# Oscillation Criteria for Third-Order Emden–Fowler Differential Equations with Unbounded Neutral Coefficients

Authors: George E. Chatzarakis; Said R. Grace; Irena Jadlovská; Tongxing Li; Ercan Tunç
Journal: Complexity
 (2019)

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## Abstract

New sufficient conditions for the oscillation of all solutions to a class of third-order Emden–Fowler differential equations with unbounded neutral coefficients are established. The criteria obtained essentially improve related results in the literature. In particular, as opposed to known results, new criteria can distinguish solutions of third-order differential equations with different behaviors. Examples are also provided to illustrate the results.


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## Body

## 1. Introduction

This paper is concerned with the oscillation of solutions of the third-order Emden–Fowler neutral differential equation(1)z′′′t+qtxλgt=0,t≥t0>0,where z(t)≔x(t)+p(t)x(η(t)), λ>0 is the ratio of odd positive integers. Throughout, the following conditions are assumed to hold: (C1)
p,q:[t0,∞)→R are continuous functions, p(t)≥1, p(t)≡1 for large t, q(t)≥0, and q(t) is not identically zero for large t;(C2)
η,g:[t0,∞)→R are continuous functions, η(t)≤t, η is strictly increasing, and limt→∞η(t)=limt→∞g(t)=∞.The study of (1) is important due to the further development of the oscillation theory and its practical reasons. Emden–Fowler differential equations have numerous applications in physics (mathematical, theoretical, and chemical physics) and engineering; see, e.g., the papers by Agarwal et al. [1], Li and Rogovchenko [2–5], and Wong [6].By a solution of (1) we mean a continuous function x:[tx,∞)→R, tx≥t0, such that z∈C3([tx,∞),R) and x(t) satisfies (1) on [tx,∞). We consider only proper solutions x(t) of (1) that satisfy supx(t):t≥T>0 for all T≥tx. Furthermore, we tacitly suppose that (1) possesses such solutions. Such a solution x(t) of (1) is said to be oscillatory if it has arbitrarily large zeros on [tx,∞); i.e., for any t1∈[tx,∞), there exists a t2≥t1 such that x(t2)=0; otherwise, it is called nonoscillatory, i.e., if it is either eventually positive or eventually negative. Equation (1) is said to be oscillatory if all its proper solutions oscillate.In recent years, there has been much research activity concerning the oscillation and asymptotic behavior of solutions to various classes of third-order neutral differential equations. We refer the reader to the papers [2, 7–17] and the references contained therein as examples of recent results on this topic. However, the sufficient conditions established in these papers except [10, 13] ensure that every solution x(t) of equations either oscillates or converges to zero as t→∞. This means that these results cannot distinguish solutions with different behaviors. On the other hand, the papers [2, 7–17] were concerned with the case where p is bounded, i.e., the cases where -1<p0≤p(t)≤0, 0≤p(t)≤p0<1, and 0≤p(t)≤p0<∞ were considered. In view of the observations above, we wish to develop new sufficient conditions which not only ensure oscillation of (1) but also can be applied to the case where p is unbounded. We would like to point out that only a few results are known regarding oscillatory and asymptotic behavior of third-order neutral differential equations for unbounded p; see, e.g., the papers [18–20], where the Riccati transformation technique and comparison method were used to obtain the results. A similar observation as above is valid for these papers as well, i.e., the sufficient conditions established in these papers cannot distinguish solutions with different behaviors too.Consequently, our work is of significance because of the above-mentioned reasons. Moreover, the results obtained in this paper can easily be extended to more general third-order differential equations with unbounded neutral coefficients to derive more general oscillation results. It is our belief that the present paper will contribute significantly to the study of oscillatory behavior of solutions of third-order neutral differential equations. In the sequel, all functional inequalities are supposed to hold eventually.


## 2. Main Results

We begin with the following lemmas that will play an important role in establishing our main results. For notational purposes, we let(2)φt≔1pη-1t1-1pη-1η-1t≥0,ψt≔1pη-1t1-1pη-1η-1tη-1η-1t2/l1η-1t2/l1≥0, where l1∈(0,1) is a constant and η-1 is the inverse function of η.Lemma 1 (see [21]).
Let the functionh satisfy h(i)(t)>0, i=0,1,…,m and h(m+1)(t)≤0 eventually. Then, for every l∈(0,1), h(t)/h′(t)≥lt/m eventually.Lemma 2.
Let conditions(C1) and (C2) be satisfied and assume that x is an eventually positive solution of (1). Then for sufficiently large t, either (I)
z(t)>0, z′(t)>0, z′′(t)>0, and z′′′(t)≤0; or(II)
z(t)>0, z′(t)<0, z′′(t)>0, and z′′′(t)≤0.Proof.
The proof is not difficult and so is omitted.Theorem 3.
In addition to conditions(C1) and (C2), assume that there exists a function σ∈C([t0,∞),R) such that g(t)≤σ(t)<η(t) for t≥t0. If for some constants l1,l2∈(0,1), the two first-order delay differential equations(3)Y′t+l1l2λ2λqtψλgtη-1gt2λYλη-1gt=0and(4)W′t+12λqtφλgtη-1σt-η-1gt2λWλη-1σt=0oscillate, then (1) oscillates.Proof.
Letx be a nonoscillatory solution of (1). Since -x is also a solution of (1), without loss of generality, we may suppose that there exists a t1∈[t0,∞) such that, for t≥t1, x(t)>0, x(η(t))>0, and x(g(t))>0. It follows from Lemma 2 that z satisfies either case (I) or case (II).
Assume first that case(I) holds. By virtue of the definition of z, we conclude that(5)xt=1pη-1tzη-1t-xη-1t=zη-1tpη-1t-zη-1η-1t-xη-1η-1tpη-1tpη-1η-1t≥zη-1tpη-1t-zη-1η-1tpη-1tpη-1η-1t.Taking into account (I) and Lemma 1 with m=2, we deduce that, for every l1∈(0,1),(6)ztz′t≥l1t2,which yields (7)ztt2/l1′=tz′t-2/l1ztt2/l1+1≤0. Hence, z(t)/t2/l1 is nonincreasing for sufficiently large t. It follows from η(t)≤t and the monotonicities of η(t) and z(t)/t2/l1 that(8)zη-1η-1t≤η-1η-1t2/l1η-1t2/l1zη-1t.Using (8) in (5), we arrive at (9)xt≥ψtzη-1t, and thus(10)xgt≥ψgtzη-1gt.Combining (1) and (10), we obtain(11)z′′′t+qtψλgtzλη-1gt≤0.It follows now from (6) and (11) that(12)z′′′t+l1λ2λqtψλgtη-1gtλz′η-1gtλ≤0.Letting w(t)≔z′(t), we have(13)wt>0,w′t>0,w′′t≤0,and inequality (12) can be written as(14)w′′t+l1λ2λqtψλgtη-1gtλwλη-1gt≤0.Combining (13) and Lemma 1 with m=1, we get, for every l2∈(0,1), (15)wtw′t≥l2t, and so(16)wη-1gt≥l2η-1gtw′η-1gt.Using (16) in (14), we deduce that (17)w′′t+l1l2λ2λqtψλgtη-1gt2λw′η-1gtλ≤0. Letting Y(t)≔w′(t), we see that Y is a positive solution of the first-order delay differential inequality (18)Y′t+l1l2λ2λqtψλgtη-1gt2λYλη-1gt≤0. Therefore, by [22, Theorem 1], we conclude that, for every l1,l2∈(0,1), (3) has a positive solution, which contradicts the fact that (3) oscillates.
Next, suppose that case(II) holds. Since z is strictly decreasing and η(t)≤t, we have(19)zη-1t≥zη-1η-1t.Using (19) in (5), we conclude that (20)xt≥φtzη-1t, and thus(21)xgt≥φgtzη-1gt.Substitution of (21) into (1) implies that(22)z′′′t+qtφλgtzλη-1gt≤0.Since z(t)>0, z′(t)<0, z′′(t)>0, and z′′′(t)≤0, for v≥u≥t2, one can easily arrive at(23)zu≥v-u22z′′v.By virtue of g(t)≤σ(t) and the fact that η is strictly increasing, we deduce that η-1g(t)≤η-1σ(t). Substitute u=η-1g(t) and v=η-1σ(t) into (23) to obtain (24)zη-1gt≥η-1σt-η-1gt22z′′η-1σt. Using (24) in (22), we get (25)z′′′t+12λqtφλgtη-1σt-η-1gt2λz′′η-1σtλ≤0. Letting W(t)≔z′′(t), we see that W is a positive solution of the first-order delay differential inequality (26)W′t+12λqtφλgtη-1σt-η-1gt2λWλη-1σt≤0. The rest of the proof is similar to that of case (I) and hence is omitted. This completes the proof.From [23], it is well known that if (27)liminft→∞∫τttRsds>1e, then the first-order delay differential equation (28)x′t+Rtxτt=0 oscillates, where R,τ∈C([t0,∞),R), R(t)≥0, τ(t)<t, and limt→∞τ(t)=∞. Therefore, by virtue of Theorem 3, we have the following result.Corollary 4.
Let conditions(C1) and (C2) be satisfied and λ=1. Assume that there exists a function σ∈C([t0,∞),R) such that g(t)≤σ(t)<η(t) for t≥t0. If for some constant l1∈(0,1),(29)liminft→∞∫η-1gttqsψgsη-1gs2ds>2l1eand(30)liminft→∞∫η-1σttqsφgsη-1σs-η-1gs2ds>2e,then (1) oscillates.Corollary 5.
Let conditions(C1) and (C2) be satisfied and λ<1. Assume that there exists a function σ∈C([t0,∞),R) such that g(t)≤σ(t)<η(t) for t≥t0. If for some constant l1∈(0,1),(31)∫t0∞qtψλgtη-1gt2λdt=∞and(32)∫t0∞qtφλgtη-1σt-η-1gt2λdt=∞,then (1) oscillates.Proof.
Applications of (31), (32), and [24, Theorem 2] imply that (3) and (4) oscillate. Hence, by Theorem 3, (1) oscillates.Next, we present the following interesting result for which we need to assume that the functiong in condition (C2) is nondecreasing.Theorem 6.
In addition to conditions(C1) and (C2), assume that the function g with g(t)<η(t) is nondecreasing on [t0,∞). If for some constant l1∈(0,1),(33)limsupt→∞η-1gt2λ∫t∞qsψλgsds>2l1,ifλ=1,=∞,ifλ<1,and(34)limsupt→∞∫η-1gttqsφλgsη-1gt-η-1gs2λds>2,ifλ=1,=∞,ifλ<1,then (1) oscillates.Proof.
Letx be a nonoscillatory solution of (1). Without loss of generality, we may suppose that there exists a t1∈[t0,∞) such that, for t≥t1, x(t)>0, x(η(t))>0, and x(g(t))>0. It follows from Lemma 2 that z satisfies either case (I) or case (II).
Assume that case(I) holds. Proceeding as in the proof of Theorem 3, we deduce that (13), (14), and (16) hold for every l1,l2∈(0,1). Integrating (14) from t to u, u≥t and letting u→∞, we obtain(35)w′t≥∫t∞l1λ2λqsψλgsη-1gsλwλη-1gsds≥∫t∞l1λ2λqsψλgsη-1gsλdswλη-1gt.Using (16) in (35), we conclude that (36)w′t≥∫t∞l1l2λ2λqsψλgsη-1gsλdsη-1gtλw′η-1gtλ, which yields(37)w′t≥∫t∞l1l2λ2λqsψλgsdsη-1gt2λw′η-1gtλ.Using (13) and the fact that η-1(g(t))≤t, we have (38)w′η-1gt≥w′t, and so inequality (37) implies that (39)w′t≥∫t∞l1l2λ2λqsψλgsdsη-1gt2λw′tλ,i.e., (40)w′t1-λ≥∫t∞l1l2λ2λqsψλgsdsη-1gt2λ. Taking limsup as t→∞ in (40), we obtain a contradiction to (33).
Next, let case(II) hold. Then, we arrive at (22) and (23). For t≥s≥t2, we see that η-1g(t)≥η-1g(s). Putting u=η-1g(s) and v=η-1g(t) into (23), we get(41)zη-1gs≥η-1gt-η-1gs22z′′η-1gt.Integrating (22) from η-1g(t) to t and using (41), we obtain (42)z′′η-1gt≥∫η-1gtt12λqsφλgsη-1gt-η-1gs2λdsz′′η-1gtλ, which can be written as (43)z′′η-1gt1-λ≥∫η-1gtt12λqsφλgsη-1gt-η-1gs2λds. Taking limsup as t→∞ in (43), we obtain a contradiction to (34). The proof is complete.We conclude this paper with the following examples and remarks to illustrate the main results. The first example is concerned with the case wherep(t)→∞ as t→∞, whereas the second example is concerned with the case where p is a bounded function.Example 7.
Consider the sublinear Emden–Fowler neutral differential equation(44)xt+txt2′′′+ktαx1/3t4=0,t≥17,where k>0 and 0<α<1 are constants. Here λ=1/3, p(t)=t, η(t)=t/2, g(t)=t/4, q(t)=k/tα, η-1g(t)=t/2, and φ(g(t))=2(1-1/t)/t. Let l1=1/2. Then ψ(g(t))=2(1-16/t)/t and condition (31) becomes (45)∫17∞ktα2t-32t21/3t22/3dt=122/3∫17∞ktα2t-321/3dt≥121/3∫17∞ktαdt=∞. Letting σ(t)=t/3, then η-1σ(t)=2t/3 and condition (32) reduces to (46)∫17∞ktα2t-2t21/3t62/3dt=162/3∫17∞ktα2t-21/3dt≥25/362/3∫17∞ktαdt=∞. Therefore, by Corollary 5, (44) oscillates.Example 8.
Consider the linear differential equation(47)xt+6t+9t+1xt2′′′+kt3xt3=0,t≥1.Here λ=1, p(t)=(6t+9)/(t+1), η(t)=t/2, g(t)=t/3, q(t)=k/t3, and k>1390 is a constant. It is easy to deduce that 6≤p(t)≤15/2, η-1g(t)=2t/3, φ(g(t))≥1/9, and ψ(g(t))≥1/25 for some constant l1∈(1/2,1). Using λ=1 in (33), we have (48)η-1gt2λ∫t∞qsψλgsds≥1252t32∫t∞ks3ds=2k225. That is, condition (33) with λ=1 holds.Next, usingλ=1 in (34), we obtain (49)∫η-1gttqsφλgsη-1gt-η-1gs2λds≥19∫2t/3tks32t-2s32ds=k81∫2t/3t4t2s3-8ts2+4sds=k814ln⁡32-32. That is, condition (34) with λ=1 holds. Therefore, by Theorem 6, (47) oscillates.Remark 9.
For a class of third-order Emden–Fowler delay differential equations with unbounded neutral coefficients (1), we established new oscillation criteria which complement and improve results in the cited papers because these criteria apply also in the case where p is unbounded and ensure that all solutions of (1) are oscillatory (that is, these results can distinguish solutions with different behaviors).Remark 10.
Using different methods, we improve results of Li and Rogovchenko [2] by removing restrictive condition η∘g=g∘η, which, in a certain sense, is a significant improvement compared to the results reported in the cited papers.Remark 11.
Combining Theorem3 and the results obtained in [25], one can derive various oscillation criteria for (1) in the linear case. To study the oscillation of (1) in the superlinear case, it would be of interest to establish oscillation criteria for (3) and (4) assuming that λ>1.Remark 12.
In the conclusion of Lemma1, the existence of the constant l∈(0,1) is necessary in some cases. For instance, for t≥2, if y(t)=t-1/t, then y′(t)>0, y′′(t)<0, and y(t)<ty′(t), and so the function y does not satisfy the conclusion of Lemma 1 provided that there is no l∈(0,1). On the basis of Lemma 1, one can easily revisit the results reported in [26–28].



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