Asymptotic Behavior of Certain Integrodifferential Equations

t n → ∞, such that x(t n ) = 0, 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).Themain 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.

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 ( ) = 0, 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][2][3][4][5][6][7][8].
Oscillation and asymptotic results for integral and integrodifferential equations are scarce; some results can be found in [5,[9][10][11][12][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.

Main Results
To obtain our main results of this paper, we need the following two lemmas.
Next, by employing Theorem 3 we present the following oscillation result for (1). Proof. Let be a nonoscillatory solution of (1), say ( ) > 0, for ≥ 1 for some 1 ≥ 0. 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 that where 1 and are positive constants. Note that we make < 1 possible by increasing the size of 1 . Finally, taking lim inf in (32) as → ∞ as well as using (30) result is a contradiction with the fact that is eventually positive.
The following corollary is immediate.
The following example is illustrative. Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodifferential equation (1) when = 1.

Theorem 7. Let = 1 and the hypotheses of Theorems 3 and 4 hold with ( ) = ℎ( ) 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 equation We will give sufficient conditions under which any nonoscillatory solution of (37) satisfies | ( )| = ( ) as → ∞.
Theorem 8. Let 0 < < 1 and let condition (ii) hold and suppose that > 1, = /( − 1), 0 < < 1, and = 2 − − 1/ , ( − 1) + 1 > 0, and ( − 1) + 1 > 0, lim sup for any 1 ≥ . If is a nonoscillatory solution of (37), then Proof. Let be a nonoscillatory solution of (37). We may assume that ( ) > 0 for ≥ 1 for some 1 ≥ . We let ( ) = ( , ( )). In view of (ii) we may then write Proceeding as in the proof of Theorem 3, we obtain Integrating inequality (42) from 1 to and interchanging the order of integration one can easily obtain Interchanging the order of integration in second integral we have The rest of the proof is similar to that of Theorem 3 and hence is omitted.
Now we give sufficient conditions for the boundedness of any nonoscillatory solution of (47).
Proof. Let be an eventually positive solution of (47). We may assume that ( ) > 0 for ≥ 1 for some 1 ≥ . We let ( ) = ( , ( )). In view of (ii) we may then write The rest of the proof is similar to that of Theorem 3 and hence is omitted.
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 = 1. We may note that results similar to Theorem 4 can be obtained for (37) and (47). The details are left to the reader.

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 ∈ ( − 1, ), ≥ 1. The details are left to the reader.
(ii) It would be of interest to study (1) when satisfies condition (iii) with > 1.