Abstract
The article deals with the existence of uncountably many positive solutions which are bounded below and above by positive functions for the first-order nonlinear neutral differential equations. Some examples are included to illustrate the results presented in this article.
MSC:34K40, 34K12.
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1 Introduction
In recent years, the study of existence and qualitative properties of solutions for various kinds of neutral delay differential equations has attracted much attention. For related results, we refer the reader to [1–13] and the references cited therein. The authors only considered the existence of solutions which are bounded by positive constants, e.g., in [8, 9, 11–13]. For example, Erbe et al. [6] established a few oscillation and nonoscillation criteria for linear neutral delay differential equation
Diblík and co-autors in [1–4] studied the existence of positive and oscillatory solutions of differential equations with delay and nonlinear systems in view of Ważievski’s retract principle and later extended to retarded functional differential equations by Rybakowski. Zhou [12] deduced the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equations and Lin et al. [9] discussed the existence of nonoscillatory solutions for a third-order nonlinear neutral delay differential equation, and by utilizing Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, they developed some sufficient conditions for the existence of uncountably many nonoscillatory solutions bounded by positive constants. Some interesting results about the existence of nonoscillatory solutions of delay differential equations can also be found in [1, 5].
In this paper, we investigate the following nonlinear neutral differential delay differential equations:
where , , , , , f is a nondecreasing function for and , .
By a solution of Eq. (1), we mean a function for some such that is continuously differentiable on and such that Eq. (1) is satisfied for .
As much as we know, in the literature there is no result for the existence of uncountably many solutions which are bounded below and above by positive functions. This problem is discussed and treated in this paper.
The following fixed point theorem will be used to prove the main results in the next section.
Lemma 1.1 ([6, 12] Krasnoselskii’s fixed point theorem)
Let X be a Banach space, let Ω be a bounded closed convex subset of X and let , be maps of Ω into X such that for every pair . If is contractive and is completely continuous, then the equation
has a solution in Ω.
2 The existence of positive solutions
In this section we consider the existence of uncountably many positive solutions for Eq. (1) which are bounded by two positive functions. We use the notation .
Theorem 2.1 Suppose that there exist bounded from below and from above by the functions u and constants , and such that
Then Eq. (1) has uncountably many positive solutions which are bounded by the functions u, v.
Proof Let be the set of all continuous bounded functions with the norm . Then is a Banach space. We define a closed, bounded and convex subset Ω of as follows:
For we define two maps and as follows:
We will show that for any , we have . For every and with regard to (4), we obtain
For we have
Furthermore, for we get
Let . With regard to (3), we get
Then, for and any , we obtain
Thus we have proved that for any .
We will show that is a contraction mapping on Ω. For and , we have
This implies that
Also, for the inequality above is valid. We conclude that is a contraction mapping on Ω.
We now show that is completely continuous. First, we show that is continuous. Let be such that as . Because Ω is closed, . For we have
According to (7), we get
Since as , by applying the Lebesgue dominated convergence theorem, we obtain that
This means that is continuous.
We now show that is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions is uniformly bounded and equicontinuous on . The uniform boundedness follows from the definition of Ω. For the equicontinuity, we only need to show, according to the Levitan result [7], that for any given , the interval can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have a change of amplitude less than ε. Then, with regard to condition (8), for and any , we take large enough so that
Then, for , , we have
For and , we get
Thus there exists , where , such that
Finally, for any , , there exists a such that
Then is uniformly bounded and equicontinuous on , and hence is a relatively compact subset of . By Lemma 1.1 there is an such that . We conclude that is a positive solution of (1).
Next we show that Eq. (1) has uncountably many bounded positive solutions in Ω. Let the constant be such that . We infer similarly that there exist mappings , satisfying (5), (6), where K, , are replaced by , , , respectively. We assume that , , , which are the bounded positive solutions of Eq. (1), that is,
From condition (8) it follows that there exists a satisfying
In order to prove that the set of bounded positive solutions of Eq. (1) is uncountable, it is sufficient to verify that . For we get
Then we have
According to (9) we get that . Since the interval contains uncountably many constants, then Eq. (1) has uncountably many positive solutions which are bounded by the functions , . This completes the proof. □
Corollary 2.1 Suppose that there exist bounded from below and from above by the functions u and constants , and such that (2), (4) hold and
Then Eq. (1) has uncountably many positive solutions which are bounded by the functions u, v.
Proof We only need to prove that condition (10) implies (3). Let and set
Then, with regard to (10), it follows that
Since and for , this implies that
Thus all the conditions of Theorem 2.1 are satisfied. □
Example 2.1 Consider the nonlinear neutral differential equation
where . We will show that the conditions of Corollary 2.1 are satisfied. The functions , satisfy (2) and also condition (10) for . For the constants , , condition (4) has the form
If the function satisfies (12), then Eq. (11) has uncountably many positive solutions which are bounded by the functions u, v.
Example 2.2 Consider the nonlinear neutral differential equation
where , . We will show that the conditions of Corollary 2.1 are satisfied. The functions , , , satisfy (2) and since
condition (10) is also satisfied. For the constants , , condition (4) has the form
For we get
If the function satisfies (14), then Eq. (13) has uncountably many solutions which are bounded by the functions u, v.
Example 2.3 Consider the nonlinear neutral differential equation
where , . We will show that the conditions of Corollary 2.1 are satisfied. The functions , , satisfy (2) and also (10)
For the constants , , where , , condition (4) has the form
For and , we have
Then for , which satisfies the inequalities
Eq. (11) has uncountably many solutions which are bounded by the functions u, v.
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The research was supported by the grant 1/0090/09 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic.
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Dorociaková, B., Kubjatková, M. & Olach, R. Uncountably many solutions of first-order neutral nonlinear differential equations. Adv Differ Equ 2013, 140 (2013). https://doi.org/10.1186/1687-1847-2013-140
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DOI: https://doi.org/10.1186/1687-1847-2013-140