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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 693890, 23 pages
http://dx.doi.org/10.1155/2011/693890
Research Article

Existence of Nonoscillatory Solutions for a Third-Order Nonlinear Neutral Delay Differential Equation

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 4 April 2011; Accepted 28 June 2011

Academic Editor: Elena Braverman

Copyright © 2011 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to study the solvability of a third-order nonlinear neutral delay differential equation of the form {??(??)[??(??)(??(??)+??(??)??(??-??))?]?}?+??(??,??(??1(??)),??(??2(??)),,??(????(??)))=0, ??=??0. By using the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we demonstrate the existence of uncountably many bounded nonoscillatory solutions for the above differential equation. Several nontrivial examples are given to illustrate our results.

1. Introduction and Preliminaries

In recent years, the study of the oscillation, nonoscillation, asymptotic behaviors and existence of solutions for various kinds of first- and second-order neutral delay differential equations and systems of differential equations have attracted much attention, for example, see [112] and the references therein. Dorociaková and Olach [2] discussed the existence of nonoscillatory solutions and asymptotic behaviors for the first-order delay differential equation ???(??)+??(??)??(??)+??(??)??(??(??))=0,??=0.(1.1) Elbert [3] and Huang [5] established a few oscillation and nonoscillation criteria for the second-order linear differential equation ????(??)+??(??)??(??)=0,??=0,(1.2) where ?????([0,+8),R+). Tang and Liu [10] studied the existence of bounded oscillation for the second-order linear delay differential equation of unstable type ????(??)=??(??)??(??-??),??=??0,(1.3) where ??>0,?????([??0,+8),R+) and ??(??)?0 on any interval of length ??. In view of the Banach fixed point theorem, Kulenovic and Hadžiomerspahic [7] deduced the existence of a nonoscillatory solution for the second-order linear neutral delay differential equation with positive and negative coefficients [??](??)+????(??-??)??+??1?(??)????-??1?-??2?(??)????-??2?=0,??=??0,(1.4) where ???R?{-1,1},??>0,??1,??2?[0,+8),??1,??2???([??0,+8),R+). Qin et al. [9] and Yang et al. [11] developed several oscillation criteria for the second-order differential equation ???(??)(??(??)+??(??)??(??-??))???+??(??)??(??(??-??))=0,??=??0,(1.5) where ?? and ?? are nonnegative constants, ??,??,?????([??0,+8),R), and ?????(R,R). By utilizing the Krasnoselskii’s fixed point theorem, Zhou [12] discussed the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation ???(??)(??(??)+??(??)??(??-??))???+?????=1???????(??)???????-??????=0,??=??0,(1.6) where ??=1 is an integer, ??>0,????=0,??,??,???????([??0,+8),R), and ???????(R,R) for ???{1,2,,??}.

However, the existence of nonoscillatory solutions of third-order neutral differential equations received much less attention, moreover, the results in [7, 11, 12] only figured out the existence of a nonoscillatory solution of (1.3)–(1.5), respectively.

Motivated by the papers mentioned above, in this paper, we investigate the following third-order nonlinear neutral delay differential equation ????(??)??(??)(??(??)+??(??)??(??-??))?????????+????,??1????(??),??2????(??),,????(??)??=0,??=??0,(1.7) where ??=1 is an integer, ??>0,??,?????([??0,+8),R+?{0}),?????([??0,+8),R), and ?????([??0,+8)×R??,R). By applying the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we obtain some sufficient conditions for the existence of uncountably many bounded nonoscillatory solutions of (1.7).

Throughout this paper, we assume that R=(-8,+8),R+=[0,+8),??([??0,+8),R) denotes the Banach space of all continuous and bounded functions on [??0,+8) with the norm ????=sup??=??0|??(??)| for each ?????([??0,+8),R) and ?????(??,??)=???????0??,+8,R:??=??(??)=??,??=??0?for??>??>0.(1.8) It is easy to see that ??(??,??) is a bounded closed and convex subset of ??([??0,+8),R).

By a solution of (1.7), we mean a function ?????([??1-??,+8),R) for some ??1=??0 such that ??(??)+??(??)??(??-??),??(??)(??(??)+??(??)??(??-??))? and ??(??)[??(??)(??(??)+??(??)??(??-??))?]?are continuously differentiable in [??1,+8) and such that (1.7) is satisfied for ??=??1. As is customary, a solution of (1.7) is called oscillatory if it has arbitrarily large zeros, and otherwise it is said to be nonoscillatory.

Definition 1.1 (see [6]). A family ?? of functions in ??([??0,+8),R) is equicontinuous on [??0,+8) if for any ??>0, the interval [??0,+8) can be decomposed into a finite number of subintervals ??1,??2,,???? such that ||||??(??)-??(??)=??,??????,??,???????,???{1,2,,??}.(1.9)

Lemma 1.2 (see Krasnoselskii’s fixed point theorem, [4]). Let ?? be a Banach space. Let O be a bounded closed convex subset of ?? and ??1 and ??2 mappings from O into ?? such that ??1??+??2???O for every pair ??,???O. If ??1 is a contraction and ??2 is completely continuous, then the equation ??1??+??2??=?? has at least one solution in O.

Lemma 1.3 (see Schauder’s fixed point theorem, [4]). Let O be a nonempty closed convex subset of a Banach space ??. Let ??:O?O be a continuous mapping such that ??O is a relatively compact subset of ??. Then ?? has at least one fixed point in O.

2. Main Results

Now we study those conditions under which (1.7) possesses uncountably many bounded nonoscillatory solutions.

Theorem 2.1. Assume that there exist constants ??,??,??1,??2,??0 and a function h???([??0,+8),R+) satisfying ???min1,??2?=0,??1+??2?<1,0<??<1-??1-??2???,(2.1)-??2=??(??)=??1,??=??0>??0;(2.2)||?????,??1,??2,,?????||???=h(??),???0?,+8,?????[]??,??,???{1,,??};(2.3)???+80???+8???+8h(??)??(??)??(??)????????????<+8.(2.4) Then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???(??1??+??,(1-??2)??). From (2.4), we pick up ??>??0 such that ???+8???+8???+8h(??)??(??)??(??)????????????<min??1-??2???-??,??-??1???-??.(2.5) Define two mappings ??1?? and ??2??:??(??,??)???([??0,+8),R) by ???1?????????(??)=??-??(??)??(??-??),??=??,1?????(??),??0=??<??,(2.6)???2?????(????????)=??+8???+8???+81????????(??)??(??)??,??1????(??),,???????(??)??????????????,??=??,2???????(??),0=??<??(2.7) for ?????(??,??).
Firstly, we prove that ??1??(??(??,??))+??2??(??(??,??))???(??,??). By (2.1)–(2.3) and (2.5)–(2.7), we get that ???1????????(??)+2?????(??)=??+??2???+??+8???+8???+8h(??)??(??)??(??)????????????=??+??2??+min??1-??2???-??,??-??1??????-??=??,??,?????(??,??),??=??,1????????(??)+2?????(??)=??-??1???-??+8???+8???+8h(??)??(??)??(??)????????????=??-??1??-min??1-??2???-??,??-??1???-??=??,??,?????(??,??),??=??,(2.8) which infer that ??1??(??(??,??))+??2??(??(??,??))???(??,??) for any ??,?????(??,??).
Secondly, we show that ??1?? is a contraction mapping. By (2.1), (2.2), and (2.6), we deduce that ||???1????????(??)-1?????||=||??||||??||=???(??)(??)(??-??)-??(??-??)1+??2????-???(2.9) for ??,?????(??,??) and ??=??, which gives that ????1????-??1??????=???1+??2????-???.(2.10)
Thirdly, we show that ??2?? is completely continuous. Let ?????(??,??) and {????}??=1???(??,??) with lim???+8????=??. By (2.7), we obtain that ||???2??????????(??)-2?????||=?(??)??+8???+8???+81||?????(??)??(??)??,???????1(???),,?????????(??????)??-????,??1????(??),,????||(??)??????????????,??=??,??=1.(2.11) Using (2.3) and (2.4), we conclude that ||?????,???????1?(??),,?????????????(??)??-????,??1????(??),,????||[?(??)??=2h(??),?????,+8),??=1,??+8||?????,???????1(???),,?????????(??????)??-????,??1(??????),,????(||???)??????=2??+8[?h(??)????,?????,8),??=1,??+8???+81||?????(??)??,???????1?(??),,?????????????(??)??-????,??1????(??),,????||?(??)??????????=2??+8???+8h(??)[??(??)????????,??,?????,8),??=1.(2.12) In light of (2.11)–(2.12), ||?????,???????1?(??),,?????????????(??)??-????,??1????(??),,????||(??)???0as???+8,(2.13) and the Lebesgue dominated convergence theorem, we conclude that lim???+8????2??????-??2??????=0,(2.14) which means that ??2?? is continuous in ??(??,??).
Now we show that ??2?? is completely continuous. By virtue of (2.3), (2.4), and (2.7), we get that ????2??????=???+8???+8???+8h(??)??(??)??(??)????????????,?????(??,??).(2.15) That is, ??2??(??(??,??)) is uniform bounded. It follows from (2.4) that for each ??>0, there exists ??*>?? such that ???+8*???+8???+8h(??)????(??)??(??)????????????<2.(2.16) In view of (2.3), (2.7), and (2.16), we infer that ||???2????????2?-???2????????1?||=???+82???+8???+8h(??)???(??)??(??)????????????+??+81???+8???+8h(??)??(??)??(??)????????????<??,?????(??,??),??2>??1=??*.(2.17) From (2.3) and (2.7), we get that ||???2????????2?-???2????????1?||=???2??1???+8???+8h(??)??(??)??(??)????????????,?????(??,??),??=??1=??2=??*,(2.18) which together with (2.4) ensures that there exists ??>0 satisfying ||???2????????2?-???2????????1?||<??,?????(??,??),??1,??2????,??*?||??with2-??1||<??.(2.19) Clearly, ||???2????????2?-???2????????1?||=0<??,?????(??,??),??0=??1=??2=??.(2.20) That is, ??2??(??(??,??)) is equicontinuous on [??0,+8). Consequently, ??2?? is completely continuous. By Lemma 1.2, there is ??0???(??,??) such that ??1??0+??2??0=??0, which is a bounded nonoscillatory solution of (1.7).
Lastly, we demonstrate that (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??). Let ??1,??2?(??1??+??,(1-??2)??) and ??1???2. For each ???{1,2}, we choose ????>??0 and the mappings ??????1 and ??????2 satisfying (2.5)–(2.7) with ?? and ?? replaced by ???? and ????, respectively, and some ??3>max{??1,??2} such that ???+83???+8???+8h(??)||????(??)??(??)????????????<1-??2||2.(2.21) Obviously, there are ??,?????(??,??) such that ??1??1??+??2??1??=?? and ??1??2??+??2??2??=??, respectively. That is, ?? and ?? are two bounded nonoscillatory solutions of (1.7) in ??(??,??). In order to prove that (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??), we prove only that ?????. In fact, by (2.6) and (2.7), we gain that for ??=??3??(??)=??1?-??(??)??(??-??)+??+8???+8???+81????????(??)??(??)??,??1????(??),,????(??)??????????????,??(??)=??2?-??(??)??(??-??)+??+8???+8???+81????????(??)??(??)??,??1(??????),,????(??)??????????????,(2.22) which together with (2.1)–(2.3) imply that ||??||=||??(??)-??(??)1-??2||-||??||||??||-?(??)(??-??)-??(??-??)??+8???+8???+81×||????????(??)??(??)??,??1(??????),,????(??????)??-????,??1(??????),,????(||=||????)??????????????1-??2||-???1+??2?????-???-2??+83???+8???+8h(??)??(??)??(??)????????????,??=??3,(2.23) which means that 1???-???=1+??1+??2?||??1-??2||?-2??+83???+8???+8h(??)???(??)??(??)????????????>0,(2.24) by (2.1) and (2.21). That is, ?????. This completes the proof.

Theorem 2.2. Assume that there exist constants ??,??,??1,??2,??0 and a function h???([??0,+8),R+) satisfying (2.3), (2.4) and ?0<1-??2????<1-??1???;(2.25)0=??2=??(??)=??1<1,??=??0>??0.(2.26) Then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???(??1??+??,??+??2??). It follows from (2.4) that there exists ??>??0 such that ???+8???+8???+8h(??)???(??)??(??)????????????<min??+??2??-??,??-??1???-??.(2.27) Define two mappings ??1?? and ??2??:??(??,??)???([??0,+8),R) by (2.6) and (2.7), respectively. By virtue of (2.3), (2.6), (2.7), (2.26), and (2.27), we obtain that ???1????????(??)+2??????(??)=??-??(??)??(??-??)+??+8???+8???+8h(??)??(??)??(??)????????????=??-??2???+min??+??2??-??,??-??1??????-??=??,??,?????(??,??),??=??,(2.28)1????????(??)+2??????(??)=??-??(??)??(??-??)-??+8???+8???+8h(??)??(??)??(??)????????????=??-??1???-min??+??2??-??,??-??1???-??=??,??,?????(??,??),??=??,(2.29) which yield that ??1??(??(??,??))+??2??(??(??,??))???(??,??).
By a similar argument used in the proof of Theorem 2.1, we gain that ??1?? is a contraction mapping ??2?? is completely continuous, and (1.7) possesses uncountably many nonoscillatory solutions. This completes the proof.

Theorem 2.3. Assume that there exist constants ??,??,??1,??2,??0 and a function h???([??0,+8),R+) satisfying (2.3), (2.4), (2.25), and -1<-??1=??(??)=-??2=0,??=??0>??0.(2.30) Then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???((1-??2)??,(1-??1)??). It follows from (2.4) that there exists ??>??0 satisfying ???+8???+8???+8h(??)??(??)??(??)????????????<min??1-??1????-??,??-1-??2????.(2.31) Let ??1?? and ??2??:??(??,??)???([??0,+8),R) be defined by (2.6) and (2.7), respectively. In view of (2.3), (2.6), (2.7), (2.30), and (2.31), we obtain that ???1??????(??)+2????(??)=??-??(??)??(??-??)+??+8???+8???+8h(??)??(??)??(??)????????????=??+??1??+min??1-??1????-??,??-1-??2???????=??,??,?????(??,??),??=??,1??????(??)+2????(??)=??-??(??)??(??-??)-??+8???+8???+8h(??)??(??)??(??)????????????=??+??2??-min??1-??1????-??,??-1-??2????=??,??,?????(??,??),??=??,(2.32) which mean that ??1??(??(??,??))+??2??(??(??,??))???(??,??).
The rest of the proof is similar to the proof of Theorem 2.1 and is omitted. This completes the proof.

Theorem 2.4. Assume that there exist constants ?? and ?? with ??>??>0 and a function h???([??0,+8),R+) satisfying (2.3) and (2.4). If ??(??)=1 for each ???[??0,+8), then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???(??,??). It follows from (2.4) that there exists ??=??0+?? satisfying ???+8???+8???+8h(??)??(??)??(??)????????????<min{??-??,??-??}.(2.33) Define a mapping ????:??(??,??)???([??0,+8),R) by ?????????????(??)=??+8???=1???+2??????+(2??-1)?????+8???+81????????(??)??(??)??,??1(??????),,????(?????)??????????????,??=??,?????(??),??0=??<??(2.34) for ?????(??,??).
Firstly, we prove that ????(??(??,??))???(??,??). By virtue of (2.3), (2.33) and (2.34), we obtain that ????????(??)=??+8???=1???+2??????+(2??-1)?????+8???+8h(??)???(??)??(??)????????????=??+??+8???+8???+8h(??)?????(??)??(??)????????????=??,?????(??,??),??=??,?????(??)=??-8???=1???+2??????+(2??-1)?????+8???+8h(??)???(??)??(??)????????????=??-??+8???+8???+8h(??)??(??)??(??)????????????=??,?????(??,??),??=??,(2.35) which imply that ????(??(??,??))???(??,??).
Secondly, we show that ???? is continuous in ??(??,??). Let ?????(??,??) and {????}??=1???(??,??) with lim???+8????=??. By (2.34), we get that ||?????????????(??)-?????||=(??)8???=1???+2??????+(2??-1)?????+8???+81||?????(??)??(??)??,???????1?(??),,?????????????(??)??-????,??1????(??),,????||=?(??)????????????????+8???+8???+81||?????(??)??(??)??,???????1?(??),,?????????????(??)??-????,??1(??????),,????(||??)??????????????,??=??,??=1.(2.36) In view of (2.12), (2.13), (2.36), and the Lebesgue dominated convergence theorem, we deduce that lim???+8??????????-????????=0,(2.37) which means that ???? is continuous in ??(??,??).
Thirdly, we show that ????(??(??,??)) is relatively compact. From (2.3), (2.33), and (2.34), we gain that ???????????=??+??+8???+8???+8h(??)??(??)??(??)????????????=2??,?????(??,??),(2.38) which means that ????(??(??,??)) is uniform bounded.
Let ??>0. It follows from (2.4) that there exist ??>??*>?? such that ???+8*???+8???+8h(??)????(??)??(??)????????????<2,(2.39)?+8?????+8???+8h(??)????(??)??(??)????????????<4.(2.40) By (2.3), (2.34), and (2.39), we deduce that ||???????????2?-???????????1?||=8???=1????2??+2????2+(2??-1)??+???1??+2????1+(2??-1)??????+8???+8h(??)???(??)??(??)????????????=2??+8*???+8???+8h(??)??(??)??(??)????????????<??,?????(??,??),??2>??1=??*.(2.41) Choose an integer ??=1 with ??+(2??+1)??=??, and put ????=max??+8???+8h(??)???(??)??(??)????????:?????+??,??*+??,??(2??-1)????=max??+8???+8h(??)???(??)??(??)????????:?????+2??,??*??,???+2??????=min,??1+4?????.1+4????(2.42) Equation (2.4) means that max{??,??}<+8. It follows from (2.3), (2.34), and (2.40)–(2.42) that for ?????(??,??),??1,??2?[??,??*]with??1=??2=??1+??||???????????2?-???????????1?||=|||||?????=1????2??+2????2+(2??-1)??-???1??+2????1+(2??-1)???×???+8???+81????????(??)??(??)??,??1(??????),,????(||||+??)??????????????8???=??+1????2??+2????2+(2??-1)??+???1??+2????1+(2??-1)??????+8???+8h(??)=|||||??(??)??(??)?????????????????=1????1??+(2??-1)??2+(2??-1)??+???2??+2????1+(2??-1)??+???1??+(2??-1)??1+2?????×???+8???+81????????(??)??(??)??,??1????(??),,????||||?(??)??????????????+2+8?????+8???+8h(??)??=(??)??(??)?????????????????=1????2??+(2??-1)??1+(2??-1)??+???2??+2????1+2????????+8???+8h(??)????(??)??(??)????????????+2???=????2-??1????+????2-??1?+??2<??.(2.43) It is not difficult to verify that ||???????????2?-???????????1?||=0<??,?????(??,??),??0=??1=??2=??.(2.44) Therefore ????(??(??,??)) is equicontinuous on [??0,+8), and consequently ???? is relatively compact. By Lemma 1.3, there is ??0???(??,??) such that ????0=??0, which together with (2.34) yields that for ??=??+????0(??)=??+8???=1???+2??????+(2??-1)?????+8???+81?????(??)??(??)??,??0???1(???),,??0?????(????)??????????????,0(??-??)=??+8???=1???+(2??-1)????+2(??-1)?????+8???+81?????(??)??(??)??,??0???1?(??),,??0?????(??)??????????????,(2.45) which mean that ??0(??)+??0?(??-??)=2??+??+8???+8???+81?????(??)??(??)??,??0???1?(??),,??0?????(??)??????????????.(2.46) It is easy to show that ??0 is a bounded nonoscillatory solution of (1.7).
Finally, we demonstrate that (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??). Let ??1,??2?(??,??) and ??1???2. For each ???{1,2}, we pick up a positive integer ????=??0+?? and the mapping ?????? satisfying (2.33) and (2.34), where ?? and ?? are replaced by ???? and ????, respectively, and some ??3>max{??1,??2} satisfying (2.21). Clearly, there are ?? and ?????(??,??) such that ????1??=?? and ????2??=??, respectively. That is, ?? and ?? are bounded nonoscillatory solutions of (1.7) in ??(??,??). By (2.34) we get that for ??=??3??(??)=??1+8???=1???+2??????+(2??-1)?????+8???+81????????(??)??(??)??,??1(??????),,????(??)??????????????,??(??)=??2+8???=1???+2??????+(2??-1)?????+8???+81????????(??)??(??)??,??1????(??),,????(??)??????????????,(2.47) which together with (2.3) and (2.21) yield that ||?????-???=1-??2||?-2??+83???+8???+8h(??)??(??)??(??)????????????>0,(2.48) which implies that ?????. This completes the proof.

Theorem 2.5. Assume that there exist constants ?? and ?? with ??>??>0 and a function h???([??0,+8),R+) satisfying (2.3) and 8???=1???+80+???????+8???+8h(??)??(??)??(??)????????????<+8.(2.49) If ??(??)=-1 for each ???[??0,+8), then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???(??,??). It follows from (2.49) that there exists ??>??0 satisfying 8???=1?+8??+???????+8???+8h(??)??(??)??(??)????????????<min{??-??,??-??}.(2.50) Define a mapping ????:??(??,??)???([??0,+8),R) by ?????????????(??)=??-8???=1?+8??+???????+8???+81????????(??)??(??)??,??1(??????),,????(?????)??????????????,??=??,?????(??),??0=??<??(2.51) for ?????(??,??).
Firstly, we prove that ????(??(??,??))???(??,??). By (2.3), (2.50), and (2.51), we obtain that ????????(??)=??+8???=1?+8??+???????+8???+8h(??)?????(??)??(??)????????????=??,?????(??,??),??=??,?????(??)=??-8???=1?+8??+???????+8???+8h(??)??(??)??(??)????????????=??,?????(??,??),??=??,(2.52) which imply that ??(??(??,??))???(??,??).
Secondly, we show that ???? is continuous in ??(??,??). Let ?????(??,??) and {????}??=1???(??,??) with lim???+8????=??. By (2.51), we get that ||?????????????(??)-?????||=(??)8???=1?+8??+???????+8???+81||?????(??)??(??)??,???????1?(??),,?????????????(??)??-????,??1????(??),,????||(??)??????????????,??=??,??=1.(2.53) Equation (2.53) together with (2.3), (2.49) and the Lebesgue dominated convergence theorem yields that lim???+8??????????-????????=0,(2.54) that is, ???? is continuous in ??(??,??).
Thirdly, we show that ????(??(??,??)) is relatively compact. From (2.3), (2.50), and (2.51), we obtain that ??????????=??+8???=1?+8??+???????+8???+8h(??)??(??)??(??)????????????=2??,?????(??,??),(2.55) which means that ????(??(??,??)) is uniform bounded. It follows from (2.49) that, for each ??>0, there exists ??*>?? such that 8???=1???+8*+???????+8???+8h(??)????(??)??(??)????????????<2.(2.56) Notice that (2.3), (2.51), and (2.56) yield that ||???????????2?-???????????1?||=28???=1???+8*+???????+8???+8h(??)??(??)??(??)????????????<??,?????(??,??),??2>??1=??*.(2.57) Choose a positive integer ?? satisfying ??+????=??*. Put ????=max??+8???+8h(??)???(??)??(??)????????:?????+??,??*??,??+??????=.1+2????(2.58) By (2.3), (2.51), and (2.58), we gain that ||???????????2?-???????????1?||=|||||8???=1????+82+????-???+81+????????+8???+81????????(??)??(??)??,??1????(??),,????|||||=(??)??????????????8???=1???2??+????1+???????+8???+8h(??)=??(??)??(??)?????????????????=1???2??+????1+???????+8???+8h(??)??(??)??(??)????????????+8???=??+1???2??+????1+???????+8???+8h(??)?????(??)??(??)????????????=????2-??1?+8???=1???+8*+???????+8???+8h(??)??(??)??(??)????????????<??,?????(??,??),??1,??2????,??*?with??1=??2=??1+??.(2.59) Clearly, ||???????????2?-???????????1?||=0<??,?????(??,??),??0=??1=??2=??.(2.60) That is, ????(??(??,??)) is equicontinuous on [??0,+8), and ???? is relatively compact. The rest argument is similar to the proof of Theorem 2.4 and is omitted. This completes the proof.

Theorem 2.6. Assume that there exist constants ??,??,??1,??2,??0 and a function h???([??0,+8),R+) satisfying (2.3), (2.4), and 1<??2<??1<??22??,0<21-??2??1????<22-??1??2??;(2.61)??2=??(??)=??1,??=??0=??0+??.(2.62) Then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???(??1??+(??1/??2)??,??2??+(??2/??1)??). It follows from (2.4) that there exists ??>??0 such that ???+8???+8???+8h(??)?????(??)??(??)????????????<min2????+2??1????-??,2??1??-??-??2???.(2.63) Define two mappings ??1?? and ??2??:??(??,??)???([??0,+8),R) by ???1????????????(??)=-??(??+??)??(??+??)?????(??+??),??=??,1?????(??),??0=??<??,(2.64)???2?????(?????1??)=???(??+??)+8??+?????+8???+81????????(??)??(??)??,??1????(??),,???????(??)??????????????,??=??,2???????(??),0=??<??(2.65) for ?????(??,??).
Firstly, we show that ??1??(??(??,??))+??2??(??(??,??))???(??,??). From (2.3), (2.61)–(2.65), we get that ???1????????(??)+2?????=??(??)??2-????1+1??2???min2????+2??1????-??,2??1??-??-??2??????=??,??,?????(??,??),??=??,1????????(??)+2?????=??(??)??1-????2-1??2???min2????+2??1????-??,2??1??-??-??2???=??,??,?????(??,??),??=??,(2.66) That is, ??1??(??(??,??))+??2??(??(??,??))???(??,??).
Secondly, by (2.61), (2.62), and (2.64), we conclude that ????1????-??1??????=1??2???-???,??,?????(??,??),(2.67) which implies that ??1?? is a contraction mapping in ??(??,??).
Thirdly, we show that ??2?? is completely continuous. Let ?????(??,??) and {????}??=1???(??,??) be such that ??????? as ???+8. By (2.61), (2.62), and (2.65), we gain that ||???2??????????(??)-2?????||=1(??)???(??+??)+8??+?????+8???+81||?????(??)??(??)??,???????1(???),,?????????(??????)??-????,??1????(??),,????||=1(??)????????????????2???+8???+8???+81??||???(??)??(??)??,???????1?(??),,?????????????(??)??-????,??1????(??),,????||(??)??????????????,??=??,??=1.(2.68) In view of (2.12), (2.13), (2.68), and the Lebesgue dominated convergence theorem, we obtain that lim???+8????2??????-??2??????=0,(2.69) that is, ??2?? is continuous in ??(??,??).
For each ??>0, it follows from (2.4) that there exists ??*=?? satisfying ???+8*???+8???+8h(??)?????(??)??(??)????????????<min2??2,??22??4??1?.(2.70) From (2.3), (2.61), (2.62), (2.65), and (2.70), we gain that ||???2????????2?-???2????????1?||=1??2???+82+?????+8???+8h(??)1??(??)??(??)????????????+??2???+81+?????+8???+8h(??)=2??(??)??(??)??????????????2???+8*???+8???+8h(??)??(??)??(??)????????????<??,?????(??,??),??2>??1=??*.(2.71) By (2.3), (2.61), (2.62), (2.65), and (2.70), we obtain that, for ?????(??,??),??=??1=??2=??*, ||???2????????2?-???2????????1?||=1?????2??+????2??+??1+?????+8???+8h(??)+||||1??(??)??(??)?????????????????2?-1+???????1?||||?+????+81+?????+8???+8h(??)=1??(??)??(??)??????????????2???2??+??1+?????+8???+8h(??)??(??)??(??)????????????+2??1??22???+8*???+8???+8h(??)??(??)??(??)????????????,(2.72) which means that there exists ??>0 such that ||???2????????2?-???2????????1?||<??,?????(??,??),??1,??2????,??*?||??with2-??1||<??.(2.73) It is easy to verify that ||???2????????2?-???2????????1?||=0<??,?????(??,??),??0=??1=??2=??.(2.74) That is, ??2??(??(??,??)) is equicontinuous on [??0,+8), and ??2?? is completely continuous. By Lemma 1.2, there is ??0???(??,??) such that ??1????0+??2????0=??0, which is a bounded nonoscillatory solution of (1.7).
Lastly, we demonstrate that (1.7) possesses uncountably many nonoscillatory solutions. Let ??1,??2?(??1??+(??1/??2)??,??2??+(??2/??1)??) and ??1???2. For each ???{1,2}, we choose ????>??0 and contraction mappings ??????1 and ??????2 satisfying (2.63)–(2.65) with ?? and ?? replaced by ???? and ????, respectively, and some ??3>max{??1,??2} satisfying ???+83???+8???+8h(??)????(??)??(??)????????????<2||??1-??2||2??1.(2.75) Obviously, there are ??,?????(??,??) such that ??1??1??+??2??1??=?? and ??1??2??+??2??2??=??, respectively, and ?? and ?? are two bounded nonoscillatory solutions of (1.7) in ??(??,??). By (2.64) and (2.65), we gain that for ??=??3????(??)=1-??(??+??)??(??+??)+1??(??+??)???(??+??)+8??+?????+8???+81????????(??)??(??)??,??1????(??),,??????(??)??????????????,??(??)=2-????(??+??)(??+??)+1??(??+??)???(??+??)+8??+?????+8???+81????????(??)??(??)??,??1(??????),,????(??)??????????????,(2.76) which together with (2.3), (2.61), and (2.62) mean that ||||=1??(??)-??(??)??||??(??+??)1-??2||-1??||||-1(??+??)??(??-??)-??(??-??)???(??+??)+8??+?????+8???+81×||????????(??)??(??)??,??1????(??),,????????(??)??-????,??1????(??),,????||=1(??)????????????????1||??1-??2||-1??2-2???-?????2???+83???+8???+8h(??)??(??)??(??)????????????,??=??3,(2.77) which together with (2.75) yields that ?????-???=21+??2?1??1||??1-??2||-2??2???+83???+8???+8h(??)???(??)??(??)????????????>0,(2.78) that is, ?????. This completes the proof.

Theorem 2.7. Assume that there exist constants ??,??,??1,??2,??0 and a function h???([??0,+8),R+) satisfying (2.3), (2.4), and ???0<1????-1??<2?-1??;(2.79)-8<-??1=??(??)=-??2<-1,??=??0=??0+??.(2.80) Then (1.7) possesses uncountably many bounded nonoscillatory solutions in ??(??,??).

Proof. Set ???((??1-1)??,(??2-1)??). It follows from (2.4) that there exists ??>??0 such that ???+8???+8???+8h(??)??????(??)??(??)????????????<min2???-1??-??,2(??+??)??1-??2???.(2.81) Define two mappings ??1?? and ??2??:??(??,??)???([??0,+8),R) by ???1??????????(??)=-??-??(??+??)??(??+??)?????(??+??),??=??,1?????(??),??0???=??<??,2??????????1(??)=???(??+??)+8??+?????+8???+81????????(??)??(??)??,??1????(??),,???????(??)??????????????,??=??,2???????(??),0=??<??(2.82) for ?????(??,??). From (2.3), (2.80)–(2.82), we obtain that ???1????????(??)+2?????=??(??)??2+????2+1??2????min2???-1??-??,2(??+??)??1-??2??????=??,??,?????(??,??),??=??,1????????(??)+2?????=??(??)??1+????1-1??2????min2???-1??-??,2(??+??)??1-??2???,=??,??,?????(??,??),??=??(2.83) which implies that ??1??(??(??,??))+??2??(??(??,??))???(??,??). The rest of the proof is similar to the proof of Theorem 2.6 and is omitted. This completes the proof.

3. Examples

In this section we construct seven examples as applications of the results presented in Section 2.

Example 3.1. Consider the following third-order nonlinear neutral delay differential equation: ???????3???????????ln(1+??)??(??)+3sin2?-2+17???(??-??)?????????+1???????2?2??2?1+3??+??2?1??-????=0,??=??0,(3.1) where ??>0 and ??0>0 are fixed. Let