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Asymptotic behavior of even-order noncanonical neutral differential equations

Asymptotic behavior of even-order noncanonical neutral differential equations 1IntroductionThe neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena, see [1,2]. It is well known that the modeling of natural and technological phenomena produces differential equations, often of higher-order; see, for instance, the papers [3,4]. Oscillation theory is a branch of qualitative theory that investigates the oscillatory and non-oscillatory behavior of solutions to differential equations.In this work, we consider the even-order NDDE (1)(a⋅(u+ρ⋅u∘τ)(m−1))′(ℓ)+h(ℓ)u(g(ℓ))=0,{(a\cdot {(u+\rho \cdot u\circ \tau )}^{(m-1)})}^{^{\prime} }(\ell )+h(\ell )u(g(\ell ))=0,where ℓ≥ℓ0\ell \ge {\ell }_{0}, m≥4m\ge 4is an even integer and aa, ρ\rho , τ\tau , hh, and ggare continuous real-valued functions on [ℓ0,∞){[}{\ell }_{0},\infty ). We also assume that a∈C1([ℓ0,∞),(0,∞))a\in {C}^{1}({[}{\ell }_{0},\infty ),(0,\infty )), a′(ℓ)>0{a}^{^{\prime} }(\ell )\gt 0, ρ(ℓ)∈[0,ρ0]\rho (\ell )\in {[}0,{\rho }_{0}], ρ0<1{\rho }_{0}\lt 1is a constant, h≥0h\ge 0, h≡0h\equiv 0on any half-line [L,∞){[}L,\infty )for all L≥ℓ0L\ge {\ell }_{0}, τ(ℓ)≤ℓ\tau (\ell )\le \ell , g(ℓ)≤ℓg(\ell )\le \ell , limℓ→∞τ(ℓ)=limℓ→∞g(ℓ)=∞{\mathrm{lim}}_{\ell \to \infty }\tau (\ell )={\mathrm{lim}}_{\ell \to \infty }g(\ell )=\infty and ∫ℓ0∞a−1(s)ds<∞.{\int }_{{\ell }_{0}}^{\infty }{a}^{-1}(s){\rm{d}}s\lt \infty .A solution of (1) is a function u∈C([ℓu,∞),R)u\in C({[}{\ell }_{u},\infty ),{\mathbb{R}}), ℓu≥ℓ0{\ell }_{u}\ge {\ell }_{0}, which satisfies the properties u+ρu(τ)∈C(m−1)([ℓu,∞),R)u+\rho u(\tau )\in {C}^{(m-1)}({[}{\ell }_{u},\infty ),{\mathbb{R}}), a(u+ρu(τ))(m−1)∈C1([ℓu,∞),R)a{(u+\rho u(\tau ))}^{(m-1)}\in {C}^{1}({[}{\ell }_{u},\infty ),{\mathbb{R}})and uusatisfies (1) on [ℓu,∞).{[}{\ell }_{u},\infty ).We consider only the proper solutions uuof (1), that is, uuis not identically zero eventually. A solution uuof (1) is called oscillatory if it is neither positive nor negative, ultimately; otherwise, it is called nonoscillatory.For several decades, an growing interest in presenting criteria for oscillation of different classes of DDE has been observed. Recently, the works [5,6,7, 8,9] have developed many techniques and approaches for studying the oscillations of delay and advanced second-order equations. However, neutral second-order equations have been studied in many techniques through works [10,11,12, 13,14,15]. The development in the study of second-order DDEs was reflected on the study of even-order equations, see for delay [16,17,18, 19,20,21] and for neutral [22,23, 24,25,26, 27,28]. On the other hand, the works ([29,30] for third-order, and [31,32,33] for odd-order) contributed to the development of the oscillatory theory of odd-order delay differential equations.Zhang et al. [19] investigated the oscillatory behavior of a higher-order differential equation (2)(a(ℓ)(u(m−1)(ℓ))α)′+h(ℓ)uβ(g(ℓ))=0,{(a(\ell ){({u}^{(m-1)}(\ell ))}^{\alpha })}^{^{\prime} }+h(\ell ){u}^{\beta }(g(\ell ))=0,where α\alpha , β\beta are ratios of odd natural numbers and (3)∫ℓ0∞a−1/α(s)ds<∞.\underset{{\ell }_{0}}{\overset{\infty }{\int }}{a}^{-1\text{/}\alpha }(s){\rm{d}}s\lt \infty .Zhang et al. [19] obtained results which ensure that every solution uuof (2) is either oscillatory or satisfies limℓ→∞u(ℓ)=0.{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0.Zhang et al. [20] studied the oscillation of (2) for α≥β\alpha \ge \beta and improved the results reported in [19].For fourth-order, Zhang et al. [21] studied (a(u‴)α)′(ℓ)+h(ℓ)uα(g(ℓ))=0{\left(a{\left({u}^{\prime\prime\prime })}^{\alpha })}^{^{\prime} }(\ell )+h(\ell ){u}^{\alpha }(g(\ell ))=0and presented some oscillation criteria (including Hille-and Nehari-type criteria). Moreover, Moaaz and Muhib [17] presented criteria for fourth-order DDE (a(u‴)α)′(ℓ)+f(ℓ,u(g(ℓ)))=0,{\left(a{\left({u}^{\prime\prime\prime })}^{\alpha })}^{^{\prime} }(\ell )+f\left(\ell ,u(g\left(\ell )))=0,under the conditions f(ℓ,u)≥h(ℓ)uβf\left(\ell ,u)\ge h(\ell ){u}^{\beta }and α,β\alpha ,\beta are the ratios of odd natural numbers.For neutral delay equations, Zhang et al. [28] studied the even-order nonlinear NDDE (u+ρ⋅u∘τ)(m)(ℓ)+h(ℓ)f(u(g(ℓ)))=0,{(u+\rho \cdot u\circ \tau )}^{(m)}(\ell )+h(\ell )f\left(u\left(g\left(\ell )))=0,under the conditions uf(u)>0uf(u)\gt 0for all u≠0,u\ne 0,and ffis nondecreasing. Moaaz et al. [25] investigated the asymptotic behavior of solutions of the higher-order NDDE (a((u+ρ⋅u∘τ)(m−1))α)′(t)+f(ℓ,u(g(ℓ)))=0,{\left(a{\left({(u+\rho \cdot u\circ \tau )}^{(m-1)})}^{\alpha })}^{^{\prime} }(t)+f\left(\ell ,u\left(g\left(\ell )))=0,where ∣f(ℓ,u)∣≥h(ℓ)∣u∣β| f(\ell ,u)| \ge h(\ell ){| u| }^{\beta }.To the best of our knowledge, the previous studies in the literature which considered the asymptotic behavior of solutions of NDDEs of mm-order were concerned only with the canonical form ∫∞a−1(s)ds=∞{\int }^{\infty }{a}^{-1}(s){\rm{d}}s=\infty . In this paper, we obtain new conditions for testing oscillation of NDDE (1) in noncanonical case, using Riccati substitution along with comparison principles with first-order DDE. Examples are presented to illustrate our new results.In the following, we present useful lemmas that will be used throughout the results.Lemma 1.1[34, Lemma 2.2.3] Assume that ϖ∈Cm([ℓ0,∞),R+)\varpi \in {C}^{m}\left({[}{\ell }_{0},\infty ),{{\mathbb{R}}}^{+}), ϖ(m){\varpi }^{(m)}is not identically zero on a subray of [ℓ0,∞){[}{\ell }_{0},\infty )and ϖ(m){\varpi }^{(m)}is of fixed sign. Suppose that ϖ(m−1)ϖ(m)≤0{\varpi }^{(m-1)}{\varpi }^{(m)}\le 0for ℓ∈[ℓ1,∞)\ell \in {[}{\ell }_{1},\infty ), where ℓ1≥ℓ0{\ell }_{1}\ge {\ell }_{0}large enough. If limℓ→∞ϖ(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }\varpi (\ell )\ne 0, then there exists a ℓλ∈[ℓ1,∞){\ell }_{\lambda }\in {[}{\ell }_{1},\infty )such thatϖ≥λ(m−1)!ℓm−1∣ϖ(m−1)∣,\varpi \ge \frac{\lambda }{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\ell }^{m-1}| {\varpi }^{(m-1)}| ,for every λ∈(0,1)\lambda \in \left(0,1)and ℓ∈[ℓλ,∞).\ell \in {[}{\ell }_{\lambda },\infty ).Lemma 1.2[35, Lemma 1] Let f∈Cm([ℓ0,∞),R)f\in {C}^{m}\left({[}{\ell }_{0},\infty ),{\mathbb{R}}), f(r)>0{f}^{(r)}\gt 0, r=0,1,…,mr=0,1,\ldots ,mand f(m+1)≤0{f}^{(m+1)}\le 0eventually. Then, for every η∈(0,1)\eta \in (0,1)f(ℓ)≥ηℓmf′(ℓ).f(\ell )\ge \frac{\eta \ell }{m}{f}^{^{\prime} }(\ell ).Lemma 1.3[36, Lemma 1.2] Assume that B≥0B\ge 0, A>0A\gt 0, w≥0w\ge 0and α>0\alpha \gt 0. Then, Bw−Aw(α+1)/α≤αα(α+1)α+1Bα+1Aα.Bw-A{w}^{(\alpha +1)\text{/}\alpha }\le \frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{{B}^{\alpha +1}}{{A}^{\alpha }}.Lemma 1.4[20, Lemma 2.1] Let f∈Cn([t0,∞),(0,∞))f\in {C}^{n}({[}{t}_{0},\infty ),(0,\infty )). If the derivative f(n)(t){f}^{(n)}(t)is eventually of one sign for all large tt, then there exist a tx{t}_{x}such that tx≥t0{t}_{x}\ge {t}_{0}and an integer l,0≤l≤nl,0\le l\le n, with n+ln+leven for f(n)(t)≥0{f}^{(n)}(t)\ge 0, or n+ln+lodd for f(n)(t)≤0{f}^{(n)}(t)\le 0such thatl>0impliesf(k)(t)>0fort≥tx,k=0,1,…,l−1l\gt 0\hspace{1em}{implies}\hspace{0.33em}{f}^{(k)}(t)\gt 0\hspace{1em}{for}\hspace{0.33em}t\ge {t}_{x},\hspace{1em}k=0,1,\ldots ,l-1andl≤n−1implies(−1)l+kf(k)(t)>0fort≥tx,k=l,l+1,…,n−1.l\le n-1\hspace{1em}{implies}\hspace{0.33em}{(-1)}^{l+k}{f}^{(k)}(t)\gt 0\hspace{1em}{for}\hspace{0.33em}t\ge {t}_{x},\hspace{1em}k=l,l+1,\ldots ,n-1.2Main resultsFor the convenience, we use notation ν≔u+ρ⋅u∘τ\nu := u+\rho \cdot u\circ \tau .Lemma 2.1Assume that u∈C([ℓ0,∞),(0,∞))u\in C({[}{\ell }_{0},\infty ),(0,\infty ))is a solution of (1), eventually. Then, ν>0\nu \gt 0, (aν(m−1))′≤0{(a{\nu }^{(m-1)})}^{^{\prime} }\le 0and ν\nu satisfies one of the following: (1)ν′{\nu }^{^{\prime} }, ν(m−1){\nu }^{(m-1)}and (−ν(m))(-{\nu }^{(m)})are positive;(2)ν′{\nu }^{^{\prime} }, ν(m−2){\nu }^{(m-2)}and (−ν(m−1))(-{\nu }^{(m-1)})are positive;(3)(−1)kν(k){(-1)}^{k}{\nu }^{(k)}are positive, for all k=1,2,…,m−1k=1,2,\ldots ,m-1,for ℓ\ell large enough.ProofAssume that uuis an eventually positive solution of (1). It follows from (1) that (a(ℓ)ν(m−1)(ℓ))′=−h(ℓ)u(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }=-h(\ell )u(g(\ell ))\le 0.Now, from above inequality and Lemma 2 that there exist three possible cases (1)–(3) for ℓ≥ℓ1\ell \ge {\ell }_{1}large enough.□Lemma 2.2Assume that u∈C([ℓ0,∞),(0,∞))u\in C({[}{\ell }_{0},\infty ),(0,\infty ))is a solution of (1), where ν\nu satisfies case (3)(3). If(4)∫ℓ0∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ=∞,\underset{{\ell }_{0}}{\overset{\infty }{\int }}\underset{\varrho }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho =\infty ,then, limℓ→∞u(ℓ)=0{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0.ProofAssume that uuis an eventually positive solution of (1), where ν\nu satisfies case (3)(3). Then, limℓ→∞ν(ℓ)=D{\mathrm{lim}}_{\ell \to \infty }\nu (\ell )=D. We claim that D=0D=0. Suppose that D>0D\gt 0, and so for all ε>0\varepsilon \gt 0, there exists ℓ1≥ℓ0{\ell }_{1}\ge {\ell }_{0}such that u(g(ℓ))≥Du(g(\ell ))\ge Dfor ℓ≥ℓ1.\ell \ge {\ell }_{1}.Integrating (1) from ℓ1{\ell }_{1}to ℓ\ell , we get a(ℓ)ν(m−1)(ℓ)=a(ℓ2)ν(m−1)(ℓ2)−∫ℓ1ℓh(s)u(g(s))ds≤−D∫ℓ1ℓh(s)ds,a(\ell ){\nu }^{(m-1)}(\ell )=a({\ell }_{2}){\nu }^{(m-1)}({\ell }_{2})-\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s)u(g(s)){\rm{d}}s\le -D\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s){\rm{d}}s,that is, (5)ν(m−1)(ℓ)<−D1a(ℓ)∫ℓ1ℓh(s)ds.{\nu }^{(m-1)}(\ell )\lt -D\frac{1}{a(\ell )}\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s){\rm{d}}s.Integrating (5) twice from ℓ\ell to ∞\infty , we obtain −ν(m−2)(ℓ)<−D∫ℓ∞1a(ς)∫ℓ1ςh(s)dsdς-{\nu }^{(m-2)}(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma and (6)ν(m−3)(ℓ)<−D∫ℓ∞∫s∞1a(ς)∫ℓ1ςh(s)dsdςds=−D∫ℓ∞(ς−ℓ)1a(ς)∫ℓ1ςh(s)dsdς.{\nu }^{(m-3)}(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}s=-D\underset{\ell }{\overset{\infty }{\int }}(\varsigma -\ell )\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma .Similarly, integrating (6) m−4m-4times from ℓ\ell to ∞\infty , we find ν′(ℓ)<−D∫ℓ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdς.{\nu }^{^{\prime} }(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma .Integrating this inequality from ℓ1{\ell }_{1}to ∞\infty , we obtain ν(ℓ1)>D∫ℓ1∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ,\nu ({\ell }_{1})\gt D\underset{{\ell }_{1}}{\overset{\infty }{\int }}\underset{\varrho }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho ,which is a contradiction with (4). Thus, D=0D=0. This completes the proof.□Theorem 2.1Let (4) hold. If there exists a λ0∈(0,1){\lambda }_{0}\in (0,1)such that the first-order delay differential equation(7)y′(ℓ)+h(ℓ)λ0(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!a(g(ℓ))y(g(ℓ))=0{y}^{^{\prime} }(\ell )+h(\ell )\frac{{\lambda }_{0}(1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(\ell ))}y(g(\ell ))=0is oscillatory and(8)limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds=∞\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\infty holds for some constant λ1∈(0,1){\lambda }_{1}\in (0,1), then every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofSuppose that (1) has a positive solution uuwhich satisfies limℓ→∞u(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }u(\ell )\ne 0. It follows from (1) that (9)(a(ℓ)ν(m−1)(ℓ))′=−h(ℓ)u(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }=-h(\ell )u(g(\ell ))\le 0.From Lemma 2.1, there are three possible cases for the behavior of ν\nu and its derivatives.Let (1)(1)hold. From Lemma 1.1, we have (10)ν(ℓ)≥λℓm−1(m−1)!ν(m−1)(ℓ)\nu (\ell )\ge \frac{\lambda {\ell }^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-1)}(\ell )for every λ∈(0,1).\lambda \in (0,1).It follows from the definition of ν(ℓ)\nu (\ell )that (11)u(ℓ)=ν(ℓ)−ρ(ℓ)u(τ(ℓ))≥(1−ρ(ℓ))ν(ℓ).u(\ell )=\nu (\ell )-\rho (\ell )u(\tau (\ell ))\ge (1-\rho (\ell ))\nu (\ell ).Combining (9) and (11), we get (12)(a(ℓ)ν(m−1)(ℓ))′≤−h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ)).{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }\le -h(\ell )(1-\rho (g(\ell )))\nu (g(\ell )).From (10), we obtain (a(ℓ)ν(m−1)(ℓ))′+h(ℓ)λ(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!ν(m−1)(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }+h(\ell )\frac{\lambda (1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-1)}(g(\ell ))\le 0.Now, we define the function y(ℓ)=a(ℓ)ν(m−1)(ℓ)y(\ell )=a(\ell ){\nu }^{(m-1)}(\ell ). Clearly, yyis a positive solution of the first-order delay differential inequality (13)y′(ℓ)+h(ℓ)λ(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!a(g(ℓ))y(g(ℓ))≤0.{y}^{^{\prime} }(\ell )+h(\ell )\frac{\lambda (1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(\ell ))}y(g(\ell ))\le 0.Thus, using [40, Theorem 1], equation (7) has also a positive solution for all λ0∈(0,1){\lambda }_{0}\in (0,1), this contradicts the assumption that (7) is oscillatory.Let (2)(2)hold. We define ω\omega by (14)ω(ℓ)=a(ℓ)ν(m−1)(ℓ)ν(m−2)(ℓ),ℓ≥ℓ1.\omega (\ell )=\frac{a(\ell ){\nu }^{(m-1)}(\ell )}{{\nu }^{(m-2)}(\ell )},\hspace{1.0em}\ell \ge {\ell }_{1}.Then, ω(ℓ)<0\omega (\ell )\lt 0for ℓ≥ℓ1\ell \ge {\ell }_{1}. Noting that (a(ℓ)ν(m−1)(ℓ))′≤0{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }\le 0, we find (15)a(s)ν(m−1)(s)≤a(ℓ)ν(m−1)(ℓ),s≥ℓ≥ℓ1.a(s){\nu }^{(m-1)}(s)\le a(\ell ){\nu }^{(m-1)}(\ell ),\hspace{1.0em}s\ge \ell \ge {\ell }_{1}.Dividing (15) by aaand integrating it from ℓ\ell to ∞\infty , we obtain 0≤ν(m−2)(ℓ)+a(ℓ)ν(m−1)(ℓ)δ(ℓ),0\le {\nu }^{(m-2)}(\ell )+a(\ell ){\nu }^{(m-1)}(\ell )\delta (\ell ),which yields −a(ℓ)ν(m−1)(ℓ)δ(ℓ)ν(m−2)(ℓ)≤1.-\frac{a(\ell ){\nu }^{(m-1)}(\ell )\delta (\ell )}{{\nu }^{(m-2)}(\ell )}\le 1.Thus, by (14), we get (16)−ω(ℓ)δ(ℓ)≤1.-\omega (\ell )\delta (\ell )\le 1.Differentiating (14), we arrive at ω′(ℓ)=(a(ℓ)ν(m−1)(ℓ))′ν(m−2)(ℓ)−a(ℓ)(ν(m−1)(ℓ))2(ν(m−2)(ℓ))2,{\omega }^{^{\prime} }(\ell )=\frac{{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }}{{\nu }^{(m-2)}(\ell )}-\frac{a(\ell ){({\nu }^{(m-1)}(\ell ))}^{2}}{{({\nu }^{(m-2)}(\ell ))}^{2}},which follows from (1) and (14) that (17)ω′(ℓ)=−h(ℓ)u(g(ℓ))ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )=-\frac{h(\ell )u(g(\ell ))}{{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.From the definition of ν(ℓ)\nu (\ell )and the fact that ν′(ℓ)>0,{\nu }^{^{\prime} }(\ell )\gt 0,we get that (11) holds. Hence, it follows from (17) that (18)ω′(ℓ)≤−h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ))ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{h(\ell )(1-\rho (g(\ell )))\nu (g(\ell ))}{{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Using Lemma 1.1, we get ν(ℓ)≥λℓm−2(m−2)!ν(m−2)(ℓ)\nu (\ell )\ge \frac{\lambda {\ell }^{m-2}}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-2)}(\ell )for every λ∈(0,1)\lambda \in (0,1)and for all sufficiently large ℓ\ell . Then, (18) becomes ω′(ℓ)≤−λh(ℓ)(1−ρ(g(ℓ)))gm−2(ℓ)ν(m−2)(g(ℓ))(m−2)!ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{\lambda h(\ell )(1-\rho (g(\ell ))){g}^{m-2}(\ell ){\nu }^{(m-2)}(g(\ell ))}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Since ℓ≥g(ℓ)\ell \ge g(\ell )and ν(m−2)(ℓ){\nu }^{(m-2)}(\ell )are decreasing, we have (19)ω′(ℓ)≤−λh(ℓ)(1−ρ(g(ℓ)))gm−2(ℓ)(m−2)!−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{\lambda h(\ell )(1-\rho (g(\ell ))){g}^{m-2}(\ell )}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Multiplying (19) by δ(ℓ)\delta (\ell )and integrating it from ℓ1{\ell }_{1}to ℓ\ell , we have 0≥δ(ℓ)ω(ℓ)−δ(ℓ1)ω(ℓ1)+∫ℓ1ℓω(s)a(s)ds+∫ℓ1ℓδ(s)a(s)ω2(s)ds+∫ℓ1ℓλh(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)ds.0\ge \delta (\ell )\omega (\ell )-\delta ({\ell }_{1})\omega ({\ell }_{1})+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\omega (s)}{a(s)}{\rm{d}}s+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\delta (s)}{a(s)}{\omega }^{2}(s){\rm{d}}s+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\lambda h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s){\rm{d}}s.Setting A=δ(s)/a(s)A=\delta (s)\hspace{0.1em}\text{/}\hspace{0.1em}a(s), B=1/a(s)B=1\hspace{0.1em}\text{/}\hspace{0.1em}a(s), and w=−ω(s)w=-\omega (s), and using Lemma 1.3, we have ∫ℓ1ℓλh(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds≤δ(ℓ1)ω(ℓ1)+1,\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left(\frac{\lambda h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s\le \delta ({\ell }_{1})\omega ({\ell }_{1})+1,due to (16), which contradicts (8).Assume that case (3)(3)holds. From Lemma 2.2 and (4), we see that limℓ→∞u(ℓ)=0{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0, which is a contradiction.This completes the proof.□Corollary 2.1Assume that (4) and (8) hold. If(20)liminfℓ→∞∫g(ℓ)ℓh(s)(1−ρ(g(s)))(g(s))m−1(m−1)!a(g(s))ds>1e,\mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\int }}h(s)\frac{(1-\rho (g(s))){(g(s))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(s))}{\rm{d}}s\gt \frac{1}{e},for some λ1∈(0,1){\lambda }_{1}\in (0,1), then every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofBy [38, Theorem 2.1.1], assumption (20) ensures that the differential equation (7) has no positive solutions. Application of Theorem 2.1 yields the result.□Remark 2.1Combining Theorem 2.1 and the results reported in [39] for the oscillation of equation (7), one can derive various oscillation criteria for equation (1).Example 2.1We consider the NDDE (21)(eℓ(u(ℓ)+ρ0u(θℓ))‴)′+h0eℓu(εℓ)=0,{\left({e}^{\ell }{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+{h}_{0}{e}^{\ell }u(\varepsilon \ell )=0,where h0>0{h}_{0}\gt 0and θ,ε∈(0,1)\theta ,\varepsilon \in (0,1). Note that, a(ℓ)=eℓa(\ell )={e}^{\ell }, ρ(ℓ)=ρ0\rho (\ell )={\rho }_{0}, τ(ℓ)=θℓ\tau (\ell )=\theta \ell , h(ℓ)=h0eℓh(\ell )={h}_{0}{e}^{\ell }and g(ℓ)=εℓg(\ell )=\varepsilon \ell . It is easy to see that δ(ℓ)=e−ℓ\delta (\ell )={e}^{-\ell }.Now, from Corollary 2.1, we have ∫ℓ0∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ=∫ℓ0∞∫ϱ∞(ς−ℓ)1eς∫ℓ1ςh0esdsdςdϱ=∞,liminfℓ→∞∫g(ℓ)ℓh(s)(1−ρ(g(s)))(g(s))m−1(m−1)!a(g(s))ds=liminfℓ→∞∫g(ℓ)ℓh0es(1−ρ0)(εs)33!eεsds=∞>1e\begin{array}{rcl}\underset{{\ell }_{0}}{\overset{\infty }{\displaystyle \int }}\underset{\varrho }{\overset{\infty }{\displaystyle \int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\displaystyle \int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho & =& \underset{{\ell }_{0}}{\overset{\infty }{\displaystyle \int }}\underset{\varrho }{\overset{\infty }{\displaystyle \int }}(\varsigma -\ell )\left(\frac{1}{{e}^{\varsigma }}\underset{{\ell }_{1}}{\overset{\varsigma }{\displaystyle \int }}{h}_{0}{e}^{s}{\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho =\infty ,\\ \mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\displaystyle \int }}h(s)\frac{(1-\rho (g(s))){(g(s))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(s))}{\rm{d}}s& =& \mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\displaystyle \int }}{h}_{0}{e}^{s}\frac{(1-{\rho }_{0}){(\varepsilon s)}^{3}}{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}{e}^{\varepsilon s}}{\rm{d}}s=\infty \gt \frac{1}{e}\end{array}and limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds=limsupℓ→∞∫ℓ0ℓλ1h0es(1−ρ0)(εs)22!e−s−14ese−sds=∞.\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}{h}_{0}{e}^{s}(1-{\rho }_{0}){(\varepsilon s)}^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e}^{-s}-\frac{1}{4{e}^{s}{e}^{-s}}\right){\rm{d}}s\hspace{6.27em}=\infty .Thus, (4), (20), and (8) are satisfied. Therefore, every solution of (21) is oscillatory or tends to zero.Example 2.2Consider the equation (22)(ℓ2(u(ℓ)+ρ0u(θℓ))‴)′+h0ℓ2u(εℓ)=0,{\left({\ell }^{2}{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+\frac{{h}_{0}}{{\ell }^{2}}u(\varepsilon \ell )=0,here h0>0{h}_{0}\gt 0and θ,ε∈(0,1)\theta ,\varepsilon \in (0,1). We note that m=4m=4, a(ℓ)=ℓ2a(\ell )={\ell }^{2}, ρ(ℓ)=ρ0\rho (\ell )={\rho }_{0}, τ(ℓ)=θℓ\tau (\ell )=\theta \ell , h(ℓ)=h0/ℓ2h(\ell )={h}_{0}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\ell }^{2}, and g(ℓ)=εℓg(\ell )=\varepsilon \ell . It is easy to see that δ(ℓ)=1/ℓ\delta (\ell )=1\hspace{-0.08em}\text{/}\hspace{-0.08em}\ell and (4) holds. Next, (20) reduces to (23)h0(1−ρ0)ln1ε>3!εe.{h}_{0}(1-{\rho }_{0})\mathrm{ln}\frac{1}{\varepsilon }\gt \frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\varepsilon e}.Moreover, (8) becomes limℓ→∞sup∫ℓ0ℓh0λ1(1−ρ0)ε22!−141sds=∞,{\mathrm{lim}}_{\ell \to \infty }\sup \underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{h}_{0}{\lambda }_{1}(1-{\rho }_{0}){\varepsilon }^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}-\frac{1}{4}\right)\frac{1}{s}{\rm{d}}s=\infty ,which is verified if (24)h0(1−ρ0)>12ε2.{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2{\varepsilon }^{2}}.Using Corollary 2.1, if h0>M≔max3!e(1−ρ0)εln1ε,12(1−ρ0)ε2,{h}_{0}\gt M:= \max \left\{\frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e(1-{\rho }_{0})\varepsilon \mathrm{ln}\frac{1}{\varepsilon }},\frac{1}{2(1-{\rho }_{0}){\varepsilon }^{2}}\right\},then every solution of (22) is oscillatory or tends to zero, where M=12(1−ρ0)ε2ifε∈(0,0.28464]M=\frac{1}{2(1-{\rho }_{0}){\varepsilon }^{2}}\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\varepsilon \in (0,0.28464]and M=3!e(1−ρ0)εln1εifε∈(0.28464,1).M=\frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e(1-{\rho }_{0})\varepsilon \mathrm{ln}\frac{1}{\varepsilon }}\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\varepsilon \in (0.28464,1).It is easy to notice that (20) does not apply in the ordinary case (g(ℓ)=ℓg(\ell )=\ell ). So, in the following theorem, we set new conditions for testing the oscillation of (1) when m=4m=4, which apply in the ordinary case.Theorem 2.2Assume that m=4m=4and (4) hold. If(25)limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))g2(s)2!δ(s)−14a(s)δ(s)ds=∞,\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{2}(s)}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\infty ,for some constant λ1∈(0,1){\lambda }_{1}\in (0,1). Assume further that there exist two positive functions ζ(ℓ),ϑ(ℓ)∈C1[ℓ0,∞)\zeta (\ell ),{\vartheta }(\ell )\in {C}^{1}{[}{\ell }_{0},\infty ), such that(26)∫ℓ0∞ζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds=∞\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\infty and(27)∫ℓ0∞ϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds=∞\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s=\infty for some constant λ2∈(0,1){\lambda }_{2}\in (0,1). Then, every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofAssume that (1) has a nonoscillatory solution uuwhich is eventually positive and limℓ→∞u(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }u(\ell )\ne 0. It follows from (1) and Lemma 2.1 that there exist four possible cases for the behavior of ν\nu and its derivatives: (i)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0, ν‴(ℓ)>0{\nu }^{\prime\prime\prime }(\ell )\gt 0and ν(4)(ℓ)≤0{\nu }^{(4)}(\ell )\le 0;(ii)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)<0{\nu }^{^{\prime\prime} }(\ell )\lt 0, ν‴(ℓ)>0{\nu }^{\prime\prime\prime }(\ell )\gt 0and ν(4)(ℓ)≤0;{\nu }^{(4)}(\ell )\le 0;(iii)ν′(ℓ)<0{\nu }^{^{\prime} }(\ell )\lt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0and ν‴(ℓ)<0{\nu }^{\prime\prime\prime }(\ell )\lt 0;(iv)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0and ν‴(ℓ)<0{\nu }^{\prime\prime\prime }(\ell )\lt 0.Let (i) hold. Define the function ϕ(ℓ)\phi (\ell )by ϕ(ℓ)=ζ(ℓ)a(ℓ)ν‴(ℓ)ν(ℓ).\phi (\ell )=\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell )}{\nu (\ell )}.Then, clearly ϕ(ℓ)\phi (\ell )is positive for ℓ≥ℓ1\ell \ge {\ell }_{1}and satisfies (28)ϕ′(ℓ)=ζ′(ℓ)ζ(ℓ)ϕ(ℓ)+ζ(ℓ)(a(ℓ)ν‴(ℓ))′ν(ℓ)−a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )=\frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )+\zeta (\ell )\left(\frac{{(a(\ell ){\nu }^{\prime\prime\prime }(\ell ))}^{^{\prime} }}{\nu (\ell )}-\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}\right).From (1) and (28), we have (29)ϕ′(ℓ)=ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)u(g(ℓ))ν(ℓ)−ζ(ℓ)a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )=\frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )\frac{h(\ell )u(g(\ell ))}{\nu (\ell )}-\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}.Using (11) and (29), we get (30)ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ))ν(ℓ)−ζ(ℓ)a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )\frac{h(\ell )(1-\rho (g(\ell )))\nu (g(\ell ))}{\nu (\ell )}-\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}.Now, it follows from Lemmas 1.1 and 1.2 that (31)ν′(ℓ)≥λ2ℓ22ν‴(ℓ){\nu }^{^{\prime} }(\ell )\ge \frac{{\lambda }_{2}{\ell }^{2}}{2}{\nu }^{\prime\prime\prime }(\ell )and so (32)ν(g(ℓ))ν(ℓ)≥g(ℓ)ℓ3/η,\frac{\nu (g(\ell ))}{\nu (\ell )}\ge {\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta },respectively. Substituting (31) and (32) into (30), we get ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η−λ2ℓ22ζ(ℓ)a(ℓ)(ν‴(ℓ))2ν2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }-\frac{{\lambda }_{2}{\ell }^{2}}{2}\frac{\zeta (\ell )a(\ell ){({\nu }^{\prime\prime\prime }(\ell ))}^{2}}{{\nu }^{2}(\ell )}.From the definition of ϕ(ℓ)\phi (\ell ), we obtain ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η−λ2ℓ22ζ(ℓ)a(ℓ)ϕ2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }-\frac{{\lambda }_{2}{\ell }^{2}}{2\zeta (\ell )a(\ell )}{\phi }^{2}(\ell ).Setting A=λ2ℓ2/2ζ(ℓ)a(ℓ)A={\lambda }_{2}{\ell }^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}2\zeta (\ell )a(\ell ), B=ζ′(ℓ)/ζ(ℓ)B={\zeta }^{^{\prime} }(\ell )\hspace{-0.08em}\text{/}\hspace{-0.08em}\zeta (\ell ), and ς=ϕ(s)\varsigma =\phi (s)and using Lemma 1.3, we have (33)ϕ′(ℓ)≤−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η+12(ζ′(ℓ))2ζ(ℓ)a(ℓ)λ2ℓ2.{\phi }^{^{\prime} }(\ell )\le -\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }+\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(\ell ))}^{2}}{\zeta (\ell )}\frac{a(\ell )}{{\lambda }_{2}{\ell }^{2}}.Integrating (33) from ℓ1{\ell }_{1}to ℓ\ell , we have ∫ℓ1ℓζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds≤ϕ(ℓ1),\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s\le \phi ({\ell }_{1}),which contradicts (26).Assume that case (ii) holds. Define the function φ(ℓ)\varphi (\ell )by φ(ℓ)=ϑ(ℓ)ν′(ℓ)ν(ℓ).\varphi (\ell )={\vartheta }(\ell )\frac{{\nu }^{^{\prime} }(\ell )}{\nu (\ell )}.Then, clearly φ(ℓ)\varphi (\ell )is positive for ℓ≥ℓ1\ell \ge {\ell }_{1}and satisfies φ′(ℓ)=ϑ′(ℓ)ϑ(ℓ)φ(ℓ)+ϑ(ℓ)ν″(ℓ)ν(ℓ)−(ν′(ℓ))2ν2(ℓ).{\varphi }^{^{\prime} }(\ell )=\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )+{\vartheta }(\ell )\left(\frac{{\nu }^{^{\prime\prime} }(\ell )}{\nu (\ell )}-\frac{{({\nu }^{^{\prime} }(\ell ))}^{2}}{{\nu }^{2}(\ell )}\right).From the definition of φ(ℓ)\varphi (\ell ), we obtain (34)φ′(ℓ)=ϑ′(ℓ)ϑ(ℓ)φ(ℓ)+ϑ(ℓ)ν″(ℓ)ν(ℓ)−φ2(ℓ)ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )=\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )+{\vartheta }(\ell )\frac{{\nu }^{^{\prime\prime} }(\ell )}{\nu (\ell )}-\frac{{\varphi }^{2}(\ell )}{{\vartheta }(\ell )}.Integrating (1) from ℓ\ell to ∞\infty , we have (35)−a(ℓ)ν‴(ℓ)=−∫ℓ∞h(s)u(g(s))ds.-a(\ell ){\nu }^{\prime\prime\prime }(\ell )=-\underset{\ell }{\overset{\infty }{\int }}h(s)u(g(s)){\rm{d}}s.Using (11) and (35), we get (36)−a(ℓ)ν‴(ℓ)=−∫ℓ∞h(s)(1−ρ(g(s)))ν(g(s))ds.-a(\ell ){\nu }^{\prime\prime\prime }(\ell )=-\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s)))\nu (g(s)){\rm{d}}s.From Lemma 1.2, we get ν(ℓ)≥ηℓν′(ℓ),\nu (\ell )\ge \eta \ell {\nu }^{^{\prime} }(\ell ),that is, (37)ν(g(ℓ))ν(ℓ)≥g(ℓ)ℓ1/η.\frac{\nu (g(\ell ))}{\nu (\ell )}\ge {\left(\frac{g(\ell )}{\ell }\right)}^{1\text{/}\eta }.Combining (37) and (36), we get −a(ℓ)ν‴(ℓ)≤−ν(ℓ)∫ℓ∞h(s)(1−ρ(g(s)))g(s)s1/ηds,-a(\ell ){\nu }^{\prime\prime\prime }(\ell )\le -\nu (\ell )\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s,that is, −ν‴(ℓ)≤−ν(ℓ)a(ℓ)∫ℓ∞h(s)(1−ρ(g(s)))g(s)s1/ηds.-{\nu }^{\prime\prime\prime }(\ell )\le -\frac{\nu (\ell )}{a(\ell )}\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s.Integrating the above inequality from ℓ\ell to ∞\infty , we have ν″(ℓ)≤−ν(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv.{\nu }^{^{\prime\prime} }(\ell )\le -\nu (\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v.Combining the above inequality with (34), we obtain φ′(ℓ)≤−ϑ(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv+ϑ′(ℓ)ϑ(ℓ)φ(ℓ)−φ2(ℓ)ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )\le -{\vartheta }(\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v+\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )-\frac{{\varphi }^{2}(\ell )}{{\vartheta }(\ell )}.Thus, we have (38)φ′(ℓ)≤−ϑ(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv+(ϑ′(ℓ))24ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )\le -{\vartheta }(\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v+\frac{{({{\vartheta }}^{^{\prime} }(\ell ))}^{2}}{4{\vartheta }(\ell )}.Integrating (38) from ℓ1{\ell }_{1}to ℓ\ell , we have ∫ℓ1ℓϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds≤φ(ℓ1),\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s\le \varphi ({\ell }_{1}),which contradicts (27).The proof of the case where (iii) or (iv) holds is the same as that of Theorem 2.1.This completes the proof.□Example 2.3Consider the equation (39)(ℓ2(u(ℓ)+ρ0u(θℓ))‴)′+h0ℓ2u(ℓ)=0,{\left({\ell }^{2}{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+\frac{{h}_{0}}{{\ell }^{2}}u(\ell )=0,here h0>0{h}_{0}\gt 0and θ∈(0,1]\theta \in (0,1]. It is easy to see that δ(ℓ)=1/ℓ\delta (\ell )=1\hspace{-0.08em}\text{/}\hspace{-0.08em}\ell and (4) holds. Let ζ(ℓ)=ϑ(ℓ)=ℓ\zeta (\ell )={\vartheta }(\ell )=\ell .Next, using Theorem 2.2, we find limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))g2(s)2!δ(s)−14a(s)δ(s)ds=limsupℓ→∞∫ℓ0ℓh0s2λ1(1−ρ0)s22!1s−14s2(1/s)ds=∞,\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{2}(s)}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{h}_{0}}{{s}^{2}}\frac{{\lambda }_{1}(1-{\rho }_{0}){s}^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\frac{1}{s}-\frac{1}{4{s}^{2}(1\hspace{0.1em}\text{/}\hspace{0.1em}s)}\right){\rm{d}}s=\infty ,which is verified if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}Moreover, ∫ℓ0∞ζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds=∫ℓ0∞sh0s2(1−ρ0)ss3/η−121ss2λ2s2ds=∞,\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(s\frac{{h}_{0}}{{s}^{2}}(1-{\rho }_{0}){\left(\frac{s}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{1}{s}\frac{{s}^{2}}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\infty ,which is verified if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}and ∫ℓ0∞ϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds=∫ℓ0∞s∫s∞1v2∫v∞h0ς2(1−ρ0)ςς1/ηdςdv−14sds=∞.\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s=\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(s\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{{v}^{2}}\underset{v}{\overset{\infty }{\int }}\frac{{h}_{0}}{{\varsigma }^{2}}(1-{\rho }_{0}){\left(\frac{\varsigma }{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{1}{4s}\right){\rm{d}}s=\infty .Thus, every solution of (39) is oscillatory or tends to zero if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}.3ConclusionIn this paper, we have presented new theorems for studying the asymptotic behavior and oscillation of (1). By using comparison principle and Riccati transformation technique, we obtained new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Suitable illustrative examples have also been provided. It will be of interest to investigate the odd-order equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

Asymptotic behavior of even-order noncanonical neutral differential equations

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de Gruyter
Copyright
© 2022 Osama Moaaz et al., published by De Gruyter
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2022-0001
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Abstract

1IntroductionThe neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena, see [1,2]. It is well known that the modeling of natural and technological phenomena produces differential equations, often of higher-order; see, for instance, the papers [3,4]. Oscillation theory is a branch of qualitative theory that investigates the oscillatory and non-oscillatory behavior of solutions to differential equations.In this work, we consider the even-order NDDE (1)(a⋅(u+ρ⋅u∘τ)(m−1))′(ℓ)+h(ℓ)u(g(ℓ))=0,{(a\cdot {(u+\rho \cdot u\circ \tau )}^{(m-1)})}^{^{\prime} }(\ell )+h(\ell )u(g(\ell ))=0,where ℓ≥ℓ0\ell \ge {\ell }_{0}, m≥4m\ge 4is an even integer and aa, ρ\rho , τ\tau , hh, and ggare continuous real-valued functions on [ℓ0,∞){[}{\ell }_{0},\infty ). We also assume that a∈C1([ℓ0,∞),(0,∞))a\in {C}^{1}({[}{\ell }_{0},\infty ),(0,\infty )), a′(ℓ)>0{a}^{^{\prime} }(\ell )\gt 0, ρ(ℓ)∈[0,ρ0]\rho (\ell )\in {[}0,{\rho }_{0}], ρ0<1{\rho }_{0}\lt 1is a constant, h≥0h\ge 0, h≡0h\equiv 0on any half-line [L,∞){[}L,\infty )for all L≥ℓ0L\ge {\ell }_{0}, τ(ℓ)≤ℓ\tau (\ell )\le \ell , g(ℓ)≤ℓg(\ell )\le \ell , limℓ→∞τ(ℓ)=limℓ→∞g(ℓ)=∞{\mathrm{lim}}_{\ell \to \infty }\tau (\ell )={\mathrm{lim}}_{\ell \to \infty }g(\ell )=\infty and ∫ℓ0∞a−1(s)ds<∞.{\int }_{{\ell }_{0}}^{\infty }{a}^{-1}(s){\rm{d}}s\lt \infty .A solution of (1) is a function u∈C([ℓu,∞),R)u\in C({[}{\ell }_{u},\infty ),{\mathbb{R}}), ℓu≥ℓ0{\ell }_{u}\ge {\ell }_{0}, which satisfies the properties u+ρu(τ)∈C(m−1)([ℓu,∞),R)u+\rho u(\tau )\in {C}^{(m-1)}({[}{\ell }_{u},\infty ),{\mathbb{R}}), a(u+ρu(τ))(m−1)∈C1([ℓu,∞),R)a{(u+\rho u(\tau ))}^{(m-1)}\in {C}^{1}({[}{\ell }_{u},\infty ),{\mathbb{R}})and uusatisfies (1) on [ℓu,∞).{[}{\ell }_{u},\infty ).We consider only the proper solutions uuof (1), that is, uuis not identically zero eventually. A solution uuof (1) is called oscillatory if it is neither positive nor negative, ultimately; otherwise, it is called nonoscillatory.For several decades, an growing interest in presenting criteria for oscillation of different classes of DDE has been observed. Recently, the works [5,6,7, 8,9] have developed many techniques and approaches for studying the oscillations of delay and advanced second-order equations. However, neutral second-order equations have been studied in many techniques through works [10,11,12, 13,14,15]. The development in the study of second-order DDEs was reflected on the study of even-order equations, see for delay [16,17,18, 19,20,21] and for neutral [22,23, 24,25,26, 27,28]. On the other hand, the works ([29,30] for third-order, and [31,32,33] for odd-order) contributed to the development of the oscillatory theory of odd-order delay differential equations.Zhang et al. [19] investigated the oscillatory behavior of a higher-order differential equation (2)(a(ℓ)(u(m−1)(ℓ))α)′+h(ℓ)uβ(g(ℓ))=0,{(a(\ell ){({u}^{(m-1)}(\ell ))}^{\alpha })}^{^{\prime} }+h(\ell ){u}^{\beta }(g(\ell ))=0,where α\alpha , β\beta are ratios of odd natural numbers and (3)∫ℓ0∞a−1/α(s)ds<∞.\underset{{\ell }_{0}}{\overset{\infty }{\int }}{a}^{-1\text{/}\alpha }(s){\rm{d}}s\lt \infty .Zhang et al. [19] obtained results which ensure that every solution uuof (2) is either oscillatory or satisfies limℓ→∞u(ℓ)=0.{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0.Zhang et al. [20] studied the oscillation of (2) for α≥β\alpha \ge \beta and improved the results reported in [19].For fourth-order, Zhang et al. [21] studied (a(u‴)α)′(ℓ)+h(ℓ)uα(g(ℓ))=0{\left(a{\left({u}^{\prime\prime\prime })}^{\alpha })}^{^{\prime} }(\ell )+h(\ell ){u}^{\alpha }(g(\ell ))=0and presented some oscillation criteria (including Hille-and Nehari-type criteria). Moreover, Moaaz and Muhib [17] presented criteria for fourth-order DDE (a(u‴)α)′(ℓ)+f(ℓ,u(g(ℓ)))=0,{\left(a{\left({u}^{\prime\prime\prime })}^{\alpha })}^{^{\prime} }(\ell )+f\left(\ell ,u(g\left(\ell )))=0,under the conditions f(ℓ,u)≥h(ℓ)uβf\left(\ell ,u)\ge h(\ell ){u}^{\beta }and α,β\alpha ,\beta are the ratios of odd natural numbers.For neutral delay equations, Zhang et al. [28] studied the even-order nonlinear NDDE (u+ρ⋅u∘τ)(m)(ℓ)+h(ℓ)f(u(g(ℓ)))=0,{(u+\rho \cdot u\circ \tau )}^{(m)}(\ell )+h(\ell )f\left(u\left(g\left(\ell )))=0,under the conditions uf(u)>0uf(u)\gt 0for all u≠0,u\ne 0,and ffis nondecreasing. Moaaz et al. [25] investigated the asymptotic behavior of solutions of the higher-order NDDE (a((u+ρ⋅u∘τ)(m−1))α)′(t)+f(ℓ,u(g(ℓ)))=0,{\left(a{\left({(u+\rho \cdot u\circ \tau )}^{(m-1)})}^{\alpha })}^{^{\prime} }(t)+f\left(\ell ,u\left(g\left(\ell )))=0,where ∣f(ℓ,u)∣≥h(ℓ)∣u∣β| f(\ell ,u)| \ge h(\ell ){| u| }^{\beta }.To the best of our knowledge, the previous studies in the literature which considered the asymptotic behavior of solutions of NDDEs of mm-order were concerned only with the canonical form ∫∞a−1(s)ds=∞{\int }^{\infty }{a}^{-1}(s){\rm{d}}s=\infty . In this paper, we obtain new conditions for testing oscillation of NDDE (1) in noncanonical case, using Riccati substitution along with comparison principles with first-order DDE. Examples are presented to illustrate our new results.In the following, we present useful lemmas that will be used throughout the results.Lemma 1.1[34, Lemma 2.2.3] Assume that ϖ∈Cm([ℓ0,∞),R+)\varpi \in {C}^{m}\left({[}{\ell }_{0},\infty ),{{\mathbb{R}}}^{+}), ϖ(m){\varpi }^{(m)}is not identically zero on a subray of [ℓ0,∞){[}{\ell }_{0},\infty )and ϖ(m){\varpi }^{(m)}is of fixed sign. Suppose that ϖ(m−1)ϖ(m)≤0{\varpi }^{(m-1)}{\varpi }^{(m)}\le 0for ℓ∈[ℓ1,∞)\ell \in {[}{\ell }_{1},\infty ), where ℓ1≥ℓ0{\ell }_{1}\ge {\ell }_{0}large enough. If limℓ→∞ϖ(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }\varpi (\ell )\ne 0, then there exists a ℓλ∈[ℓ1,∞){\ell }_{\lambda }\in {[}{\ell }_{1},\infty )such thatϖ≥λ(m−1)!ℓm−1∣ϖ(m−1)∣,\varpi \ge \frac{\lambda }{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\ell }^{m-1}| {\varpi }^{(m-1)}| ,for every λ∈(0,1)\lambda \in \left(0,1)and ℓ∈[ℓλ,∞).\ell \in {[}{\ell }_{\lambda },\infty ).Lemma 1.2[35, Lemma 1] Let f∈Cm([ℓ0,∞),R)f\in {C}^{m}\left({[}{\ell }_{0},\infty ),{\mathbb{R}}), f(r)>0{f}^{(r)}\gt 0, r=0,1,…,mr=0,1,\ldots ,mand f(m+1)≤0{f}^{(m+1)}\le 0eventually. Then, for every η∈(0,1)\eta \in (0,1)f(ℓ)≥ηℓmf′(ℓ).f(\ell )\ge \frac{\eta \ell }{m}{f}^{^{\prime} }(\ell ).Lemma 1.3[36, Lemma 1.2] Assume that B≥0B\ge 0, A>0A\gt 0, w≥0w\ge 0and α>0\alpha \gt 0. Then, Bw−Aw(α+1)/α≤αα(α+1)α+1Bα+1Aα.Bw-A{w}^{(\alpha +1)\text{/}\alpha }\le \frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{{B}^{\alpha +1}}{{A}^{\alpha }}.Lemma 1.4[20, Lemma 2.1] Let f∈Cn([t0,∞),(0,∞))f\in {C}^{n}({[}{t}_{0},\infty ),(0,\infty )). If the derivative f(n)(t){f}^{(n)}(t)is eventually of one sign for all large tt, then there exist a tx{t}_{x}such that tx≥t0{t}_{x}\ge {t}_{0}and an integer l,0≤l≤nl,0\le l\le n, with n+ln+leven for f(n)(t)≥0{f}^{(n)}(t)\ge 0, or n+ln+lodd for f(n)(t)≤0{f}^{(n)}(t)\le 0such thatl>0impliesf(k)(t)>0fort≥tx,k=0,1,…,l−1l\gt 0\hspace{1em}{implies}\hspace{0.33em}{f}^{(k)}(t)\gt 0\hspace{1em}{for}\hspace{0.33em}t\ge {t}_{x},\hspace{1em}k=0,1,\ldots ,l-1andl≤n−1implies(−1)l+kf(k)(t)>0fort≥tx,k=l,l+1,…,n−1.l\le n-1\hspace{1em}{implies}\hspace{0.33em}{(-1)}^{l+k}{f}^{(k)}(t)\gt 0\hspace{1em}{for}\hspace{0.33em}t\ge {t}_{x},\hspace{1em}k=l,l+1,\ldots ,n-1.2Main resultsFor the convenience, we use notation ν≔u+ρ⋅u∘τ\nu := u+\rho \cdot u\circ \tau .Lemma 2.1Assume that u∈C([ℓ0,∞),(0,∞))u\in C({[}{\ell }_{0},\infty ),(0,\infty ))is a solution of (1), eventually. Then, ν>0\nu \gt 0, (aν(m−1))′≤0{(a{\nu }^{(m-1)})}^{^{\prime} }\le 0and ν\nu satisfies one of the following: (1)ν′{\nu }^{^{\prime} }, ν(m−1){\nu }^{(m-1)}and (−ν(m))(-{\nu }^{(m)})are positive;(2)ν′{\nu }^{^{\prime} }, ν(m−2){\nu }^{(m-2)}and (−ν(m−1))(-{\nu }^{(m-1)})are positive;(3)(−1)kν(k){(-1)}^{k}{\nu }^{(k)}are positive, for all k=1,2,…,m−1k=1,2,\ldots ,m-1,for ℓ\ell large enough.ProofAssume that uuis an eventually positive solution of (1). It follows from (1) that (a(ℓ)ν(m−1)(ℓ))′=−h(ℓ)u(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }=-h(\ell )u(g(\ell ))\le 0.Now, from above inequality and Lemma 2 that there exist three possible cases (1)–(3) for ℓ≥ℓ1\ell \ge {\ell }_{1}large enough.□Lemma 2.2Assume that u∈C([ℓ0,∞),(0,∞))u\in C({[}{\ell }_{0},\infty ),(0,\infty ))is a solution of (1), where ν\nu satisfies case (3)(3). If(4)∫ℓ0∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ=∞,\underset{{\ell }_{0}}{\overset{\infty }{\int }}\underset{\varrho }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho =\infty ,then, limℓ→∞u(ℓ)=0{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0.ProofAssume that uuis an eventually positive solution of (1), where ν\nu satisfies case (3)(3). Then, limℓ→∞ν(ℓ)=D{\mathrm{lim}}_{\ell \to \infty }\nu (\ell )=D. We claim that D=0D=0. Suppose that D>0D\gt 0, and so for all ε>0\varepsilon \gt 0, there exists ℓ1≥ℓ0{\ell }_{1}\ge {\ell }_{0}such that u(g(ℓ))≥Du(g(\ell ))\ge Dfor ℓ≥ℓ1.\ell \ge {\ell }_{1}.Integrating (1) from ℓ1{\ell }_{1}to ℓ\ell , we get a(ℓ)ν(m−1)(ℓ)=a(ℓ2)ν(m−1)(ℓ2)−∫ℓ1ℓh(s)u(g(s))ds≤−D∫ℓ1ℓh(s)ds,a(\ell ){\nu }^{(m-1)}(\ell )=a({\ell }_{2}){\nu }^{(m-1)}({\ell }_{2})-\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s)u(g(s)){\rm{d}}s\le -D\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s){\rm{d}}s,that is, (5)ν(m−1)(ℓ)<−D1a(ℓ)∫ℓ1ℓh(s)ds.{\nu }^{(m-1)}(\ell )\lt -D\frac{1}{a(\ell )}\underset{{\ell }_{1}}{\overset{\ell }{\int }}h(s){\rm{d}}s.Integrating (5) twice from ℓ\ell to ∞\infty , we obtain −ν(m−2)(ℓ)<−D∫ℓ∞1a(ς)∫ℓ1ςh(s)dsdς-{\nu }^{(m-2)}(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma and (6)ν(m−3)(ℓ)<−D∫ℓ∞∫s∞1a(ς)∫ℓ1ςh(s)dsdςds=−D∫ℓ∞(ς−ℓ)1a(ς)∫ℓ1ςh(s)dsdς.{\nu }^{(m-3)}(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}s=-D\underset{\ell }{\overset{\infty }{\int }}(\varsigma -\ell )\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma .Similarly, integrating (6) m−4m-4times from ℓ\ell to ∞\infty , we find ν′(ℓ)<−D∫ℓ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdς.{\nu }^{^{\prime} }(\ell )\lt -D\underset{\ell }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma .Integrating this inequality from ℓ1{\ell }_{1}to ∞\infty , we obtain ν(ℓ1)>D∫ℓ1∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ,\nu ({\ell }_{1})\gt D\underset{{\ell }_{1}}{\overset{\infty }{\int }}\underset{\varrho }{\overset{\infty }{\int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho ,which is a contradiction with (4). Thus, D=0D=0. This completes the proof.□Theorem 2.1Let (4) hold. If there exists a λ0∈(0,1){\lambda }_{0}\in (0,1)such that the first-order delay differential equation(7)y′(ℓ)+h(ℓ)λ0(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!a(g(ℓ))y(g(ℓ))=0{y}^{^{\prime} }(\ell )+h(\ell )\frac{{\lambda }_{0}(1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(\ell ))}y(g(\ell ))=0is oscillatory and(8)limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds=∞\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\infty holds for some constant λ1∈(0,1){\lambda }_{1}\in (0,1), then every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofSuppose that (1) has a positive solution uuwhich satisfies limℓ→∞u(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }u(\ell )\ne 0. It follows from (1) that (9)(a(ℓ)ν(m−1)(ℓ))′=−h(ℓ)u(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }=-h(\ell )u(g(\ell ))\le 0.From Lemma 2.1, there are three possible cases for the behavior of ν\nu and its derivatives.Let (1)(1)hold. From Lemma 1.1, we have (10)ν(ℓ)≥λℓm−1(m−1)!ν(m−1)(ℓ)\nu (\ell )\ge \frac{\lambda {\ell }^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-1)}(\ell )for every λ∈(0,1).\lambda \in (0,1).It follows from the definition of ν(ℓ)\nu (\ell )that (11)u(ℓ)=ν(ℓ)−ρ(ℓ)u(τ(ℓ))≥(1−ρ(ℓ))ν(ℓ).u(\ell )=\nu (\ell )-\rho (\ell )u(\tau (\ell ))\ge (1-\rho (\ell ))\nu (\ell ).Combining (9) and (11), we get (12)(a(ℓ)ν(m−1)(ℓ))′≤−h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ)).{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }\le -h(\ell )(1-\rho (g(\ell )))\nu (g(\ell )).From (10), we obtain (a(ℓ)ν(m−1)(ℓ))′+h(ℓ)λ(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!ν(m−1)(g(ℓ))≤0.{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }+h(\ell )\frac{\lambda (1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-1)}(g(\ell ))\le 0.Now, we define the function y(ℓ)=a(ℓ)ν(m−1)(ℓ)y(\ell )=a(\ell ){\nu }^{(m-1)}(\ell ). Clearly, yyis a positive solution of the first-order delay differential inequality (13)y′(ℓ)+h(ℓ)λ(1−ρ(g(ℓ)))(g(ℓ))m−1(m−1)!a(g(ℓ))y(g(ℓ))≤0.{y}^{^{\prime} }(\ell )+h(\ell )\frac{\lambda (1-\rho (g(\ell ))){(g(\ell ))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(\ell ))}y(g(\ell ))\le 0.Thus, using [40, Theorem 1], equation (7) has also a positive solution for all λ0∈(0,1){\lambda }_{0}\in (0,1), this contradicts the assumption that (7) is oscillatory.Let (2)(2)hold. We define ω\omega by (14)ω(ℓ)=a(ℓ)ν(m−1)(ℓ)ν(m−2)(ℓ),ℓ≥ℓ1.\omega (\ell )=\frac{a(\ell ){\nu }^{(m-1)}(\ell )}{{\nu }^{(m-2)}(\ell )},\hspace{1.0em}\ell \ge {\ell }_{1}.Then, ω(ℓ)<0\omega (\ell )\lt 0for ℓ≥ℓ1\ell \ge {\ell }_{1}. Noting that (a(ℓ)ν(m−1)(ℓ))′≤0{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }\le 0, we find (15)a(s)ν(m−1)(s)≤a(ℓ)ν(m−1)(ℓ),s≥ℓ≥ℓ1.a(s){\nu }^{(m-1)}(s)\le a(\ell ){\nu }^{(m-1)}(\ell ),\hspace{1.0em}s\ge \ell \ge {\ell }_{1}.Dividing (15) by aaand integrating it from ℓ\ell to ∞\infty , we obtain 0≤ν(m−2)(ℓ)+a(ℓ)ν(m−1)(ℓ)δ(ℓ),0\le {\nu }^{(m-2)}(\ell )+a(\ell ){\nu }^{(m-1)}(\ell )\delta (\ell ),which yields −a(ℓ)ν(m−1)(ℓ)δ(ℓ)ν(m−2)(ℓ)≤1.-\frac{a(\ell ){\nu }^{(m-1)}(\ell )\delta (\ell )}{{\nu }^{(m-2)}(\ell )}\le 1.Thus, by (14), we get (16)−ω(ℓ)δ(ℓ)≤1.-\omega (\ell )\delta (\ell )\le 1.Differentiating (14), we arrive at ω′(ℓ)=(a(ℓ)ν(m−1)(ℓ))′ν(m−2)(ℓ)−a(ℓ)(ν(m−1)(ℓ))2(ν(m−2)(ℓ))2,{\omega }^{^{\prime} }(\ell )=\frac{{(a(\ell ){\nu }^{(m-1)}(\ell ))}^{^{\prime} }}{{\nu }^{(m-2)}(\ell )}-\frac{a(\ell ){({\nu }^{(m-1)}(\ell ))}^{2}}{{({\nu }^{(m-2)}(\ell ))}^{2}},which follows from (1) and (14) that (17)ω′(ℓ)=−h(ℓ)u(g(ℓ))ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )=-\frac{h(\ell )u(g(\ell ))}{{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.From the definition of ν(ℓ)\nu (\ell )and the fact that ν′(ℓ)>0,{\nu }^{^{\prime} }(\ell )\gt 0,we get that (11) holds. Hence, it follows from (17) that (18)ω′(ℓ)≤−h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ))ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{h(\ell )(1-\rho (g(\ell )))\nu (g(\ell ))}{{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Using Lemma 1.1, we get ν(ℓ)≥λℓm−2(m−2)!ν(m−2)(ℓ)\nu (\ell )\ge \frac{\lambda {\ell }^{m-2}}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\nu }^{(m-2)}(\ell )for every λ∈(0,1)\lambda \in (0,1)and for all sufficiently large ℓ\ell . Then, (18) becomes ω′(ℓ)≤−λh(ℓ)(1−ρ(g(ℓ)))gm−2(ℓ)ν(m−2)(g(ℓ))(m−2)!ν(m−2)(ℓ)−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{\lambda h(\ell )(1-\rho (g(\ell ))){g}^{m-2}(\ell ){\nu }^{(m-2)}(g(\ell ))}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}{\nu }^{(m-2)}(\ell )}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Since ℓ≥g(ℓ)\ell \ge g(\ell )and ν(m−2)(ℓ){\nu }^{(m-2)}(\ell )are decreasing, we have (19)ω′(ℓ)≤−λh(ℓ)(1−ρ(g(ℓ)))gm−2(ℓ)(m−2)!−ω2(ℓ)a(ℓ).{\omega }^{^{\prime} }(\ell )\le -\frac{\lambda h(\ell )(1-\rho (g(\ell ))){g}^{m-2}(\ell )}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}-\frac{{\omega }^{2}(\ell )}{a(\ell )}.Multiplying (19) by δ(ℓ)\delta (\ell )and integrating it from ℓ1{\ell }_{1}to ℓ\ell , we have 0≥δ(ℓ)ω(ℓ)−δ(ℓ1)ω(ℓ1)+∫ℓ1ℓω(s)a(s)ds+∫ℓ1ℓδ(s)a(s)ω2(s)ds+∫ℓ1ℓλh(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)ds.0\ge \delta (\ell )\omega (\ell )-\delta ({\ell }_{1})\omega ({\ell }_{1})+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\omega (s)}{a(s)}{\rm{d}}s+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\delta (s)}{a(s)}{\omega }^{2}(s){\rm{d}}s+\underset{{\ell }_{1}}{\overset{\ell }{\int }}\frac{\lambda h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s){\rm{d}}s.Setting A=δ(s)/a(s)A=\delta (s)\hspace{0.1em}\text{/}\hspace{0.1em}a(s), B=1/a(s)B=1\hspace{0.1em}\text{/}\hspace{0.1em}a(s), and w=−ω(s)w=-\omega (s), and using Lemma 1.3, we have ∫ℓ1ℓλh(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds≤δ(ℓ1)ω(ℓ1)+1,\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left(\frac{\lambda h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s\le \delta ({\ell }_{1})\omega ({\ell }_{1})+1,due to (16), which contradicts (8).Assume that case (3)(3)holds. From Lemma 2.2 and (4), we see that limℓ→∞u(ℓ)=0{\mathrm{lim}}_{\ell \to \infty }u(\ell )=0, which is a contradiction.This completes the proof.□Corollary 2.1Assume that (4) and (8) hold. If(20)liminfℓ→∞∫g(ℓ)ℓh(s)(1−ρ(g(s)))(g(s))m−1(m−1)!a(g(s))ds>1e,\mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\int }}h(s)\frac{(1-\rho (g(s))){(g(s))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(s))}{\rm{d}}s\gt \frac{1}{e},for some λ1∈(0,1){\lambda }_{1}\in (0,1), then every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofBy [38, Theorem 2.1.1], assumption (20) ensures that the differential equation (7) has no positive solutions. Application of Theorem 2.1 yields the result.□Remark 2.1Combining Theorem 2.1 and the results reported in [39] for the oscillation of equation (7), one can derive various oscillation criteria for equation (1).Example 2.1We consider the NDDE (21)(eℓ(u(ℓ)+ρ0u(θℓ))‴)′+h0eℓu(εℓ)=0,{\left({e}^{\ell }{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+{h}_{0}{e}^{\ell }u(\varepsilon \ell )=0,where h0>0{h}_{0}\gt 0and θ,ε∈(0,1)\theta ,\varepsilon \in (0,1). Note that, a(ℓ)=eℓa(\ell )={e}^{\ell }, ρ(ℓ)=ρ0\rho (\ell )={\rho }_{0}, τ(ℓ)=θℓ\tau (\ell )=\theta \ell , h(ℓ)=h0eℓh(\ell )={h}_{0}{e}^{\ell }and g(ℓ)=εℓg(\ell )=\varepsilon \ell . It is easy to see that δ(ℓ)=e−ℓ\delta (\ell )={e}^{-\ell }.Now, from Corollary 2.1, we have ∫ℓ0∞∫ϱ∞(ς−ℓ)m−31a(ς)∫ℓ1ςh(s)dsdςdϱ=∫ℓ0∞∫ϱ∞(ς−ℓ)1eς∫ℓ1ςh0esdsdςdϱ=∞,liminfℓ→∞∫g(ℓ)ℓh(s)(1−ρ(g(s)))(g(s))m−1(m−1)!a(g(s))ds=liminfℓ→∞∫g(ℓ)ℓh0es(1−ρ0)(εs)33!eεsds=∞>1e\begin{array}{rcl}\underset{{\ell }_{0}}{\overset{\infty }{\displaystyle \int }}\underset{\varrho }{\overset{\infty }{\displaystyle \int }}{(\varsigma -\ell )}^{m-3}\left(\frac{1}{a(\varsigma )}\underset{{\ell }_{1}}{\overset{\varsigma }{\displaystyle \int }}h(s){\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho & =& \underset{{\ell }_{0}}{\overset{\infty }{\displaystyle \int }}\underset{\varrho }{\overset{\infty }{\displaystyle \int }}(\varsigma -\ell )\left(\frac{1}{{e}^{\varsigma }}\underset{{\ell }_{1}}{\overset{\varsigma }{\displaystyle \int }}{h}_{0}{e}^{s}{\rm{d}}s\right){\rm{d}}\varsigma {\rm{d}}\varrho =\infty ,\\ \mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\displaystyle \int }}h(s)\frac{(1-\rho (g(s))){(g(s))}^{m-1}}{(m-1)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}a(g(s))}{\rm{d}}s& =& \mathop{\mathrm{lim}\inf }\limits_{\ell \to \infty }\underset{g(\ell )}{\overset{\ell }{\displaystyle \int }}{h}_{0}{e}^{s}\frac{(1-{\rho }_{0}){(\varepsilon s)}^{3}}{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}{e}^{\varepsilon s}}{\rm{d}}s=\infty \gt \frac{1}{e}\end{array}and limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))gm−2(s)(m−2)!δ(s)−14a(s)δ(s)ds=limsupℓ→∞∫ℓ0ℓλ1h0es(1−ρ0)(εs)22!e−s−14ese−sds=∞.\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{m-2}(s)}{(m-2)\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}{h}_{0}{e}^{s}(1-{\rho }_{0}){(\varepsilon s)}^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e}^{-s}-\frac{1}{4{e}^{s}{e}^{-s}}\right){\rm{d}}s\hspace{6.27em}=\infty .Thus, (4), (20), and (8) are satisfied. Therefore, every solution of (21) is oscillatory or tends to zero.Example 2.2Consider the equation (22)(ℓ2(u(ℓ)+ρ0u(θℓ))‴)′+h0ℓ2u(εℓ)=0,{\left({\ell }^{2}{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+\frac{{h}_{0}}{{\ell }^{2}}u(\varepsilon \ell )=0,here h0>0{h}_{0}\gt 0and θ,ε∈(0,1)\theta ,\varepsilon \in (0,1). We note that m=4m=4, a(ℓ)=ℓ2a(\ell )={\ell }^{2}, ρ(ℓ)=ρ0\rho (\ell )={\rho }_{0}, τ(ℓ)=θℓ\tau (\ell )=\theta \ell , h(ℓ)=h0/ℓ2h(\ell )={h}_{0}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\ell }^{2}, and g(ℓ)=εℓg(\ell )=\varepsilon \ell . It is easy to see that δ(ℓ)=1/ℓ\delta (\ell )=1\hspace{-0.08em}\text{/}\hspace{-0.08em}\ell and (4) holds. Next, (20) reduces to (23)h0(1−ρ0)ln1ε>3!εe.{h}_{0}(1-{\rho }_{0})\mathrm{ln}\frac{1}{\varepsilon }\gt \frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{\varepsilon e}.Moreover, (8) becomes limℓ→∞sup∫ℓ0ℓh0λ1(1−ρ0)ε22!−141sds=∞,{\mathrm{lim}}_{\ell \to \infty }\sup \underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{h}_{0}{\lambda }_{1}(1-{\rho }_{0}){\varepsilon }^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}-\frac{1}{4}\right)\frac{1}{s}{\rm{d}}s=\infty ,which is verified if (24)h0(1−ρ0)>12ε2.{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2{\varepsilon }^{2}}.Using Corollary 2.1, if h0>M≔max3!e(1−ρ0)εln1ε,12(1−ρ0)ε2,{h}_{0}\gt M:= \max \left\{\frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e(1-{\rho }_{0})\varepsilon \mathrm{ln}\frac{1}{\varepsilon }},\frac{1}{2(1-{\rho }_{0}){\varepsilon }^{2}}\right\},then every solution of (22) is oscillatory or tends to zero, where M=12(1−ρ0)ε2ifε∈(0,0.28464]M=\frac{1}{2(1-{\rho }_{0}){\varepsilon }^{2}}\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\varepsilon \in (0,0.28464]and M=3!e(1−ρ0)εln1εifε∈(0.28464,1).M=\frac{3\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}{e(1-{\rho }_{0})\varepsilon \mathrm{ln}\frac{1}{\varepsilon }}\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\varepsilon \in (0.28464,1).It is easy to notice that (20) does not apply in the ordinary case (g(ℓ)=ℓg(\ell )=\ell ). So, in the following theorem, we set new conditions for testing the oscillation of (1) when m=4m=4, which apply in the ordinary case.Theorem 2.2Assume that m=4m=4and (4) hold. If(25)limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))g2(s)2!δ(s)−14a(s)δ(s)ds=∞,\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{2}(s)}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\infty ,for some constant λ1∈(0,1){\lambda }_{1}\in (0,1). Assume further that there exist two positive functions ζ(ℓ),ϑ(ℓ)∈C1[ℓ0,∞)\zeta (\ell ),{\vartheta }(\ell )\in {C}^{1}{[}{\ell }_{0},\infty ), such that(26)∫ℓ0∞ζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds=∞\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\infty and(27)∫ℓ0∞ϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds=∞\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s=\infty for some constant λ2∈(0,1){\lambda }_{2}\in (0,1). Then, every nonoscillatory solution uuof (1) satisfies limt→∞u(ℓ)=∞.{\mathrm{lim}}_{t\to \infty }u(\ell )=\infty .ProofAssume that (1) has a nonoscillatory solution uuwhich is eventually positive and limℓ→∞u(ℓ)≠0{\mathrm{lim}}_{\ell \to \infty }u(\ell )\ne 0. It follows from (1) and Lemma 2.1 that there exist four possible cases for the behavior of ν\nu and its derivatives: (i)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0, ν‴(ℓ)>0{\nu }^{\prime\prime\prime }(\ell )\gt 0and ν(4)(ℓ)≤0{\nu }^{(4)}(\ell )\le 0;(ii)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)<0{\nu }^{^{\prime\prime} }(\ell )\lt 0, ν‴(ℓ)>0{\nu }^{\prime\prime\prime }(\ell )\gt 0and ν(4)(ℓ)≤0;{\nu }^{(4)}(\ell )\le 0;(iii)ν′(ℓ)<0{\nu }^{^{\prime} }(\ell )\lt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0and ν‴(ℓ)<0{\nu }^{\prime\prime\prime }(\ell )\lt 0;(iv)ν′(ℓ)>0{\nu }^{^{\prime} }(\ell )\gt 0, ν″(ℓ)>0{\nu }^{^{\prime\prime} }(\ell )\gt 0and ν‴(ℓ)<0{\nu }^{\prime\prime\prime }(\ell )\lt 0.Let (i) hold. Define the function ϕ(ℓ)\phi (\ell )by ϕ(ℓ)=ζ(ℓ)a(ℓ)ν‴(ℓ)ν(ℓ).\phi (\ell )=\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell )}{\nu (\ell )}.Then, clearly ϕ(ℓ)\phi (\ell )is positive for ℓ≥ℓ1\ell \ge {\ell }_{1}and satisfies (28)ϕ′(ℓ)=ζ′(ℓ)ζ(ℓ)ϕ(ℓ)+ζ(ℓ)(a(ℓ)ν‴(ℓ))′ν(ℓ)−a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )=\frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )+\zeta (\ell )\left(\frac{{(a(\ell ){\nu }^{\prime\prime\prime }(\ell ))}^{^{\prime} }}{\nu (\ell )}-\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}\right).From (1) and (28), we have (29)ϕ′(ℓ)=ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)u(g(ℓ))ν(ℓ)−ζ(ℓ)a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )=\frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )\frac{h(\ell )u(g(\ell ))}{\nu (\ell )}-\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}.Using (11) and (29), we get (30)ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))ν(g(ℓ))ν(ℓ)−ζ(ℓ)a(ℓ)ν‴(ℓ)ν′(ℓ)ν2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )\frac{h(\ell )(1-\rho (g(\ell )))\nu (g(\ell ))}{\nu (\ell )}-\zeta (\ell )\frac{a(\ell ){\nu }^{\prime\prime\prime }(\ell ){\nu }^{^{\prime} }(\ell )}{{\nu }^{2}(\ell )}.Now, it follows from Lemmas 1.1 and 1.2 that (31)ν′(ℓ)≥λ2ℓ22ν‴(ℓ){\nu }^{^{\prime} }(\ell )\ge \frac{{\lambda }_{2}{\ell }^{2}}{2}{\nu }^{\prime\prime\prime }(\ell )and so (32)ν(g(ℓ))ν(ℓ)≥g(ℓ)ℓ3/η,\frac{\nu (g(\ell ))}{\nu (\ell )}\ge {\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta },respectively. Substituting (31) and (32) into (30), we get ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η−λ2ℓ22ζ(ℓ)a(ℓ)(ν‴(ℓ))2ν2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }-\frac{{\lambda }_{2}{\ell }^{2}}{2}\frac{\zeta (\ell )a(\ell ){({\nu }^{\prime\prime\prime }(\ell ))}^{2}}{{\nu }^{2}(\ell )}.From the definition of ϕ(ℓ)\phi (\ell ), we obtain ϕ′(ℓ)≤ζ′(ℓ)ζ(ℓ)ϕ(ℓ)−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η−λ2ℓ22ζ(ℓ)a(ℓ)ϕ2(ℓ).{\phi }^{^{\prime} }(\ell )\le \frac{{\zeta }^{^{\prime} }(\ell )}{\zeta (\ell )}\phi (\ell )-\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }-\frac{{\lambda }_{2}{\ell }^{2}}{2\zeta (\ell )a(\ell )}{\phi }^{2}(\ell ).Setting A=λ2ℓ2/2ζ(ℓ)a(ℓ)A={\lambda }_{2}{\ell }^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}2\zeta (\ell )a(\ell ), B=ζ′(ℓ)/ζ(ℓ)B={\zeta }^{^{\prime} }(\ell )\hspace{-0.08em}\text{/}\hspace{-0.08em}\zeta (\ell ), and ς=ϕ(s)\varsigma =\phi (s)and using Lemma 1.3, we have (33)ϕ′(ℓ)≤−ζ(ℓ)h(ℓ)(1−ρ(g(ℓ)))g(ℓ)ℓ3/η+12(ζ′(ℓ))2ζ(ℓ)a(ℓ)λ2ℓ2.{\phi }^{^{\prime} }(\ell )\le -\zeta (\ell )h(\ell )(1-\rho (g(\ell ))){\left(\frac{g(\ell )}{\ell }\right)}^{3\text{/}\eta }+\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(\ell ))}^{2}}{\zeta (\ell )}\frac{a(\ell )}{{\lambda }_{2}{\ell }^{2}}.Integrating (33) from ℓ1{\ell }_{1}to ℓ\ell , we have ∫ℓ1ℓζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds≤ϕ(ℓ1),\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s\le \phi ({\ell }_{1}),which contradicts (26).Assume that case (ii) holds. Define the function φ(ℓ)\varphi (\ell )by φ(ℓ)=ϑ(ℓ)ν′(ℓ)ν(ℓ).\varphi (\ell )={\vartheta }(\ell )\frac{{\nu }^{^{\prime} }(\ell )}{\nu (\ell )}.Then, clearly φ(ℓ)\varphi (\ell )is positive for ℓ≥ℓ1\ell \ge {\ell }_{1}and satisfies φ′(ℓ)=ϑ′(ℓ)ϑ(ℓ)φ(ℓ)+ϑ(ℓ)ν″(ℓ)ν(ℓ)−(ν′(ℓ))2ν2(ℓ).{\varphi }^{^{\prime} }(\ell )=\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )+{\vartheta }(\ell )\left(\frac{{\nu }^{^{\prime\prime} }(\ell )}{\nu (\ell )}-\frac{{({\nu }^{^{\prime} }(\ell ))}^{2}}{{\nu }^{2}(\ell )}\right).From the definition of φ(ℓ)\varphi (\ell ), we obtain (34)φ′(ℓ)=ϑ′(ℓ)ϑ(ℓ)φ(ℓ)+ϑ(ℓ)ν″(ℓ)ν(ℓ)−φ2(ℓ)ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )=\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )+{\vartheta }(\ell )\frac{{\nu }^{^{\prime\prime} }(\ell )}{\nu (\ell )}-\frac{{\varphi }^{2}(\ell )}{{\vartheta }(\ell )}.Integrating (1) from ℓ\ell to ∞\infty , we have (35)−a(ℓ)ν‴(ℓ)=−∫ℓ∞h(s)u(g(s))ds.-a(\ell ){\nu }^{\prime\prime\prime }(\ell )=-\underset{\ell }{\overset{\infty }{\int }}h(s)u(g(s)){\rm{d}}s.Using (11) and (35), we get (36)−a(ℓ)ν‴(ℓ)=−∫ℓ∞h(s)(1−ρ(g(s)))ν(g(s))ds.-a(\ell ){\nu }^{\prime\prime\prime }(\ell )=-\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s)))\nu (g(s)){\rm{d}}s.From Lemma 1.2, we get ν(ℓ)≥ηℓν′(ℓ),\nu (\ell )\ge \eta \ell {\nu }^{^{\prime} }(\ell ),that is, (37)ν(g(ℓ))ν(ℓ)≥g(ℓ)ℓ1/η.\frac{\nu (g(\ell ))}{\nu (\ell )}\ge {\left(\frac{g(\ell )}{\ell }\right)}^{1\text{/}\eta }.Combining (37) and (36), we get −a(ℓ)ν‴(ℓ)≤−ν(ℓ)∫ℓ∞h(s)(1−ρ(g(s)))g(s)s1/ηds,-a(\ell ){\nu }^{\prime\prime\prime }(\ell )\le -\nu (\ell )\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s,that is, −ν‴(ℓ)≤−ν(ℓ)a(ℓ)∫ℓ∞h(s)(1−ρ(g(s)))g(s)s1/ηds.-{\nu }^{\prime\prime\prime }(\ell )\le -\frac{\nu (\ell )}{a(\ell )}\underset{\ell }{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s.Integrating the above inequality from ℓ\ell to ∞\infty , we have ν″(ℓ)≤−ν(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv.{\nu }^{^{\prime\prime} }(\ell )\le -\nu (\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v.Combining the above inequality with (34), we obtain φ′(ℓ)≤−ϑ(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv+ϑ′(ℓ)ϑ(ℓ)φ(ℓ)−φ2(ℓ)ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )\le -{\vartheta }(\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v+\frac{{{\vartheta }}^{^{\prime} }(\ell )}{{\vartheta }(\ell )}\varphi (\ell )-\frac{{\varphi }^{2}(\ell )}{{\vartheta }(\ell )}.Thus, we have (38)φ′(ℓ)≤−ϑ(ℓ)∫ℓ∞1a(v)∫v∞h(s)(1−ρ(g(s)))g(s)s1/ηdsdv+(ϑ′(ℓ))24ϑ(ℓ).{\varphi }^{^{\prime} }(\ell )\le -{\vartheta }(\ell )\underset{\ell }{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{1\text{/}\eta }{\rm{d}}s\right){\rm{d}}v+\frac{{({{\vartheta }}^{^{\prime} }(\ell ))}^{2}}{4{\vartheta }(\ell )}.Integrating (38) from ℓ1{\ell }_{1}to ℓ\ell , we have ∫ℓ1ℓϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds≤φ(ℓ1),\underset{{\ell }_{1}}{\overset{\ell }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s\le \varphi ({\ell }_{1}),which contradicts (27).The proof of the case where (iii) or (iv) holds is the same as that of Theorem 2.1.This completes the proof.□Example 2.3Consider the equation (39)(ℓ2(u(ℓ)+ρ0u(θℓ))‴)′+h0ℓ2u(ℓ)=0,{\left({\ell }^{2}{(u(\ell )+{\rho }_{0}u(\theta \ell ))}^{\prime\prime\prime })}^{^{\prime} }+\frac{{h}_{0}}{{\ell }^{2}}u(\ell )=0,here h0>0{h}_{0}\gt 0and θ∈(0,1]\theta \in (0,1]. It is easy to see that δ(ℓ)=1/ℓ\delta (\ell )=1\hspace{-0.08em}\text{/}\hspace{-0.08em}\ell and (4) holds. Let ζ(ℓ)=ϑ(ℓ)=ℓ\zeta (\ell )={\vartheta }(\ell )=\ell .Next, using Theorem 2.2, we find limsupℓ→∞∫ℓ0ℓλ1h(s)(1−ρ(g(s)))g2(s)2!δ(s)−14a(s)δ(s)ds=limsupℓ→∞∫ℓ0ℓh0s2λ1(1−ρ0)s22!1s−14s2(1/s)ds=∞,\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{\lambda }_{1}h(s)(1-\rho (g(s))){g}^{2}(s)}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\delta (s)-\frac{1}{4a(s)\delta (s)}\right){\rm{d}}s=\mathop{\mathrm{lim}\sup }\limits_{\ell \to \infty }\underset{{\ell }_{0}}{\overset{\ell }{\int }}\left(\frac{{h}_{0}}{{s}^{2}}\frac{{\lambda }_{1}(1-{\rho }_{0}){s}^{2}}{2\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}}\frac{1}{s}-\frac{1}{4{s}^{2}(1\hspace{0.1em}\text{/}\hspace{0.1em}s)}\right){\rm{d}}s=\infty ,which is verified if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}Moreover, ∫ℓ0∞ζ(s)h(s)(1−ρ(g(s)))g(s)s3/η−12(ζ′(s))2ζ(s)a(s)λ2s2ds=∫ℓ0∞sh0s2(1−ρ0)ss3/η−121ss2λ2s2ds=∞,\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(\zeta (s)h(s)(1-\rho (g(s))){\left(\frac{g(s)}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{{({\zeta }^{^{\prime} }(s))}^{2}}{\zeta (s)}\frac{a(s)}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(s\frac{{h}_{0}}{{s}^{2}}(1-{\rho }_{0}){\left(\frac{s}{s}\right)}^{3\text{/}\eta }-\frac{1}{2}\frac{1}{s}\frac{{s}^{2}}{{\lambda }_{2}{s}^{2}}\right){\rm{d}}s=\infty ,which is verified if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}and ∫ℓ0∞ϑ(s)∫s∞1a(v)∫v∞h(ς)(1−ρ(g(ς)))g(ς)ς1/ηdςdv−(ϑ′(s))24ϑ(s)ds=∫ℓ0∞s∫s∞1v2∫v∞h0ς2(1−ρ0)ςς1/ηdςdv−14sds=∞.\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left({\vartheta }(s)\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{a(v)}\underset{v}{\overset{\infty }{\int }}h(\varsigma )(1-\rho (g(\varsigma ))){\left(\frac{g(\varsigma )}{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{{({{\vartheta }}^{^{\prime} }(s))}^{2}}{4{\vartheta }(s)}\right){\rm{d}}s=\underset{{\ell }_{0}}{\overset{\infty }{\int }}\left(s\underset{s}{\overset{\infty }{\int }}\left(\frac{1}{{v}^{2}}\underset{v}{\overset{\infty }{\int }}\frac{{h}_{0}}{{\varsigma }^{2}}(1-{\rho }_{0}){\left(\frac{\varsigma }{\varsigma }\right)}^{1\text{/}\eta }{\rm{d}}\varsigma \right){\rm{d}}v-\frac{1}{4s}\right){\rm{d}}s=\infty .Thus, every solution of (39) is oscillatory or tends to zero if h0(1−ρ0)>12{h}_{0}(1-{\rho }_{0})\gt \frac{1}{2}.3ConclusionIn this paper, we have presented new theorems for studying the asymptotic behavior and oscillation of (1). By using comparison principle and Riccati transformation technique, we obtained new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Suitable illustrative examples have also been provided. It will be of interest to investigate the odd-order equations.

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 2022

Keywords: differential equations; even-order; oscillatory behavior; noncanonical case; 34C10; 34K11

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