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Existence of positive solutions of higher-order nonlinear neutral equations
Journal of Inequalities and Applications volume 2013, Article number: 573 (2013)
Abstract
In this work, we consider the existence of positive solutions of higher-order nonlinear neutral differential equations. In the special case, our results include some well-known results. In order to obtain new sufficient conditions for the existence of a positive solution, we use Schauder’s fixed point theorem.
1 Introduction
The purpose of this article is to study higher-order neutral nonlinear differential equations of the form
and
where is an integer, , , , , r, , , , , , f is a nondecreasing function with , .
The motivation for the present work was the recent work of Culáková et al. [1] in which the second-order neutral nonlinear differential equation of the form
was considered. Note that when in (1), we obtain (4). Thus, our results contain the results established in [1] for (1). The results for (2) and (3) are completely new.
Existence of nonoscillatory or positive solutions of higher-order neutral differential equations was investigated in [2–5], but in this work our results contain not only existence of solutions but also behavior of solutions. For books, we refer the reader to [6–11].
Let . By a solution of (1) we understand a function , for some , such that is times continuously differentiable, is continuously differentiable on and (1) is satisfied for . Similarly, let . By a solution of (2) we understand a function , for some , such that is times continuously differentiable, is continuously differentiable on and (2) is satisfied for . Finally, let . By a solution of (3) we understand a function , for some , such that is times continuously differentiable, is continuously differentiable on and (3) is satisfied for .
The following fixed point theorem will be used in proofs.
Theorem 1 (Schauder’s fixed point theorem [9])
Let A be a closed, convex and nonempty subset of a Banach space Ω. Let be a continuous mapping such that SA is a relatively compact subset of Ω. Then S has at least one fixed point in A. That is, there exists such that .
2 Main results
Theorem 2 Let
Assume that and there exists such that
Then (1) has a positive solution which tends to zero.
Proof Let Ω be the set of all continuous and bounded functions on with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
where and are nonnegative functions such that
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator as
We show that S satisfies the assumptions of Schauder’s fixed point theorem.
First, S maps A into A. For and , using (7) and (8), we have
and
For and , we obtain
and in order to show , consider
By making use of (6), it follows that
Since and for , we conclude that
Then and for any ,
Hence, S maps A into A.
Second, we show that S is continuous. Let be a convergent sequence of functions in A such that as . Since A is closed, we have . It is obvious that for and , S is continuous. For ,
Since as , by making use of the Lebesgue dominated convergence theorem, we see that
and therefore S is continuous.
Third, we show that SA is relatively compact. In order to prove that SA is relatively compact, it suffices to show that the family of functions is uniformly bounded and equicontinuous on . Since uniform boundedness of is obvious, we need only to show equicontinuity. For and any , we take large enough such that . For and , we have
Note that
For and , by using (9) we obtain
Thus there exits such that
Finally, for and , there exits such that
Therefore SA is relatively compact. In view of Schauder’s fixed point theorem, we can conclude that there exists such that . That is, x is a positive solution of (1) which tends to zero. The proof is complete. □
Theorem 3 Let
where . Assume that and there exists such that
Then (2) has a positive solution which tends to zero.
Proof Let Ω be the set of all continuous and bounded functions on with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
where and are nonnegative functions such that
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator as follows:
Since the remaining part of the proof is similar to those in the proof of Theorem 2, it is omitted. Thus the theorem is proved. □
Theorem 4 Suppose that (10) and (11) hold. In addition, assume that
where . Then (3) has a positive solution which tends to zero.
Proof Let Ω be the set of all continuous and bounded functions on with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
where and are nonnegative functions such that
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator as
We show that S satisfies the assumptions of Schauder’s fixed point theorem.
First of all, S maps A into A. For and , using (12), (13), the decreasing nature of and , we have
and
For and , we obtain
and to show , consider
By making use of (11), it follows that
Since and for , we conclude that
Then and for any ,
Hence, S maps A into A.
Next, we show that S is continuous. Let be a convergent sequence of functions in A such that as . Since A is closed, we have . It is obvious that for and , S is continuous. For ,
Since as and , by making use of the Lebesgue dominated convergence theorem, we see that
Thus S is continuous.
Finally, we show that SA is relatively compact. In order to prove that SA is relatively compact, it suffices to show that the family of functions is uniformly bounded and equicontinuous on . Since uniform boundedness of is obvious, we need only to show equicontinuity. For and any , we take large enough such that . For and , we have
For and , by using (9) we obtain
Thus there exits such that
For and , there exits such that
Therefore SA is relatively compact. In view of Schauder’s fixed point theorem, we can conclude that there exists such that . That is, x is a positive solution of (1) which tends to zero. The proof is complete. □
Example 1 Consider the neutral differential equation
where and
Note that for , , and , we have
and
If fulfils the last inequality above, a straightforward verification yields that the conditions of Theorem 2 are satisfied and therefore (14) has a positive solution which tends to zero.
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Candan, T. Existence of positive solutions of higher-order nonlinear neutral equations. J Inequal Appl 2013, 573 (2013). https://doi.org/10.1186/1029-242X-2013-573
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DOI: https://doi.org/10.1186/1029-242X-2013-573