Oscillation Criteria for Some Higher Order Integrodynamic Equations on Timescales

and Applied Analysis 3 Integrating (15) (n − 1)-times from t 1 to t and then using Lemma 3, we obtain (rxΔ) (t) ≤ h0 (t, t1) (rx Δ) (t 1 ) + h 1 (t, t 1 ) (rxΔ) Δ (t 1 ) + ⋅ ⋅ ⋅ + h n−2 (t, t 1 ) (rxΔ) Δ n−2 (t 1 ) + kh n−1 (t, t 1 ) + ∫ t


Introduction
Integrodynamic equations on timescales are an important topic with applications in many physical systems. For general basic ideas and background, we refer to [1]. Oscillation results of integral equations of Volterra type are scant and only few results exist on this subject. Related studies can be found in [2][3][4][5][6]. In this paper, we investigate the oscillatory behavior of the solutions of some higher order integrodynamic equations on timescale in the form To the best of our knowledge, there appear to be no such results on the oscillation of (1). Therefore, our main goal here is to initiate such a study by establishing some new criteria for the oscillation of (1) and other related equations. This work is an extension to the analysis done in [7]. The nonoscillatory behavior for some higher order integrodynamic equations was studied recently in [8].
We take ⊆ to be an arbitrary timescale with 0 ∈ and Sup = ∞.

Auxiliary Results
We employ the following lemmas.
Lemma 1 (see [9]). If and are nonnegative, then where equality holds if and only if = .

Main Results
In this section we present the following main results.
Proof. Let ( ) be a nonoscillatory solution of (1). Hence either ( ) is eventually positive or ( ) is eventually negative. First assume ( ) is eventually positive. Fix 0 ≥ 0 and suppose ( ) > 0 for ≥ 1 for some 1 ≥ 0 . From (1), we see that Let By assumption (3), we have Hence, from (12), we get Integrating (15) ( − 1)-times from 1 to and then using Lemma 3, we obtain From the properties of the functions ℎ and the definition of the function −1 ( , 1 ) for all 1 ≥ 0, we get where Dividing (17) by ( ) and hence integrating from 1 to we obtain Dividing (19) by ( , 0 ) and taking lim inf of both sides of (19) as → ∞, we obtain a contradiction to the fact that ( ) > 0 for ≥ 1 . The proof of the case when ( ) is eventually negative is similar. This completes the proof.
From the proof of Theorem 4, one can easily extract the following result on the asymptotic behavior of the nonoscillatory solutions of (1).

Theorem 5. Let conditions (I) and (II) hold with 2 = 0 and suppose
for all 0 ≥ 0. If ( ) is nonoscillatory solution of (1), then Next, we present the following result.
Proof. Let ( ) be a bounded nonoscillatory solution of (1) and assume that ( ) is eventually positive. Fix 0 ≥ 0 and suppose ( ) > 0 for ≥ 1 for some 1 ≥ 0 and ( ) ≤ 1 for some constant 1 > 0. From (1), we have Proceeding as in the proof of Theorem 4, we get (14). Thus, The rest of the proof is similar to that of Theorem 4 and hence it is omitted.

Theorem 7. Let conditions (I) and (II) hold with > 1 and = 1 and suppose that conditions (11) hold and
Proof. Let ( ) be a nonoscillatory solution of (1), Hence either ( ) is eventually positive or ( ) is eventually negative. First, assume ( ) is eventually positive. Fix 0 ≥ 0 and assume ( ) > 0 for ≥ 1 for some 1 ≥ 0 . Using conditions (I) and (II) with > 1 and = 1 in (1), we have Abstract and Applied Analysis for ≥ 1 . Proceeding as in the proof of Theorem 4, we get (14) and hence By applying (5) with = , we obtain Using (29) in (26), we find where ( ) = ( − 1) Integrating (30) -times from 1 to and then using Lemma 3, we have where is given in (18). The rest of the proof is similar to that of the proof of Theorem 4 and hence is omitted.
Proof. Let ( ) be a nonoscillatory solution of (1). First, assume ( ) is eventually positive. Fix 0 ≥ 0 and suppose ( ) > 0 for ≥ 1 for some 1 ≥ 0 . Using conditions (I) and (II) with = 1 and < 1 in (1), we find where is defined as in the proof of Theorem 4. By applying (6) with = , we obtain Using (37) in (35), we find The rest of the proof is similar to the proof of Theorem 4 and hence is omitted. Abstract and Applied Analysis 5 Next, we present the following result with different nonlinearities, that is, with > 1 and < 1.

Theorem 9. Let conditions (I) and (II) hold with > 1 and
< 1 and suppose that there exists a positive rd-continuous function : → + such that for all 0 ≥ 0, where If conditions (11) hold for all 0 ≥ 0, then (1) is oscillatory.
Proof. Let ( ) be a nonoscillatory solution of (1). First, assume ( ) is eventually positive. Fix 0 ≥ 0 and suppose ( ) > 0 for ≥ 1 for some 1 ≥ 0 . Using conditions (I) and (II) in (1), we have As in the proof of Theorems 7 and 8, one can easily find The rest of the proof is similar to that of Theorem 4 and hence is omitted.
For the cases when both functions 1 and 2 are superlinear, that is, > > 1, or sublinear, that is, 1 > > > 0, we present the following result.
Proof. Let ( ) be a nonoscillatory solution of (1). First, assume ( ) is eventually positive. Fix 0 ≥ 0 and suppose ( ) > 0 for ≥ 1 for some 1 ≥ 0 . Using conditions (I) and (II) in (1) with < 1 and > 1, we have for ≥ 1 . By applying Lemma 2 with = > 1, we find Using (46) in (44), we have for ≥ 1 . The rest of the proof is similar to the proof of Theorem 4 and hence is omitted.
Remark 11. The results of this section will remain the same if we replace condition (3) of assumption (I) by with = 1 2 .
Remark 12. We note that we can obtain criteria on the asymptotic behavior of the nonoscillatory solutions of (1) similar to Theorem 5. The details are left to the reader.

Further Oscillation Results
This section is devoted to the study of the oscillatory properties of (1) with 1 = 0.
Similar to the above results, one can easily prove the following theorems for the integrodynamic equation (1)
Equation (76) can be converted to two simultaneous firstorder ODEs by substituting (1 + ) 3 ( ) = . This will lead to the following system: Many numerical techniques can be used to solve (77). In the current work, the second-order accurate modified Euler technique is considered. The time interval [ 0 , ] will be divided into equal subdivisions with Δ width for each one. The prediction and correction steps of the modified Euler technique will be where The integral in (79) can be approximated numerically at each time instant using the trapezoidal rule which has accuracy of (Δ ) 2 . Solving (76) with = 1, 2, 3, and 4 and = 1/3 with initial conditions (0) = 0.1 and (0) = 0.0, to get Figures 1 and 2. In Figure 1, = 1, 2, the solution is not oscillatory as conditions (11) are not satisfied. In Figure 2, = 3, 4, we get oscillatory solution and this example validates numerically Theorem 4. Similar results are obtained for = 1 and = 3.

General Remarks
(1) The results presented in this paper are new for = and = .
(2) The results of this paper are presented in a form which is essentially new for (1) with different nonlinearities.