#### Abstract

We study the oscillation behavior for some higher order integrodynamic equations on timescales. We establish some new sufficient conditions guaranteeing that all solutions of theses equations are oscillatory. Some numerical examples in the continuous case are given to validate the theoretical results.

#### 1. 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 [2–6]. In this paper, we investigate the oscillatory behavior of the solutions of some higher order integrodynamic equations on timescale in the formTo 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 and .

Whenever we write , we mean .

We assume throughout the following:(I), and are rd-continuous functions, , and for ; and there exist rd-continuous functions , such that(II) is continuous and assume that there exist , continuous functions such that for and for , and ;(III)there exist constants and which are the ratios of positive odd integers and , , such thatBy a solution of (1) we mean a -differential function defined on that is nontrivial in every neighborhood of infinity. A solution of (1) is said to be oscillatory if there exists such that or ; otherwise, it is called nonoscillatory.

#### 2. Auxiliary Results

We employ the following lemmas.

Lemma 1 (see [9]). *If and are nonnegative, thenwhere equality holds if and only if .*

Lemma 2 (Young’s inequality). *Let , , and , thenand equality holds if and only if .*

Lemma 3 (see [3, corollary 1]). *Assume that , , and is rd-continuous function, and then*

In Lemma 3, stand for the Taylor monomials (see [1, section 1.6]) which are defined recursively byIt follows that for any timescale, but simple formulas, in general, do not hold for .

For , we define

#### 3. Main Results

In this section we present the following main results.

Theorem 4. *Let conditions (I) and (II) hold with . Iffor all , then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1). Hence either is eventually positive or is eventually negative.

First assume is eventually positive. Fix and suppose for for some . From (1), we see thatLetBy assumption (3), we haveHence, from (12), we getIntegrating (15) -times from to and then using Lemma 3, we obtainFrom the properties of the functions and the definition of the function for all , we get whereDividing (17) by and hence integrating from to we obtainDividing (19) by and taking of both sides of (19) as , we obtain a contradiction to the fact that for . 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 and supposefor all . If is nonoscillatory solution of (1), then*

Next, we present the following result.

Theorem 6. *Let conditions (I) and (II) hold with and is nondecreasing in the second variable. If conditions (11) hold for every constant ,for any , then every bounded solution of (1) is oscillatory.*

*Proof. *Let be a bounded nonoscillatory solution of (1) and assume that is eventually positive. Fix and suppose for for some and for some constant . From (1), we haveProceeding 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 and and suppose that conditions (11) hold andfor all . Then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1), Hence either is eventually positive or is eventually negative. First, assume is eventually positive. Fix and assume for for some .

Using conditions (I) and (II) with and in (1), we havefor . Proceeding as in the proof of Theorem 4, we get (14) and henceBy applying (5) withwe obtainUsing (29) in (26), we find whereIntegrating (30) -times from to and then using Lemma 3, we havewhere is given in (18). The rest of the proof is similar to that of the proof of Theorem 4 and hence is omitted.

Theorem 8. *Let conditions (I) and (II) hold with and . If, in addition to conditions (11), we supposefor any , then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1). First, assume is eventually positive. Fix and suppose for for some . Using conditions (I) and (II) with and in (1), we findfor . Hencewhere is defined as in the proof of Theorem 4. By applying (6) withwe obtainUsing (37) in (35), we findThe rest of the proof is similar to the proof of Theorem 4 and hence is omitted.

Next, we present the following result with different nonlinearities, that is, with and .

Theorem 9. *Let conditions (I) and (II) hold with and and suppose that there exists a positive rd-continuous function such thatfor all , whereIf conditions (11) hold for all , then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1). First, assume is eventually positive. Fix and suppose for for some . Using conditions (I) and (II) in (1), we havefor

As in the proof of Theorems 7 and 8, one can easily findThe rest of the proof is similar to that of Theorem 4 and hence is omitted.

For the cases when both functions and are superlinear, that is, , or sublinear, that is, , we present the following result.

Theorem 10. *Let conditions (I) and (II) hold with . If, in addition to conditions (11), we supposefor all , then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1). First, assume is eventually positive. Fix and suppose for for some . Using conditions (I) and (II) in (1) with and , we havefor . By applying Lemma 2 withwe findUsing (46) in (44), we havefor . 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) bywith .

*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.

#### 4. Further Oscillation Results

This section is devoted to the study of the oscillatory properties of (1) with .

Theorem 13. *Let conditions (I) and (II) hold with and . Assume that there exists a rd-continuous function such thatIffor all , then every nonoscillatory solution of (1) satisfies*

*Proof. *Let be a nonoscillatory solution of (1). First, assume is eventually positive. Fix and suppose for for some . Using conditions (I) and (II) in (1), we haveAs in the proof of Theorem 4, we obtain (14) and hence (55) becomesBy applying (6) withwe haveUsing (58) in (56), we findIntegrating this inequality -times from to and using Lemma 3, we haveAs in the proof of Theorem 4, one can easily findwhere is given by (18). Using condition (50), we see that is bounded for ; say by , and we see thatIntegrating this inequality from to and employing Lemma 3 in [10] to interchange the order of integration we obtainUsing conditions (49), (50), and (53), there exist positive constants and such thatApplying Gronwall’s inequality [1, Corollary 6.7] to inequality (64) and then using condition (51), we haveIf is eventually negative, we set to see that satisfies (1) with replaced by and by .

It follows in a similar manner thatWe conclude from (65) and (66) that (54) holds.

Next by employing Theorem 13, we present the following oscillation result for (1) with .

Theorem 14. *Let conditions (I) and (II) hold with and . Suppose that conditions (50), (52), and (53) hold. Iffor all , then (1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1). First, assume is eventually positive. Fix and suppose for for some . The proof when is eventually negative is similar.

Proceeding as in the proof of Theorem 13, we arrive at (58). ThereforeConditions (67) and (68) imply conditions (50) and (49), respectively, and so the conclusion of Theorem 13 holds. This together with condition (50) and (68) shows that the second term and last integral are bounded.

Taking for both sides of (71) as and using (67) and (70) result in a contradiction with the fact that is eventually positive.

Similar to the sublinear case one can easily prove the following theorems for the integrodynamic equation (1) when .

Theorem 15. *Let conditions (I) and (II) hold with and . Moreover, assumeand conditions (50), (52), and (53) hold for any . Then every nonoscillatory solution of (1) satisfies (54).*

Theorem 16. *Let conditions (I) and (II) hold with and . In addition, suppose conditions (50), (52), (53), (69), and (70) hold. Iffor all , then (1) is oscillatory.*

Similar to the above results, one can easily prove the following theorems for the integrodynamic equation (1) with .

Theorem 17. *Let conditions (I) and (II) hold with and . Assume that there exists a rd-continuous function such thatIf conditions (50)–(53) hold for all , then every nonoscillatory solution of (1) satisfies (54).*

Theorem 18. *Let conditions (I) and (II) hold with and . Assume that there exists a rd-continuous function such thatIf conditions (50), (52), (53), (68), and (69) hold for all , then (1) is oscillatory.*

#### 5. Illustrative Examples

As we already mentioned, the results of the present paper are new for the cases when , that is, the continuous case, or when , that is, the discrete case. As a numerical illustration of our results in Section 3 with , we consider the following equation:with initial conditions and . Compare (76) with (1) to get that , , , , and . We can easily show that conditions (I) and (II) are satisfied. Conditions (11) are satisfied only for .

Equation (76) can be converted to two simultaneous first-order ODEs by substituting . 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 will be divided into equal subdivisions with width for each one. The prediction and correction steps of the modified Euler technique will bewhere The integral in (79) can be approximated numerically at each time instant using the trapezoidal rule which has accuracy of . Solving (76) with , and and with initial conditions and , to get Figures 1 and 2. In Figure 1, , the solution is not oscillatory as conditions (11) are not satisfied. In Figure 2, , we get oscillatory solution and this example validates numerically Theorem 4. Similar results are obtained for and .

As another example, consider the following equation:with initial conditions , , and . Follow the same procedure above and substitute to get the following system:The prediction and correction steps of the modified Euler technique will bewhere We will consider the numerical solution of (80) with and with the initial conditions: .

For and , the oscillatory solution in Figure 3 is obtained and this validates Theorem 7. Similar results are obtained to validate Theorem 8 (with and ) and Theorems 9 and 10 (with and ). Theorems 13–18 can be validated in a similar way.

*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.(3)The techniques offered in this paper can be employed to Volterra integral equations on timescales of the form The formulation of our results to the above equation is left to the reader.(4)The results of this paper can be extended easily to delay integrodynamic equations of the form where is rd-continuous such that and for and . The formulation of the results is left to the reader.(5)We note that we can formulate the obtained results for the timescales (the continuous case), (the discrete case), with (the quantum calculus case), with , , and so forth. The details are left to the reader.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.