Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Oscillatory Behavior of Solutions of Certain Integrodynamic Equations of Second Order on Time Scales
This paper deals with the oscillatory behavior of forced second-order integrodynamic equations on time scales. The results are new for the continuous and discrete cases and can be applied to Volterra integral equation on time scale. We also provide a numerical example in the continuous case to illustrate the results.
Dynamic equations on time scales is a fairly new topic and for general basic ideas and background, we refer the reader to . Oscillation and nonoscillation results for integral equations of Volterra type have been discussed only in a few papers in the literature; see [2–5].
In this paper, we are concerned with the oscillatory behavior of solutions of the second-order integrodynamic equation on time scales of the form We take to be an arbitrary time scale with and . Whenever we write , we mean . We assume throughout the following.(I) and are rd-continuous and and for and there exist rd-continuous functions such that (II) is continuous and assume that there exists rd-continuous function and a real number with such that We only consider those solutions of (1) which are nontrivial and delta-differentiable on . The term solution henceforth applies to such solutions of (1).
A solution of (1) is said to be oscillatory if, for every , we have and it is said to be nonoscillatory otherwise.
To the best of our knowledge, there are no results on the oscillatory behavior of solutions of (1). Therefore, the main goal of this paper is to establish some new criteria for the oscillatory behavior of (1) and other related equations. Also we provide some numerical examples to illustrate the results when .
2. Main Results
We shall employ the following lemmas.
Lemma 1 (see ). If and are nonnegative, then where equality holds if and only if .
Theorem 2. Let , conditions (I) and (II) hold, and for all , the function is bounded: If then every nonoscillatory solution of (1) satisfies
Proof. Let be a nonoscillatory solution of (1). First, assume is eventually positive, say for for some . Using condition (4) in (1) we have
By assumption (3) we obtain
Hence, from (13) we have
Applying Lemma 1 to with we have Thus, we get Integrating (19) from to we have Employing Lemma 3 in  to interchange the order of integration, we obtain and so Integrating this inequality from to and using (9) and the fact that the function is bounded for , say by , we see that Once again, using Lemma 3 in  we have or and so where and is an upper bound for the expression
Applying Gronwall’s inequality [1, Corollary 6.7] to inequality (26) and then using condition (10) we have If is eventually negative, we can set to see that satisfies (1) with replaced by and replaced by . It follows in a similar manner that We conclude from (28) and (29) that (12) holds.
Proof. Let be a nonoscillatory solution of (1), say for for some . The proof when is eventually negative is similar. Proceeding as in the proof of Theorem 2 we arrive at Condition (30) implies condition (10) and so the conclusion of Theorem 2 holds. This, together with (9), shows that the second and the last two integrals in (32) are bounded. Finally, taking and using conditions (30) and (31) result in a contradiction with the fact that is eventually positive.
Corollary 4. Let and condition (I), (II), (9), and (30) hold. In addition, assume that If for all , then (1) is oscillatory. Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodynamic (1) when .
Remark 7. The technique in this paper can be employed to Volterra integral equation on time scales of the form
Remark 8. We note that the results in this section can be obtained by using the additional assumption that the function is nonincreasing with respect to the first variable. In this case the function which appeared in the proofs and the function which appeared in the statements of the theorems are replaced by . The details are left to the reader.
3. Illustrative Examples
As a numerical illustration of our results of Section 2, we consider the following equation: with initial conditions and . Equation (39) can be converted into two first order ordinary differential equations by substituting . This will lead to the following system: Many numerical techniques can be used to solve (40). In the current work, the second-order accurate modified Euler technique is considered. The time interval will be subdivided into equal subdivisions with width for each subdivision. The prediction and correction steps of the modified Euler technique will be where The integral (43) can be approximated numerically at each time instant using the trapezoidal rule, which has accuracy of .
Let , and , with . For , all conditions of Theorems 3 and 6 are satisfied and hence (39) is oscillatory (see Figure 1). In Figure 2, the initial conditions are changed to be and with ; the solution is also oscillatory.
The oscillatory behavior of a second-order integrodynamic equation on time scales is considered. The results in the paper establish new criteria and it is shown that these results can be employed for Volterra integral equation on time scale. A numerical example in the continuous case is solved to show the validity of the results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
G. H. Hardy, I. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, Mass, USA, 1959.