Abstract
This paper deals with asymptotic behavior of nonoscillatory solutions of certain forced integrodifferential equations of the form: From the obtained results, we derive a technique which can be applied to some related integrodifferential as well as integral equations.
1. Introduction
In this paper, we consider the integrodifferential equationIn the sequel, we assume that (i);(ii) and also there exists such that for all ;(iii) and also there exist and real numbers , , and such that for and .
We only consider solutions of (1) which are continuable and nontrivial in any neighborhood of . Such a solution is said to be oscillatory if there exists a sequence , , such that , and it is nonoscillatory otherwise.
In the last few decades, integral, integrodifferential, and fractional differential equations have gained considerable attention due to their applications in many engineering and scientific disciplines as the mathematical models for systems and processes in fields such as physics, mechanics, chemistry, aerodynamics, and the electrodynamics of complex media. For more details one can refer to [1–8].
Oscillation and asymptotic results for integral and integrodifferential equations are scarce; some results can be found in [5, 9–13]. It seems that there are no such results for integral equations of type (1). The main objective of this paper is to establish some new criteria on the oscillatory and the asymptotic behavior of all solutions of (1). From the obtained results, we derive a technique which can be applied to some related integrodifferential as well as integral equations.
2. Main Results
To obtain our main results of this paper, we need the following two lemmas.
Lemma 1 (see [5, 7]). Let , , and be positive constants such that and . Then where , , , and .
Lemma 2 (see [14]). If and are nonnegative, thenwhere equality holds if and only if .
In what follows, we letand , for some , where .
Now we give sufficient conditions under which any solution of (1) satisfies as .
Theorem 3. Let and conditions (i)–(iii) hold and suppose that , , , , , , andIffor any , then every nonoscillatory solution of (1) satisfies
Proof. Let be a nonoscillatory solution of (1). We may assume that for for some . We let . In view of (i)–(iii) we may then writeand so Applying (4) of Lemma 2 to with and we have and hence we obtain orwhere and are the upper bounds of the functions and , respectively. Integrating inequality (15) from to we have Interchanging the order of integration in the last integral, we havewhere is the upper bound of the function . Integrating (17) from to and interchanging the order of integration in the last integral we findNow, one can easily see thatorwhere is the upper bound of the function and . Applying Holder’s inequality and Lemma 1 we obtain where , and and so Thus, inequality (20) becomesUsing (24) and the elementary inequality we obtain from (24)If we denote , that is, , , and , thenThe conclusion follows from Gronwall’s inequality and we conclude thatIf is eventually negative, we can set to see that satisfies (1) with being replaced by and by . It follows in a similar manner thatFrom (28) and (29) we get (10). This completes the proof.
Next, by employing Theorem 3 we present the following oscillation result for (1).
Theorem 4. Let and conditions (i)–(iii), (6)–(9) hold and suppose that , , , , , and . If for every , ,for all , then (1) is oscillatory.
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 3 we arrive at (19). Therefore, Clearly, the conclusion of Theorem 3 holds. This together with (7) and (8) implies that the first, second, and fourth integrals on the above inequality are bounded and hence one can easily see thatwhere and are positive constants. Note that we make possible by increasing the size of . Finally, taking in (32) as as well as using (30) result is a contradiction with the fact that is eventually positive.
The following corollary is immediate.
Corollary 5. Let and conditions (i)–(iii), (6)–(9) hold for some . In addition, assume thatIf for every , ,for all , then (1) is oscillatory.
The following example is illustrative.
Example 6. Let , , , and . Clearly, Let the functions and be as in (i) and (ii) with being a bounded function and let , , and , where , with , , and Condition (34) is also fulfilled. Thus, all conditions of Theorem 3 are satisfied and hence every nonoscillatory solution of (1) satisfies .
Now if , , we see that all the hypotheses of Corollary 5 are satisfied and hence (1) is oscillatory.
Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodifferential equation (1) when .
Theorem 7. Let and the hypotheses of Theorems 3 and 4 hold with and . Then the conclusion of Theorems 3 and 4 holds, respectively.
From the obtained results, we apply the employed technique to some related integrodifferential equations.
Now, we consider the integrodifferential equationWe will give sufficient conditions under which any nonoscillatory solution of (37) satisfies as .
Theorem 8. Let and let condition (ii) hold and suppose that , , , and , , and ,for any . If is a nonoscillatory solution of (37), then
Proof. Let be a nonoscillatory solution of (37). We may assume that for for some . We let . In view of (ii) we may then write Proceeding as in the proof of Theorem 3, we obtainIntegrating inequality (42) from to and interchanging the order of integration one can easily obtain Interchanging the order of integration in second integral we haveThe rest of the proof is similar to that of Theorem 3 and hence is omitted.
Example 9. Let , , , and . Clearly, Let the functions and , where , with , , and Condition (39) is also fulfilled. Thus, all conditions of Theorem 8 are satisfied and hence every nonoscillatory solution of (37) satisfies .
Finally, we consider the integral equationNow we give sufficient conditions for the boundedness of any nonoscillatory solution of (47).
Theorem 10. Let and let condition (ii) hold and suppose that , , , and , , and ,where is defined as in (5) for any . If is a nonoscillatory solution of (47), then is bounded.
Proof. Let be an eventually positive solution of (47). We may assume that for for some . We let . In view of (ii) we may then write orThe rest of the proof is similar to that of Theorem 3 and hence is omitted.
Example 11. Let , , , and . Clearly, Let the functions and , where , with , , and Condition (49) is also fulfilled. Thus, all conditions of Theorem 10 are satisfied and hence every nonoscillatory solution of (37) is bounded.
Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodifferential equations (37) and (47) when .
Theorem 12. Let and the hypotheses of Theorems 8 and 10 hold with . Then the conclusion of Theorems 8 and 10 holds.
We may note that results similar to Theorem 4 can be obtained for (37) and (47). The details are left to the reader.
3. General Remarks
(i)The results of this paper are presented in a form which is essentially new and it can also be employed to investigate the asymptotic and oscillatory behavior of certain integrodifferential equations of higher order , . The details are left to the reader.(ii)It would be of interest to study (1) when satisfies condition (iii) with .
Competing Interests
The authors declare that they have no competing interests.