Forced oscillation of conformable fractional partial delay differential equations with impulses

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.


Introduction.
In recent years, many researchers found that fractional differential equations are more accurate in describing the mathematical modeling of systems and processes in the field of chemical processes, electrodynamics of computer medium, polymer rheology, mathematical biology, etc. The applications of fractional calculus to biomedical problems are done in the areas of membrane biophysics and polymer viscoelasticity, where the experimentally observed power law dynamics for current-voltage and stress-strain relationships are concisely captured by fractional order differential equations. But the most frequently used definitions involve integration which is nonlocal: Riemann-Liouville derivative & Caputo derivative [5,9,13,22,27]. Fractional calculus is the study of derivatives and integrals of non-integer order and is the generalized form of classical derivatives and integrals. Those fractional derivatives in the fractional calculus have seemed complicated and lacked some basic properties, like the product rule and the chain rule. But in 2014, Khalil et al. [12] introduced a new fractional derivative called the conformable derivative which closely resembles the classical derivative.
In order to describe dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so forth, some authors have used impulsive differential systems to describe the models. For the basic theory on impulsive differential equations, the reader can refer to the monographs and references [2,16,17,20]. The study of the qualitative behavior of partial differential equations has rapidly expanded in the last few decades, see for example [11,14,15,24,25,26,29,30,32] and the references they are cited. In particular, the problem of interval oscillation criteria for integer and fractional order impulsive differential equations have been investigated by few authors, we refer the reader to [3,4,19,28,31] and the references cited therein.
Recently, the theory of fractional differential equations has been intensively studied by many authors. For example, we mention to the problem of anomalous diffusion [7,8], the nonlinear oscillation of earthquake which can be modeled with fractional derivative [6], and fluid-dynamic traffic model with fractional derivatives [10] also can be used to eliminate the deficiency arising from the assumption to continuum traffic flow and many other, see also [18,23] and the references they are cited for recent developments in the description of anomalous transport by fractional dynamics. Following this trend, our aim in this paper is to study oscillation properties of partial differential equation of fractional order of the form where Ω is a bounded domain in R N with a piecewise smooth boundary ∂Ω, ∆ is the Laplacian in the Euclidean space R N and R + = [0, +∞), and ∂ α ∂t α denotes the conformable partial fractional derivative of order α, 0 < α ≤ 1. Equation (1.1) is the enhancement with the boundary condition where γ is the outer surface normal vector to ∂Ω and µ(x, t) ∈ C(∂Ω × [0, +∞), [0, +∞)).
In this paper, we assume that the following hypotheses are satisfied: , a j (t) ∈ P C (R + , R + ), j = 1, 2, . . . , m, where P C represents the class of functions which are piecewise continuous in t with discontinuities of first kind only at t = t k , k = 1, 2, . . . , and left continuous at t = t k , k = 1, 2, . . . . (H 4 ) u(x, t) and its derivative ∂ α ∂t α u(x, t) are piecewise continuous in t with discontinuities of first kind only at t = t k , k = 1, 2, . . . , and left continuous at For simplicity, we denote For two constants c, d = t k with c < d and a function ψ ∈ C([c, d], R), we define the operator Θ : .
The paper is organized as follows: in Section 2, we present some definitions and results that will be needed later. In Section 3, we establish some interval oscillation criteria for the problem (1.1)-(1.2). An example to illustrate our main results is given. To the best of authors' knowledge there has been no work done on the interval oscillation of conformable fractional impulsive partial delay differential equations.

Preliminaries.
In this section, we present the basic definitions and the basic lemma that will be used in the proof of the main results.
is said to be oscillatory in the domain G, if it has arbitrary large zeros. Otherwise it is non-oscillatory.
Conformable fractional derivatives have the following properties: For convenience, we introduce the following notations: The following lemma will be the basic tool in proving the main results.
Lemma 2.1. If the impulsive conformable fractional differential inequality has no eventually positive solution, then every solution of the problem (1.1)- Proof. Assume that there exists a nonoscillatory solution u(x, t) of the problem (1.1)-(1.2) and u(x, t) > 0. By the assumptions, there exists By multiplying both sides of the main equation in (1.1) by 1 |Ω| and integrating with respect to x over the domain Ω, we obtain t ≥ t 1 and t = t k , From Green's formula and boundary condition (1.2) we see that where dS is the surface element on ∂Ω. Moreover by (H 1 ), it follows that for i = 1, 2, . . . , n, In view of (2.3)-(2.6), we obtain For t ≥ t 0 , t = t k , k = 1, 2, . . . , from the boundary conditions we see that Multiplying both sides by 1 |Ω| , integrating with respect to x over the domain Ω, and using (H 5 ), we obtain Hence we see that Y (t) is an eventually positive solution of (2.2). This contradicts our assumption and complete the proof.

Main Results.
In this section, we establish some new interval oscillation criteria for (1.1) by using the Riccati transformation technique and integral average method. For simplicity, we define for I(c l ) < I(d l ), l = 1, 2, then every solution of the problem (1.1)-(1.2) is oscillatory.
Proof. Assume to the contrary that u(x, t) is a non-oscillatory solution of (1.1)-(1.2). Without loss of generality we may assume that u(x, t) is an eventually positive solution of (1.1)-(1.2). Then there exists t 1 ≥ t 0 such that u(x, t) > 0 for t ≥ t 1 . Therefore, from (2.2) it follows that Thus T α (Y (t)) ≥ 0 or T α (Y (t)) < 0, t ≥ t 1 , for some t 1 ≥ t 0 . We now claim that Suppose not, then T α (Y (t)) < 0 and there exists where c > 0 is a constant. For t ∈ [t 2 , ∞), after integrating the above inequality from t 2 to t, we have Letting t → ∞, we get lim t→+∞ Y (t) = −∞. This contradiction shows that (3.3) holds. Define the Riccati transformation It follows from (2.2) that w(t) satisfies By assumption (H 6 ), we can choose 2), we can easily see that For t = t k , k = 1, 2, . . . , we have First, we consider the case when I(c 1 ) < I(d 1 ). In this case, all the impulsive moments in [c 1 , d 1 ] are t I(c 1 )+1 , t I(c 1 )+2 , . . . , t I(d 1 ) . Choose v(t) ∈ L v (c 1 , d 1 ).
Multiplying both sides of (3.5) by v 2 (t) and integrating the resulting inequality from c 1 to d 1 , we obtain Using integration by parts on the left-hand side, and noting that v(c 1 ) = v(d 1 ) = 0, we get (3.7) We consider several cases to estimate Y (t−σ) Y (t) .
, since t k+1 − t k > σ, we consider two subcases: and there are no impulsive moments in (t − σ, t). Then, for any t ∈ [t k + σ, t k+1 ] we have From this, Integrating it from t − σ to t, we have and there is an impulsive moment t k in (t − σ, t). Similarly to Case 1.1, we obtain Integrating it from t − σ to t, we get For any t ∈ (t k , t k + σ) we have Using the impulsive conditions in equation (1.1), we get Using Tα(Y (t k )) That is, .

Case 2.2:
and there is an impulsive moment t I(c 1 ) in (t−σ, t). Making a similar analysis as in Case 1.2, we have and there are no impulsive moments in (t − σ, t). Working as in Case 1.1, we get Case 3: For t ∈ (t I(d 1 ) , d 1 ] we consider three subcases: and there are no impulsive moments in (t − σ, t). Using a similar analysis as in Case 2.1, we have

Case 3.2:
If t I(d 1 ) + σ < d 1 and t ∈ [t I(d 1 ) , t I(d 1 ) + σ), then t − σ ∈ [t I(d 1 ) − σ, t I(d 1 ) ) and there is an impulsive moment t I(d 1 ) in (t − σ, t). Using a similar analysis as in Case 2.2, we obtain Proceeding as in Case 3.2, we get Combining all these cases, we have Hence, and since r(t)g(T α (Y (t))) is non-increasing in (c 1 , t I(c 1 )+1 ], by (3.7) we have (3.10) Thus Letting t → t − I(c 1 )+1 , it follows that Similarly, we can prove that on (t k−1 , t k ], k = I(c 1 ) + 2, . . . , I(d 1 ), Hence, from (3.11) and (3.12), we have (3.13) Thus we have Therefore, using (3.10), we get Similarly to the proof of (3.10), we obtain This again contradicts our assumption. Finally, if u(x, t) is eventually negative, we can consider [c 2 , d 2 ] instead of [c 1 , d 1 ] and get a contradiction. The proof is complete.
Next, we establish new oscillation criteria for (1.1)-(1.2) by using the integral average method used in [21] for ordinary differential equations. Let D = {(t, s) : t 0 ≤ s ≤ t}, then the functions H 1 , H 2 ∈ C(D, R) are said to belong to the class H if (H 7 ) H 1 (t, t) = H 2 (t, t) = 0, H 1 (t, s) > 0, H 2 (t, s) > 0 for t > s and (H 8 ) H 1 and H 2 have partial derivatives ∂H 1 ∂t and ∂H 2 ∂s on D such that where h 1 , h 2 ∈ L loc (D, R). We put Applying integration by parts on the R.H.S of first integral we get, As in the proof of Theorem 3.1, we divide the interval [c 1 , λ 1 ] into several parts and calculate the estimation of Y (t − σ)/Y (t), to obtain On the other hand, multiplying both sides of (3.5) by H 2 (d 1 , t), integrating from λ 1 to d 1 and following a similar procedure as above, we get which is a contradiction to condition (3.14). In the case when u(x, t) < 0, we take the interval [c 2 , d 2 ] instead of the interval [c 1 , d 1 ] and proceeding as in the proof of this case, we get a contradiction. The proof is complete.