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Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations
Journal of Inequalities and Applications volume 2014, Article number: 302 (2014)
Abstract
This paper deals with the solvability of the higher-order nonlinear neutral delay differential equation , , where , , , and satisfying , , . With respect to various ranges of the function p, we investigate the existence of uncountably many bounded nonoscillatory solutions for the equation. The main tools used in this paper are the Krasnoselskii and Schauder fixed point theorems together with some new techniques. Six nontrivial examples are given to illustrate the superiority of the results presented in this paper.
MSC:39A10, 39A20, 39A22.
1 Introduction and preliminaries
In the past two decades, the oscillation, nonoscillation, and existence of solutions for some kinds of neutral delay differential equations have been extensively studied by many authors. See, for example, [1–16] and the references cited therein.
Recently, Zhang et al. [14] and Öcalan [10] got several existence results of a nonoscillatory solution or positive solution for the first-order neutral delay differential equations
and
where , , and for . Shen and Debnath [12] obtained some sufficient conditions for the oscillations of (1.1) and Luo and Shen [9] established a few oscillation and nonoscillation criteria for (1.2). Liu and Huang [6] used the coincidence degree theory to get the existence and uniqueness results of a T-periodic solution for the first-order neutral functional differential equation with a deviating argument of the form
where c, τ are constants, , , , f, α are T-periodic and , are T-periodic in the first argument. Using the Banach fixed point theorem, Kulenović and Hadžiomerspahić [4] studied the existence of a nonoscillatory solution for the second-order neutral delay differential equation with positive and negative coefficients
where , and . Kong et al. [3] established a complete classification of nonoscillatory solutions for the higher-order neutral differential equation
and gave conditions for each type of nonoscillatory solutions to exist, where n is an odd number, , , and . Zhou and Zhang [16] extended the results in [5] to higher-order neutral functional differential equation with positive and negative coefficients
where , and . Liu et al. [8] investigated the higher-order neutral delay differential equation with positive and negative coefficients
where , , , , and . Utilizing the Banach fixed point theorem, they obtained the existence of bounded nonoscillatory solutions for (1.7), suggested some algorithms for approximating these bounded nonoscillatory solutions, and discussed the convergence and stability of iteration sequences generated by the algorithms. Parhi [11] discussed the oscillation of solutions for the higher-order neutral delay linear differential equation
where , , , and for . Li et al. [5] considered the following higher-order neutral delay differential equation with unstable type:
proved bounded oscillation and nonoscillation criteria and the existence of an unbounded positive solution for (1.9), where n is an even integer, , , , . Zhou and Zhang [15] used the Krasnoselskii and Schauder fixed point theorems to prove the existence of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation
where , for and . Liu et al. [7] got the existence of infinitely many nonoscillatory solutions for the n th-order neutral delay differential equation
where , , for , , and .
However, to the best of our knowledge, there exist no results for the existence of solutions of the higher-order nonlinear neutral delay differential equation
where , , , and satisfying
It is clear that (1.12) includes (1.1)-(1.11) as special cases. The purpose of this paper is to study the solvability of (1.12) under various ranges of the function p. Utilizing the Krasnoselskii and Schauder fixed point theorems and some new techniques, we study sufficient conditions of the existence of uncountably many bounded nonoscillatory solutions for (1.12) relative to various ranges of the function p. The results presented in this paper extend, improve, and unify the corresponding results in [4] and [13–16]. Six nontrivial examples are also given to illustrate the importance of the results obtained in this paper.
Throughout this paper, we assume that ℝ, and ℕ denote the sets of all real numbers, nonnegative numbers and positive integers, respectively, and
Let stand for the Banach space of all continuous and bounded functions on with norm for all and
where with . It is easy to see that is a nonempty bounded closed convex subset of .
By a solution of (1.12), we mean a function for some , such that is n times continuously differentiable on and (1.12) holds for . As is customary, a solution of (1.12) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise.
The following lemmas are well known.
Lamma 1.1 (Krasnoselskii fixed point theorem [1])
Let X be a nonempty bounded closed convex subset of a Banach space E and let U, S be maps of X into E such that for every pair . If U is a contraction and S is completely continuous, then the equation has a solution in X.
Lamma 1.2 (Schauder fixed point theorem [1])
Let X be a nonempty closed convex subset of a Banach space E. Let be a continuous mapping such that SX is a relatively compact subset of X. Then S has at least one fixed point in X.
2 The existence of uncountably many bounded nonoscillatory solutions
Now we investigate sufficient conditions for the existence of uncountably many bounded nonoscillatory solutions of (1.12) under various ranges of the function p. The proofs of the results presented in this section are based on the Krasnoselskii and Schauder fixed point theorems and a few new and key techniques, one of which is to construct the mappings and satisfying the conditions in the cited fixed point theorems for each constant L, which belongs to certain interval. Let
Theorem 2.1 Assume that there exist and constants M, N, , and c satisfying
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.2) and (2.3) that there exist constants and sufficiently large satisfying
and
Define two mappings and by
for each .
First of all we show that
Let and . By (2.3)-(2.6), we get
and
which imply (2.7).
Second, we show that is continuous in and is relatively compact. Let be an arbitrary sequence in with
Since is a closed subset of , it follows that . Put
From (2.8) and the continuity of f and for we infer that
which together with (2.6) and the Lebesgue dominated convergence theorem yields for any
which gives
which implies that is continuous in .
Using (2.1), (2.5), and (2.6), we conclude that for any and
which yields
That is, is uniformly bounded in . In order to prove that is relatively compact, we have to prove that is equicontinuous in . Let be given. Equation (2.2) ensures that there exists satisfying
Put
Now we consider the following possible cases:
-
(i)
with . By (2.1), (2.6), and (2.10) we have
(2.12) -
(ii)
with and . By means of (2.1), (2.6), and (2.11) we infer that
(2.13) -
(iii)
with and . In light of (2.1), (2.5), and (2.6), we get
which together with the mean value theorem and (2.11) yields
-
(iv)
with . Clearly (2.6) means that
(2.15)
It follows from (2.12)-(2.15) that is equicontinuous in . Thus Lemma 1.1 means that there exists such that . That is,
which implies that
that is, is a bounded nonoscillatory solution of (1.12) in .
Finally, we show (1.12) has uncountably many bounded nonoscillatory solutions in . Let with . As in the above proof we can deduce that for each , there exist constants , , and mappings satisfying (2.4)-(2.6), where θ, T, L, and are replaced by , , , , , respectively, and has a fixed point . That is, and are also bounded nonoscillatory solutions of (1.12) in . We now need to show that . In view of (2.2) there exists satisfying
Note that (2.6) means that for ,
It follows from (2.1), (2.4), and (2.17) that for
which together with (2.16) implies that
which yields . This completes the proof. □
Theorem 2.2 Assume that there exist and constants M, N, and c satisfying (2.1), (2.2), and
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . By (2.2) and (2.18), we choose constants and satisfying (2.4) and
Define two mappings and by (2.6).
Let and . In terms of (2.6), (2.18), and (2.19), we arrive at
which yields for any . The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □
Theorem 2.3 Assume that there exist and constants M, N, , , and c satisfying (2.1), (2.2), and
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.2) and (2.20) that there exist constants and satisfying
and
Define two mappings and by
We show that (2.7) holds. In fact, for every and , by (2.1) and (2.20)-(2.23), we get
and
which means that we have (2.7).
Next we show that is equicontinuous in . For any given , (2.2) guarantees that (2.10) holds for some sufficiently large . Set
It follows from the uniform continuity of p in that there exists such that
whenever with . Put
We have to consider the following possible cases:
-
(i)
with . It follows from (2.1), (2.10), (2.20), (2.23), and (2.26) that
(2.27) -
(ii)
with . For each , it follows from the mean value theorem that there exists satisfying
which together with (2.1), (2.20), (2.22), and (2.26) yields for each
which implies that for each
By means of (2.1), (2.20), (2.22)-(2.26), and (2.28), we get
-
(iii)
with . Obviously, (2.23) guarantees that
(2.30)
Using (2.27), (2.29), and (2.30), we conclude that is equicontinuous in . As in the proof of Theorem 2.1, we prove similarly that is continuous in and is uniformly bounded. It follows that is relatively compact. Consequently, Lemma 1.1 shows that there is such that . That is,
which yields
which implies that
that is, is a bounded nonoscillatory solution of (1.12) in .
Finally we show (1.12) has uncountably many bounded nonoscillatory solutions. Let with . As in the above proof, we infer that for each , there exist constants , , and mappings satisfying (2.21)-(2.23), where θ, T, L, , are replaced by , , , , , respectively, and (1.12) possesses a bounded nonoscillatory solution . In terms of (2.2), we select satisfying
It follows from (2.21), (2.23), and (2.31) that for
which yields
that is, . This completes the proof. □
Theorem 2.4 Assume that there exist and constants M, N, , , and c satisfying (2.1), (2.2), and
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.2) and (2.32) that there exist constants and satisfying (2.21) and
Let the mappings and be defined by (2.23).
Note that (2.1), (2.21), (2.23), and (2.33) imply that for each , and
and
which yields for any . The rest of the proof is similar to that of Theorem 2.3 and is omitted. This completes the proof. □
Theorem 2.5 Let . Assume that there exist and constants M, N, and c satisfying (2.1), (2.2), and
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.2) and (2.34) that there exists a constant satisfying
Define a mapping by
For every and , by (2.1), (2.35), and (2.36), we deduce that
which yield and hence is uniformly bounded in .
Let be a sequence in and satisfying (2.8) and let
Using (2.8), (2.37), and the continuity of f, , , and for and , we obtain for all . In light of (2.36) and the Lebesgue dominated convergence theorem, we conclude that for any
which means that
which implies that is continuous in .
Let ε be an arbitrary positive number. It follows from (2.2) that there exists large enough such that
Set
where satisfies
We consider the following possible cases:
-
(i)
with . From (2.1), (2.36), and (2.38), we conclude immediately that
(2.41) -
(ii)
with . In terms of (2.1) and (2.36)-(2.40), we deduce that
(2.42) -
(iii)
with . Equation (2.36) means that
(2.43)
It follows from (2.41)-(2.43) that is equicontinuous in . Thus Lemma 1.2 means that has a fixed point , that is, for any
and
which give for any
which implies that
that is, is a bounded nonoscillatory solution of (1.12). The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □
Theorem 2.6 Let . Assume that there exist and constants M, N, and c satisfying (2.1), (2.2), and (2.34). Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.2) and (2.34) that there exists a constant satisfying
Define a mapping by
for each .
Let and . By (2.44) and (2.45), we get
which gives that and is uniformly bounded in .
Let satisfy (2.8) for some and be defined by (2.37). Using (2.8), (2.37), (2.45), the continuity of f and for , and the Lebesgue dominated convergence theorem, we conclude that for any
which yields
that is, is continuous in .
Next we show that is equicontinuous in . Let . Equation (2.34) ensures that there exists large enough satisfying
Let B and K be defined by (2.39) and (2.40), respectively. Put
Now we have to consider the following possible cases:
-
(i)
with . In terms of (2.45)-(2.47), we know that
(2.48) -
(ii)
with . By means of (2.39), (2.40), (2.44)-(2.47), and the mean value theorem, we conclude that
(2.49) -
(iii)
with . It is easy to see that
(2.50)
It follows from (2.48)-(2.50) that is equicontinuous in . Consequently is relatively compact. Thus Lemma 1.2 ensures that possesses a fixed point , that is,
and
which imply that
which yields
which together with (2.34) means that x is a bounded nonoscillatory solution of (1.12) in .
Let with . For each , there exist a constant and a mapping satisfying (2.44) and (2.45), where T, L, and are replaced by , , and , respectively, and (1.12) possesses a bounded nonoscillatory solution . By (2.2), we choose some with
Using (2.1), (2.45), and (2.51), we infer that for
that is, . Consequently (1.12) has uncountably bounded nonoscillatory solutions in . This completes the proof. □
Theorem 2.7 Assume that there exist and constants M, N, and c satisfying (2.1),
and
Then (1.12) has uncountably many bounded nonoscillatory solutions in .
Proof Let . It follows from (2.52) that there exists a constant satisfying
Let and denote the largest integer not exceeding . Note that
and (2.52) implies that
which yields
Define a mapping by
Notice that (2.1) and (2.55) mean that the mapping is well defined. Let . In view of (2.1), (2.54), and (2.56), we conclude that for any
which shows that and is uniformly bounded in .
Let be a sequence in and satisfying (2.8) and let be defined by (2.37). By means of (2.37), (2.56), the continuity of f, , and for and , and the Lebesgue dominated convergence theorem, we deduce that for any
which gives
that is, is continuous in .
Next we prove that is equicontinuous in . Given a positive number ε. It follows from (2.52) that there exists large enough satisfying
Let B and K be defined by (2.39) and (2.40), respectively. Put
Now we have to consider the following possible cases:
-
(i)
with . In view of (2.1) and (2.56)-(2.58), we get
(2.59) -
(ii)
with . Let . In light of (2.1), (2.54), (2.56)-(2.58), and the mean value theorem, we conclude that
(2.60)
Let . By means of (2.1), (2.54), and (2.56)-(2.58), we get
-
(iii)
with . It is easy to see that
(2.62)
It is easy to see that (2.59)-(2.62) ensure that is equicontinuous in . Consequently is relatively compact. Thus Lemma 1.2 implies that possesses a fixed point , that is,
and
which imply that
which together with (2.53) means that x is a bounded nonoscillatory solution of (1.12) in .
Let and be two different numbers in . Similar to the above proof, we infer that there exist constants , and mappings defined by (2.56), where L, T, and are replaced by , , and , respectively, and for , such that and have, respectively, fixed points and , which are bounded nonoscillatory solutions of (1.12) in , that is,
Note that (2.52) ensures that there exists satisfying
Combining (2.1), (2.63), and (2.64), we deduce that for any
which implies that
that is, . Consequently, (1.12) has uncountably bounded nonoscillatory solutions in . This completes the proof. □
Remark 2.1 Theorems 2.1-2.7 extend, improve, and unify the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16].
3 Examples and applications
Now we construct six nontrivial examples to show the superiority and applications of the results presented in the second section.
Example 3.1 Consider the higher-order neutral delay differential equation:
where is a constant. Let , , , , , , ,
It is easy to verify that (2.1)-(2.3) are fulfilled. Thus Theorem 2.1 ensures that (3.1) has uncountably many bounded nonoscillatory solutions in . But the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are invalid for (3.1).
Example 3.2 Consider the higher-order neutral delay differential equation:
where is a constant, , , , , , ,
Obviously, (2.1), (2.2), and (2.18) hold. It follows from Theorem 2.2 that (3.2) has uncountably many bounded nonoscillatory solutions in . However, the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are unapplicable for (3.2).
Example 3.3 Consider the higher-order neutral delay differential equation:
where is a constant. Let , , , , ,
It is easy to verify that (2.1), (2.2), and (2.20) are fulfilled. Consequently Theorem 2.3 means that (3.3) has uncountably many bounded nonoscillatory solutions in . But the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are not applicable for (3.3).
Example 3.4 Consider the higher-order neutral delay differential equation:
where is a constant. Let , , , , , , ,
Obviously, (2.1), (2.2), and (2.32) hold. Consequently Theorem 2.4 shows that (3.4) has uncountably many bounded nonoscillatory solutions in . But the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are useless for (3.4).
Example 3.5 Consider the higher-order neutral delay differential equation:
where is a constant. Let , , , , , ,
It is clear that (2.1), (2.2), and (2.34) hold. Consequently Theorems 2.5 and 2.6 ensure that (3.5) has uncountably many bounded nonoscillatory solutions in . But the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are null for (3.5).
Example 3.6 Consider the higher-order neutral delay differential equation:
where is a constant. Let , , , , ,
It is clear that (2.1), (2.2), and (2.52) hold. Consequently Theorem 2.7 shows that (3.6) has uncountably many bounded nonoscillatory solutions in . But the theorem in [4], Theorems 2 and 3 in [13], Theorems 1-3 in [14], Theorems 1-6 in [15], and Theorems 1-4 in [16] are void for (3.6).
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Acknowledgements
This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2057665).
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Liu, Z., Jia, M., Ume, J.S. et al. Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations. J Inequal Appl 2014, 302 (2014). https://doi.org/10.1186/1029-242X-2014-302
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DOI: https://doi.org/10.1186/1029-242X-2014-302