Nonoscillatory Solutions of Second-Order Superlinear Dynamic Equations with Integrable Coefficients
The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time scales , is discussed under the condition that exists and for large .
Consider the second-order superlinear dynamic equation where exists and is finite. for large .
When , , (1.1) is the second-order superlinear differential equation
When , , (1.1) is the second-order superlinear difference equation
The following condition is introduced in .
Condition (H). We say that satisfies Condition (H) provided one of the following holds.(1)There exists a strictly increasing sequence with and for each either or the real interval ; (2) for some .
We note that time scales which satisfy Condition (H) include most of the important time scales, such as , , and , where and is the nonnegative integers and harmonic numbers [2, Example 1.45].
In , Naito proved the following result.
Theorem 1.1. If for large , then a nonoscillatory solution of (1.3) satisfies exactly one of the following three asymptotic properties:
In this paper, we extend Theorem 1.1 to superlinear dynamic equation (1.1) on time scale. As an application, we get the asymptotic behavior of each nonoscillation solution of the difference equation where , , and .
2. Main Theorems
Consider the second-order nonlinear dynamic equation where , , , and . exists and is finite.
Lemma 2.1. Suppose that satisfies Condition . If is a positive solution of (1.1) on , then the integral equation is satisfied for , where is a nonnegative constant.
Proof. Suppose that is a positive solution of (1.1) on .
In the first place, we will prove where .
Multiplying both sides of (1.1) by , we get that Integrating from to , If (2.3) fails to hold, that is, from (2.5), we have Without loss of generality, we can assume that for Otherwise, let . By (2.7), we can take a large such that . So we have So we can replace by such that (2.8) still holds.
From (2.5) and (2.8), we get for In particular, we have Therefore, is strictly decreasing.
Assume that . Then, , and so If the real interval , then, for , we have Let Hence, from (2.10), we get that From (2.14) and (2.15), we get that Assume that . From (2.16) and (2.12), we get that So, that is, If the real interval , then, for , it follows from (2.16) and (2.13) that that is, Integrating from to , we get that Let , and let . Then, there is an such that . From (2.22) and (2.19), we get that Multiplying, we get that Using (2.15) again, we get If we set , we get Integrating from to , we get that which contradicts .
In (2.5), letting , replacing by , and denoting , we get that
We need to show that .
Suppose that . Then, there exists a large such that, for , we have So,
Thus, Assume that . From (2.12) and , we have If the real interval , from (2.13) we have, for , From (2.31), (2.32), (2.33) and the additivity of the integral, it is easy to get So, for large , we have Thus, By (2.30) and noticing that , we get that Integrating (2.37), we get that , which is a contradiction.
This completes the proof of the lemma.
Consider the second-order superlinear dynamic equation where , , exists and is finite, and for .
Theorem 2.2. Suppose that satisfies Condition and for . Then each nonoscillatory solution of (2.38) satisfies exactly one of the following three asymptotic properties:
Proof. Let be a nonoscillatory solution of (2.38), say, for . From Lemma 2.1, it is known that satisfies the equality
for . Therefore, we have
for . Since for , it follows that for . An integration by parts of (2.38) gives
where . Let be fixed. Since is nonnegative, the integral term in (2.45) has a finite limit or diverges to as . If the latter case occurs, then as , which is a contradiction to (2.44). Thus, the former case occurs, that is,
Define the function as
and the finite constant . Then, equality (2.45) yields
for . Observe by (2.44) that . From (2.44) and (2.46), it follows that
Next, we define the function by (noticing that )
for . Dividing (2.48) by and integrating from to , we get
By Schwartz’s inequality and the fact that for , the second integral term in (2.51) can be estimated as follows:
for . From (2.51) and (2.52) and noticing that is decreasing, we get that
The above inequality may be regarded as a quadratic inequality in . Then, we have
for , where
It is obvious that as , and, consequently, there exists a positive constant such that
for . Let be an arbitrary number. It is clear that
for . Arguing as in (2.52), we find
for , which when combined with (2.56) yields
for . Using (2.57) and (2.59) and noticing that , we obtain
Since is arbitary and as , letting in (2.60), we get
Using L’Hospital’s rule of time scale (see Theorem 1.119 of ), we have
In view of (2.51), (2.61), and (2.62), we find .
Recall that is nondecreasing for . Now, there are three cases to consider:(i) and is bounded above,(ii) and is unbounded,(iii) (and hence is unbounded).
Case (i) implies (2.40) with , while case (iii) implies (2.42) with . It is also clear that case (ii) implies (2.41). This completes the proof.
The following lemma is from .
Lemma 2.3. Suppose that satisfies Condition (H). is a solution of (1.1). Then, one has
Using Lemma 2.1, we can prove the following corollary.
The following theorem can be regarded as a time scale version of [4, Theorem 1].
Theorem 2.5. Suppose that satisfies Condition (H), with , and suppose that exists and is finite. Let . Then, the superlinear dynamic equation (1.1) is oscillatory if
Proof. Suppose that is a nonoscillatory solution of (1.1) on . Without loss of generality, assume that is positive for . From Corollary 2.4, satisfies the integral equation (2.64). Dropping the last integral term in (2.64), we have the inequality Dividing (2.68) by , integrating from to , and using Lemma 2.3, we find This contradicts (2.67), and so (1.1) is oscillatory.
Consider the second-order superlinear dynamic equation with forced term where , , and exists and is finite.
Lemma 2.6. Suppose that If is a positive solution of (2.70) and , then are satisfied for sufficiently large , where , is a nonnegative constant.
Proof. The fact that implies the existence of and such that for . Then, using (2.72), we find
where is some finite positive constant.
So, exists and is finite.
Similar to the proof of Lemma 2.1 and Corollary 2.4, it is easy to know that (2.73) and (2.74) hold.
For subsequent results, we define where is a positive constant. It is noted that, if (2.71) and (2.72) hold, then is finite for any . Assume that for sufficiently large . Define, for a positive integer and a positive constant , the following functions: We introduce the following condition.
Condition (A). For every , there exists a positive integer such that is finite for and is infinite.
Proof. Suppose on the contrary that is a nonoscillatory solution of (2.70) and . Without loss of generality, let be eventually positive. By Lemma 2.6, satisfies (2.73) and (2.74). Further, there exist and such that for . Let
Then, from (2.74) we find
for . From (2.80), we get
Applying (2.81) and noticing that , we find for
If in Condition (A), then the right side of (2.82) is infinite. This is a contradiction to (2.73).
Next, it follows from (2.80) and (2.82) that Using a similar technique and relations (2.83), we get If in Condition (A), then the right side of (2.84) is infinite. This again contradicts (2.73).
A similar argument yields a contradiction for any integer . This completes the proof of the theorem.
Example 2.8. We have where , , and . It is easy to see that By , we have . So, Using (2.86) and (2.87), we get that, for large , By Theorem 2.2, each nonoscillatory solution of (2.85) satisfies exactly one of the following three asymptotic properties:
The paper is supported by the National Natural Science Foundation of China (no. 10971232).
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