the sake of convenience:
ξ = tα /Γ (1 + α);
Let h1, h2, H ∈ C ([ξ0, ∞), R) satisfy
H has continuous partial derivatives ∂H (ξ, s) /∂ξ and ∂H (ξ, s) /∂s on [ξ0, ∞) such that
This paper is organized next as follows: in Section 2, we establish new oscillation criteria for (1) using the Riccati transformation, inequalities and the integration average technique and in Section 3, we present some examples that apply the results established. Finally, we give a conclusion.
2 Oscillatory criteria
Assume x (t) is an eventually positive solution of (1), and
Then, there exist a sufficiently large T such that
Suppose x (t) is an eventually positive solution of (1). Let a (t) = ã̃ (ξ), r(t) = r̃ (ξ), x (t) = x̃ (ξ), p (t) = p̃ (ξ), q (t) = q̃ (ξ) where ξ = tα /Γ (1 + α). Then by using (5), we obtain
Similarly we have
Then x̃ (ξ) is an eventually positive solution of (13), and there exists ξ1 >ξ0 such that x̃ (ξ) > 0 on [ξ1, ∞). So, f (x̃ (ξ)) > 0 and we have
Therefore, we get
Then, A(ξ) ã(ξ)(r̃(ξ)x̃′(ξ))′ is strictly decreasing on [ξ1, ∞) , thus we know that (r̃(ξ)x̃′(ξ))′ is eventually of one sign. For ξ2 > ξ1 is sufficiently large, we claim (r̃(ξ)x̃′(ξ))′ > 0 on [ξ2, ∞). Otherwise, assume that there exists a sufficiently large ξ3 > ξ2 such that (r̃(ξ)x̃′(ξ))′ < 0 on [ξ3, ∞). Thus, (r̃(ξ)x̃′(ξ))′ is strictly decreasing on [ξ3, ∞), and we get that
Therefore, we get
By (9), we have limξ→∞r̃(ξ) x̃′(ξ) = −∞. So there exists a sufficiently large ξ4 > ξ3 such that x̃′(ξ) < 0, ξ ∈ [ξ4, ∞). Then, we have
and so
By (10), we deduce that limξ→∞x̃(ξ) = −∞, which contradicts the fact that x̃ (ξ) is an eventually positive solution of (13). Thus, r̃(ξ)x̃(ξ)) > 0 on [ξ2, ∞) , and then
substitutingξ with in (17), and integrating it with respect to s from ξ to ∞ yields
which means
substituting ξ with τ in (19), and integrating it with respect to τ from ξ to ∞ yields
where ϰ = A(τ) ã(τ). That is,
substituting ξ with ζ in (21), and integrating it with respect to ζ from ξ5 to ξ yields
By (11), we have limt→∞x̃ (ξ) = −∞, which causes a contradiction. So, the proof is complete. □
Assume that x(t)is an eventually positive solution of (1) such that
on [t1, ∞), where t1 > t0is sufficiently large. Then, for t ≥ t1, we have
Assume that x is an eventually positive solution of (1). So, by (14), we obtain that A(ξ) ã (ξ) r̃ (ξ) x̃′ (ξ)) is strictly decreasing on [ξ1, ∞). Then,
and so
multiplying both sides of (26) by 1/r(t), we obtain
On the other hand, we have
Using (26), we obtain
That is
So, the proof is complete. □
Assume that (9)-(11) hold and f(x)/x ≥ k > 0 for all x ≠ = 0. If there exists ϕ ∈ Cα ([t0, ∞) , R+) such that for any sufficiently large T ≥ ξ0, there exist a, b, c with T ≤ a < c < b satisfying
where k ∈ ℝ+, ϕ̃(ξ)
Suppose the contrary that x(t) is non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t1, ∞), where t1 is sufficiently large. By Lemma 1, we have
For t ∈ [t2, ∞) , we have
So,
If we use
and so,
Using f (x(t)) /x(t) ≥ k,
Now, let ω (t) = ω̃ (ξ). Then we have
We can choose a, b, c arbitrary in [ξ2, ∞) with b >c > a. Substituting ξ with s, we multiply both sides of (33) by H (ξ, s ) and integrating it with respect to s from c to ξ for ξ ∈ [c, b), then we get that
using the method of integration by parts
where
Letting ξ → b− in (34) and dividing both sides by H (ξ, c), we obtain,
On the other hand, substituting ξ with s, multiplying both sides of (33) by H (s, ξ) and integrating it with respect to s from ξ to c for ξ ∈ (a, c ], we deduce that
Letting ξ → a+ in (36) and dividing both sides of it by H(c, ξ) and we obtain
A combination of (35) and (37) yields the inequality
which contradicts (29). Thus, the proof is complete. □
Under the conditions of Theorem 4, if for any sufficiently large l ξ0,
then (1) is oscillatory.
For any sufficiently large T ≥ ξ0, let a = T. If we choose l = a in (39), then there exists c > a such that
If we choose l = c > a in (40), then there exists b > c such that
Finally, we combine (41) and (42), to obtain (29). Thus, the proof is complete from Theorem 4. □
If we choose H(ξ, s) = (ξ − s), ξ s ξ0, where λ > 1 is a constant in Theorem 4 and Theorem 5, then we obtain the following corollaries.
Under the conditions of Theorem 4, if for any sufficiently large T ≥ ξ0, there exist a, b, c with T ≤ a < c < b satisfying
then (1) is oscillator
Under the conditions of Theorem 5, if for any sufficiently large l ≥ ξ0,
then (1) is oscillatory.
If (9)-(11) hold, ffi is defined as in Teorem 4 and
Then every solution of (1) is oscillatory or satisfies limt→∞x(t) = 0.
Suppose the contrary that x(t) is a non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t1, 1) , where t1 is suffciently large. By Lemma 1, we have
and thus,
Substituting ξ with s in (46) and integrating it with respect to s from ξ2 to ξ, then we get that
which contradicts (45). So, the proof is complete. ⎕
Assume (9)-(11) hold, and there exists a function G ∈ C([ξ0, ∞),ℝ) such that G(ξ,ξ) = 0, for ξ ≥ ξ0, G(ξ, s) ≥ 0, for ξ > sξ0, and G has a non-positive continuous partial derivative Gs (ξ, s).Ifϕ̃ is defined as in Theorem 4 and
where
Suppose the contrary that x(t) is a non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t1, ∞), where t1 is suffciently large. By Lemma 1, we have
Substituting ξ with s in (49), multiplying both sides by G(ξ, s) and then integrating it with respect to s from ξ2 to ξ, we get that
and thus,
Then,
and
So,
which contradicts (48). So the proof is complete.
3 Applications of the results
Consider the nonlinear fractional differential equation with damping term
This corresponds to (1) with t0 = 2;
which implies limξ→∞δ̃ (ξ, ξ2) = ∞, and so, (9) holds. Then, there exists a suffciently large T > ξ2such thatδ̃1(ξ, ξ2) > 1 on [T, ∞). In (10),
In (11),
Letting ff (ξ) =ξ in (45),
Consider the nonlinear fractional di˙erential equation with damping term
This corresponds to (1) with t0 = 2;
which implies limξ→∞δ̃1(ξ, ξ2) = ∞ and so (9) holds. Then, there exists a sufficiently large T > ξ2such thatδ̃1 (ξ, ξ2) > 1
on [T ∞). In(10),
In (11),
Letting ϕ(ξ) = 1 andλ = 2 in (44), for any suffciently large l, we have
So (44) holds, and then we deduce that (56) is oscillatory by Corollary 2.
4 Conclusion
In this paper, we are concerned with the oscillation of solutions to nonlinear fractional di˙erential equations with a damping term. Based on the variable transformation used in ξ, the fractional di˙erential equations are converted into another di˙erential equation of integer order. Then, some new oscillation criteria for the equations are established by using inequalities, the integration average technique and the Riccati transformation. Consequently, it can be seen that this approach can also be applied to the oscillation of other fractional di˙erential equations involving the modified Riemann-Liouville derivative.
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