1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China 2Department of Mathematics, Kunming University, Kunming, Yunnan 650214, China 3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 4Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea
This paper is concerned with the th-order forced nonlinear neutral differential equation . Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.
Consider the following th-order forced nonlinear neutral differential equation:
where and are constants for . In what follows, we assume that and satisfy that
and there exists such that is positive eventually: is strictly increasing and in ; satisfies that
During the last decades, the oscillation criteria and the existence results of nonoscillatory solutions for various linear and nonlinear differential equations have been studied extensively, for example, see [1–28] and the references cited therein. In particular, Zhang and Yan  obtained some sufficient conditions for the oscillation of the first-order linear neutral delay differential equation with positive and negative coefficients:
where , and . Das and Misra  studied the nonhomogeneous neutral delay differential equation:
where , for , is nondecreasing, Lipschitzian, and satisfies for every , and they obtained a necessary and sufficient condition for the solutions of (1.5) to be oscillatory or tend to zero asymptotically. Parhi and Rath  extended Das and Misra’s result to the following forced first-order neutral differential equation with variable coefficients:
where , and they got necessary and sufficient conditions which ensures every solution of (1.6) is oscillatory or tends to zero or to as . By using Banach’s fixed point theorem, Zhang et al.  proved the existence of a nonoscillatory solution for the first-order linear neutral delay differential equation:
where , and for . Çakmak and Tiryaki  showed several sufficient conditions for the oscillation of the forced second-order nonlinear differential equations with delayed argument in the form:
where , , and . Travis  investigated the oscillatory behavior of the second-order differential equation with functional argument:
where and satisfies that has the same sign of and when they have the same sign. Lin  got some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation:
where with eventually, , is nondecreasing and for . Kulenović and Hadžiomerspahić  deduced the existence of a nonoscillatory solution for the neutral delay differential equation of second order with positive and negative coefficients:
where , , and for . Utilizing the fixed point theorems due to Banach, Schauder and Krasnoselskii, and Zhou and Zhang , and Zhou et al.  established some sufficient conditions for the existence of a nonoscillatory solution of the following higher-order neutral functional differential equations:
where , and for . Li et al.  investigated the existence of an unbounded positive solution, bounded oscillation, and nonoscillation criteria for the following even-order neutral delay differential equation with unstable type:
where , and . Zhang and Yan  obtained some sufficient conditions for oscillation of all solutions of the even-order neutral differential equation with variable coefficients and delays:
where is even, and for , and . Yilmaz and Zafer  discussed sufficient conditions for the existence of positive solutions and the oscillation of bounded solutions of the th-order neutral type differential equations:
where , , and . Bolat and Akin [4, 5] got sufficient criteria for oscillatory behaviour of solutions for the higher-order neutral type nonlinear forced differential equations with oscillating coefficients:
where , is nondecreasing and for for , for , and and are oscillating functions. Zhou and Yu  attempted to extend the result of Bolat and Akin  and established a necessary and sufficient condition for the oscillation of bounded solutions of the higher-order nonlinear neutral forced differential equation of the form:
where , and for and ; and are oscillating functions; and for ; is nondecreasing function, for and .
That is, they claimed the following result.
Theorem 1.1 (see [26, Theorem 2.1]). Assume that there is an oscillating function such that and ; is an oscillating function and ; , .Then, every bounded solution of (1.17) either oscillates or tends to zero if and only if
We, unfortunately, point out that the necessary part in Theorem 1.1 is false, see Remark 4.2 and Example 4.7 below. It is clear that (1.1) includes (1.4)–(1.17) as special cases. To the best of our knowledge, there is no literature referred to the oscillation and existence of uncountably many bounded nonoscillatory solutions of (1.1). The aim of this paper is to establish the bounded oscillation and the existence of uncountably many bounded positive and negative solutions for (1.1) without the monotonicity of the nonlinear term . Our results extend and improve substantially some known results in [4, 5, 9, 10, 20, 24, 26–28] and correct Theorem 2.1 in .
The paper is organized as follows. In Section 2, a few notation and lemmas are introduced and proved, respectively. In Section 3, by employing Krasnoselskii’s fixed point theorem and some techniques, the existence of uncountably many bounded positive and negative solutions for (1.1) are given, and some necessary and sufficient conditions for all bounded solutions of (1.1) to be oscillatory or tend to zero as are provided. In Section 4, a number of examples which clarify advantages of our results are constructed.
It is assumed throughout this paper that and
By a solution of (1.1), we mean a function for some , such that is times continuously differentiable in and such that (1.1) is satisfied for . As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is nonoscillatory, that is, if it is eventually positive or eventually negative. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Let stand for the Banach space of all bounded continuous functions in with the norm for each and
It is easy to see that is a bounded closed and convex subset of the Banach space .
Lemma 2.1. Let and be bounded. If eventually, then(a) exists and for ; furthermore, there exists for odd and for even such that(b) eventually for ;(c) is nonincreasing eventually for .
Proof. Now, we consider two possible cases below. Case 1. Assume that . Let . Note that eventually. It follows that there exists a constant satisfying , for all , which yields that is nonincreasing in . Since is bounded in , it follows that exists. Case 2. Assume that . Notice that is odd. It follows that eventually, which implies that there exists a constant satisfying
which means that
Suppose that there exists a constant satisfying , which together with (2.4) gives that
which guarantees that is increasing in and
which means that
which contradicts the boundedness of . Consequently, we have
Combining (2.4) and (2.9), we conclude easily that there exists a constant with
Next, we claim that . Otherwise, there exists a constant satisfying
which yields that
which gives that
which means that
which contradicts the boundedness of in . Hence, , that is,
Repeating the proof of (2.3)–(2.15), we deduce similarly that
which together with the boundedness of implies that is nonincreasing in and exists. Thus, (2.3) and (2.16) yield (a)–(c). This completes the proof.
Lemma 2.2. Let satisfy and
where is a fixed constant. Then, .
Proof. Since is a strictly increasing continuous function, in and , it follows that the inverse function of is also strictly increasing continuous, in and , where for all . Equation (2.18) implies that there exists a constant with
Using (2.18) and (2.19), we deduce that, for any , there exist sufficiently large numbers and satisfying
In view of (2.17), (2.20), and (2.21), we infer that for all
which gives that . This completes the proof.
Lemma 2.3. Let , and be in satisfying , (2.17), (2.18), and
where , and are constants. Then, there exists such that eventually.
Proof. Obviously, (2.20) holds. It follows from (2.18), (2.23), and (2.24) that for , there exist and satisfying
Put . In light of (2.17), we conclude that for each
which together with (2.20) and (2.25) yields that for any
This completes the proof.
Similar to the proof of Lemma 3.2 in , we have the following two lemmas.
Lemma 2.4. Let , and be in satisfying , (2.17), (2.18), and
where is a constant. Then, .
Lemma 2.5. Let , , , , and be in satisfying , (2.17), (2.18), (2.23), and (2.29). Then, there exists such that eventually.
Lemma 2.6 (Krasnoselskii’s fixed point theorem). Let be a Banach space, let be a nonempty bounded closed convex subset of , and let , be mappings of into such that for every pair . If is a contraction mapping and is completely continuous, then the mapping has a fixed point in .
3. Main Results
First, we use the Krasnoselskii’s fixed point theorem to show the existence and multiplicity of bounded positive and negative solutions of (1.1).
Theorem 3.1. Let , and hold. Assume that there exist , and satisfying
Then, the following hold:(a)for arbitrarily positive constants and with
equation (1.1) has uncountably many bounded positive solutions with
(b)for arbitrarily positive constants and with
equation (1.1) has uncountably many bounded negative solutions with
Proof. It follows from (3.1) and (3.2) that there exists an enough large constant with satisfying
(a) Assume that and are arbitrary positive constants satisfying (3.4). Let . First of all, we prove that there exist two mappings and a constant such that has a fixed point , which is also a bounded positive solution of (1.1) with . Put
In light of (3.3), (3.9), and , we infer that there exists a sufficiently large number satisfying
Define two mappings by
for each . In view of (3.1), (3.8), and (3.10)–(3.12), we conclude that for any and
which ensures that
It follows from (3.11), (3.12), (3.15), and (3.16) that and map into , respectively. Now, we show that is continuous in . Let and with , given . It follows from the uniform continuity of in for and that there exist and satisfying
In view of (3.8), (3.12), (3.17), we arrive at
which means that is continuous in . Next, we show that is equicontinuous in . Let . Taking into account (3.3) and , we know that there exists satisfying
It follows from the uniform continuity of and in that there exists satisfying
Let and with . We consider three possible cases. Case 1. Let . In view of (3.8), (3.9), (3.12), and (3.19), we conclude that
Case 2. Let . In terms of (3.8), (3.9), (3.12), (3.21), we arrive at
Case 3. Let . By (3.12), we have
Thus, is equicontinuous in . Consequently, is relatively compact by (3.16) and the continuity of . By means of (3.14), (3.15), and Lemma 2.6, we infer that possesses a fixed point , that is,
which gives that
which mean that is a bounded positive solution of (1.1) with
Let and be two arbitrarily different numbers in