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Existence of Nonoscillatory Solutions of First-Order Neutral Differential Equations
This paper contains some sufficient conditions for the existence of positive solutions which are bounded below and above by positive functions for the first-order nonlinear neutral differential equations. These equations can also support the existence of positive solutions approaching zero at infinity
This paper is concerned with the existence of a positive solution of the neutral differential equations of the form where is nondecreasing function, and .
The problem of the existence of solutions of neutral differential equations has been studied by several authors in the recent years. For related results we refer the reader to [1–11] and the references cited therein. However there is no conception which guarantees the existence of positive solutions which are bounded below and above by positive functions. In this paper we have presented some conception. The method also supports the existence of positive solutions approaching zero at infinity.
As much as we know, for (1.1) in the literature, there is no result for the existence of solutions which are bounded by positive functions. Only the existence of solutions which are bounded by constants is treated, for example, in [6, 10, 11]. It seems that conditions of theorems are rather complicated, but cannot be simpler due to Corollaries 2.3, 2.6, and 3.2.
The following fixed point theorem will be used to prove the main results in the next section.
Lemma 1.1 ([see [6, 10] Krasnoselskii's fixed point theorem]). Let be a Banach space, let be a bounded closed convex subset of , and let be maps of into such that for every pair . If is contractive and is completely continuous, then the equation has a solution in .
2. The Existence of Positive Solution
In this section we will consider the existence of a positive solution for (1.1). The next theorem gives us the sufficient conditions for the existence of a positive solution which is bounded by two positive functions.
Theorem 2.1. Suppose that there exist bounded functions , constant and such that Then (1.1) has a positive solution which is bounded by functions .
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 of as follows:
We now define two maps and as follows:
We will show that for any we have . For every and , we obtain
For , we have
Furthermore, for , we get
Let . With regard to (2.2), we get
Then for and any , we obtain
Thus we have proved that for any .
We will show that is a contraction mapping on . For and we have This implies that Also for , the previous inequality is valid. We conclude that is a contraction mapping on .
We now show that is completely continuous. First we will show that is continuous. Let be such that as . Because is closed, . For we have According to (2.8), we get Since as , by applying the Lebesgue dominated convergence theorem, we obtain 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 . For the equicontinuity we only need to show, according to Levitans result , 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 . Then with regard to condition (2.14), 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 relatively compact subset of . By Lemma 1.1 there is an such that . We conclude that is a positive solution of (1.1). The proof is complete.
Proof. We only need to prove that condition (2.21) implies (2.2). Let and set Then with regard to (2.21), it follows that Since and for , this implies that Thus all conditions of Theorem 2.1 are satisfied.
Corollary 2.3. Suppose that there exists a function , constant and such that Then (1.1) has a solution .
Proof. We put and apply Theorem 2.1.
Proof. The proof is similar to that of Theorem 2.1 and we omit it.
Corollary 2.5. Suppose that there exist functions , constant and such that (2.1), (2.3), (2.21), and (2.26) hold. Then (1.1) has a positive solution which is bounded by the functions and tends to zero.
Proof. The proof is similar to that of Corollary 2.2, and we omitted it.
Proof. We put and apply Theorem 2.4.
3. Applications and Examples
In this section we give some applications of the theorems above.
Theorem 3.1. Suppose that and there exist constants such that Then (1.1) has a positive solution which tends to zero.
Proof. We put and apply Theorem 3.1.
Example 3.3. Consider the nonlinear neutral differential equation
where . We will show that the conditions of Theorem 3.1 are satisfied. Condition (3.1) obviously holds and (3.2) has a form
. For function , we obtain
For , condition (3.9) is satisfied and
If the function satisfies (3.11), then (3.8) has a solution which is bounded by the functions .
For example if , from (3.11) we obtain and the equation has the solution which is bounded by the function and .
The research was supported by the Grant 1/0090/09 and the Grant 1/1260/12 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and Project APVV-0700-07 of the Slovak Research and Development Agency.
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