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Research article

Postural orthostatic tachycardia syndrome in patients of orthostatic intolerance symptoms: an ambispective study

  • Background A Postural orthostatic tachycardia syndrome (POTS) is infrequently diagnosed in routine practice because of the variable range of symptoms that could be seen in cardiac rhythm disorders, vertigo, chronic fatigue syndrome and anxiety panic disorder. POTS is a chronic debilitating condition that affects day to day efficient working of an individual. We have planned a study to look for POTS in patients who are having orthostatic intolerance symptoms and underwent a head-up tilt table test (HUTT).
    Aim To study the prevalence of POTS in patients of orthostatic intolerance (OI) symptoms and to analyze symptomatology, its association with neurocardiogenic syncope (NCS), and its outcome.
    Methods We reviewed the medical records of 246 patients presented with symptoms of OI seen at our centre from January 2010 till March 2019. Out of them, 40 patients included, those qualifying the criteria for POTS on HUTT.
    Results The mean age of the cohort was 25.90 ± 10.33 years with a range of 15 to 55 years, and males comprised 52.5% (21/40) of total patients. The most frequent presenting orthostatic symptoms of POTS patients are loss of consciousness (77.5%), lightheadedness (75%), and palpitation (67.5%). A total of 18 patients (45%) had coexisting neurocardiogenic syncope.
    Conclusion POTS is a prevalent condition and have a significant impact on the quality of life, and the majority of patients may not present with OI symptoms during HUTT. We have to keep this possibility in young patients of transient loss of consciousness because it may coexist with NCS.

    Citation: Dinesh Chouksey, Pankaj Rathi, Ajoy Sodani, Rahul Jain, Hashash Singh Ishar. Postural orthostatic tachycardia syndrome in patients of orthostatic intolerance symptoms: an ambispective study[J]. AIMS Neuroscience, 2021, 8(1): 74-85. doi: 10.3934/Neuroscience.2021004

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  • Background A Postural orthostatic tachycardia syndrome (POTS) is infrequently diagnosed in routine practice because of the variable range of symptoms that could be seen in cardiac rhythm disorders, vertigo, chronic fatigue syndrome and anxiety panic disorder. POTS is a chronic debilitating condition that affects day to day efficient working of an individual. We have planned a study to look for POTS in patients who are having orthostatic intolerance symptoms and underwent a head-up tilt table test (HUTT).
    Aim To study the prevalence of POTS in patients of orthostatic intolerance (OI) symptoms and to analyze symptomatology, its association with neurocardiogenic syncope (NCS), and its outcome.
    Methods We reviewed the medical records of 246 patients presented with symptoms of OI seen at our centre from January 2010 till March 2019. Out of them, 40 patients included, those qualifying the criteria for POTS on HUTT.
    Results The mean age of the cohort was 25.90 ± 10.33 years with a range of 15 to 55 years, and males comprised 52.5% (21/40) of total patients. The most frequent presenting orthostatic symptoms of POTS patients are loss of consciousness (77.5%), lightheadedness (75%), and palpitation (67.5%). A total of 18 patients (45%) had coexisting neurocardiogenic syncope.
    Conclusion POTS is a prevalent condition and have a significant impact on the quality of life, and the majority of patients may not present with OI symptoms during HUTT. We have to keep this possibility in young patients of transient loss of consciousness because it may coexist with NCS.


    Studying the properties of solutions to differential equations (DEs) is the main aim of qualitative theory. This theory is concerned with investigating features such as stability, periodicity, bifurcation, oscillation, synchronization, etc. This theory emerged from researchers' attempts to obtain sufficient information about the nonlinear models that appear when modeling biological, physical, and other phenomena; see [1,2,3,4]. The qualitative theory was also extended to include functional, fractional, and partial differential equations.

    A functional differential equation (FDE) is a DE with a deviating argument. That is, the FDE contains a dependent variable and some of its derivatives for various argument values. These equations are distinguished by the fact that they take into account the previous and subsequent times when studying any phenomenon or system. In fact, the majority of studies on FDEs until the time of Volterra [5] focused only on some properties of some very special equations. A few books on FDEs were published. In the latter portion of the 1940s and the initial phase of the 1950s. Mishkis [6] laid the groundwork for a broad theory of linear systems in his book by introducing a general class of delayed-argument equations. In 1954, Bellman and Danskin [7] noted the many uses of equations containing past information in disciplines such as economics and biology. Bellman and Cook [8] provide a more comprehensive treatment of the theoretical framework concerning linear equations and the fundamentals of stability theory.

    Oscillation theory, as part of qualitative theory, revolves around creating the criteria for the presence of oscillatory and non-oscillatory solutions to FDEs, investigating zero-order distribution laws, estimating the number of zeros in a certain period of time and the distance between neighboring zeros, and other basics. Oscillation theory has now become an important mathematical instrument for numerous advanced fields and technological applications. Obtaining oscillation conditions for specific FDEs has been a widely studied area in the last few decades; see [9,10,11,12,13].

    Delay DEs that have the derivative of the solutions of the highest order with and without delay are known as neutral differential equations. These equations are apparent in the investigation of oscillatory masses and the modeling of electrical circuitry with ideal transmission lines (see [14]). Understanding the qualitative characteristics of delay differential equations is becoming more and more important as new models and developments in biology, economics, physics, and engineering keep appearing.

    In the canonical scenario, we find new conditions to test the oscillation of nonlinear neutral FDEs of second order. Namely, we consider the FDE

    ddt(aψ(x)[ddtz]α)(t)+q(t)F(x(g(t)))=0, (1.1)

    where tI:=[t0,), αQ+ is a ratio of odd numbers, and

    z:=x+px(h).

    In our study, we consider the following hypotheses:

    A1: aC1(I,R+), qC(I,R+), and pC(I,[0,p0]), where p0<1.

    A2: h, gC(I,R), ht, gt, g0, limth(t)=, and limtg(t)=.

    A3: ψC1(R,[m,M]), where 0<mM, and κ=m1/αM1/α.

    A4: FC1(R,R), ϰF(ϰ)>0 for ϰ0, F(ϰ)0, and F(ϰw)F(ϰw)F(ϰ)F(w) for ϰw>0.

    For a solution of Eq (1.1), we define a function xC1([tx,),R), txI, which has the properties: aψ(z)αC1([tx,),R), sup{|x(t)|:tt}>0 for all ttx, and satisfies Eq (1.1) on [tx,). If a solution x of FDE (1.1) has arbitrarily large zeros, it is referred to as oscillatory; if not, it is referred to as non-oscillatory. The FDE (1.1) is said to be in the canonical case if

    t0a1/α(s)ds=. (1.2)

    We can divide previous related works in the literature into two main parts. The first section consists of studies that are concerned with using different techniques to study oscillation, and the second section consists of studies that focus on improving the relationships and inequalities used in studying oscillation.

    Oscillation criteria are often only sufficient conditions to test the oscillation and are not necessary. Therefore, finding different techniques and methods for studying oscillation is inevitable and necessary for the purpose of application to the largest area of special cases and also to get rid of some of the restrictions that may be imposed by some techniques.

    In 2000, Džurina and Mihalíková [15] used Riccati substitution to provide some oscillatory conditions for the solutions of the equation

    (az)(t)+q(t)(xh)(t)=0 (1.3)

    with h(t)=tk, k>0 and p(t)=p0<1. Han et al. [16] acquired some Kamenev-type oscillation conditions for (1.3) when h(t)=h0>0. By comparison method, Džurina [17] studied the oscillation of (1.3) with an advanced neutral term, i.e., when h(t)t.

    Şahiner [18] presented Philos-type conditions for oscillation of (1.1) when α=1, F(υ)K>0, and ±F(±υw)LF(υ)F(w), for all υw>0 and for some L>0.

    Theorem 1. ([18], Theorem 2.1) Assume that

    D0={(ϰ,ς):ϰ>ς>ϰ0} and D={(ϰ,ς):ϰςt0}.

    By P, we mean that the continuous real-valued function P with domain D belongs to class and satisfies

    (i)P(ϰ,ς)=0forϰt0;(ii)P(ϰ,ς)>0on the domainD0;(iii)P(ϰ,ς)possesses a continuous and non-positive partial derivativeP/ς    on the domainD0where                P(ϰ,ς)ς=k(ϰ,ς)P(ϰ,ς)for all(ϰ,ς)D0   holds forkC(D,R).

    If there is a function ρC(I,R+) such that

    limsupt1P(t,t0)tt0(P(t,s)ρ(s)Q(s)Mρ(s)a(g(s))4LKg(s)G2(t,s))ds=,

    then all solutions of (1.1) are oscillatory, where Q:=q[1p(g)], and

    G(ϰ,ς)=k(ϰ,ς)ρ(ς)ρ(ς)P(ϰ,ς).

    Putting P(ϰ,ς)=(ϰς)n, n>1, in the results of Theorem 1.1, we obtain a Kamenev-type criterion of Eq (1.1).

    On the other hand, in the most recent period, there was a significant surge in research activity concentrated on improving the inequalities that are used in the study of oscillation. One of the influential relationships in investigating the oscillation of neutral equations involves examining the interplay between x and its z, as well as the relationship between the corresponding function and its derivatives.

    The classical substitution x>(1p)z is often used in studying the oscillation of neutral equations. Moaaz et al. [19] devised improved conditions for the oscillation of

    (a[z]α)(t)+Li=1qi(t)xγ(gi(t))=0, (1.4)

    They provided the following relationship as an improvement on the classical relationship:

    x>zν/2j=11p2j10(11p0ηt1(h[2j])ηt1(h[(2j1)]))

    for p0>1 and ν is an even natural number, and

    x>z(1p0)(ν1)/2j=0p2j0ηt1(h[2j+1])ηt1

    for p<1, and ν is an odd natural number, where h[±l]=h±1(h[±(l1)]), for l=1,2,..., and

    ηt1(t):=tt1a1/α(s)ds.

    For the non-canonical situation, Hassan et al. [20] used the improved relationship

    x>z(n1)/2r=0p2r0(1p0μ0(h[2r+1])μ0(h[2r](t))),

    when z<0.

    A research study for the fourth-order DE

    (az)(t)+q(t)x(g(t))=0,

    Moaaz et al. [21] devised improved substitution for x by z in all cases of positive solutions.

    Lemma 1. ([21], Lemma 1) Let x be a solution of (1.1) and x>0 eventually. Then,

    x(t)>mr=0(2rl=0p(h[l](t)))[z(h[2r](t))p(h[2r](t))z(h[2r+1](t))],

    for any integer m0, where

    h[0](t):=t,h[l](t)=(hh[l1])(t), forl=1,2,....

    See [22,23,24] for other intriguing findings on the oscillatory nature of solutions to third- and fourth-order DEs that have been published more recently.

    The oscillation criteria of neutral DEs depend on several factors:

    The substitution x by z;

    The monotonic and asymptotic features of non-oscillatory solutions;

    The technique used to obtain the oscillation criteria.

    Improving any of these factors directly affects the oscillation criteria. In our paper, we test the extent to which the oscillation criteria are affected by improving the relationship between x and z. We used a well-known technique that produces criteria known as Kamenev criteria; but the improvement lies in the new relationships used in the study.

    Here, we define a class of functions to obtain the oscillation condition of the Kamenev-type criteria. Suppose that

    K:={(u,v,w)I3:wvu}.

    A φC(K,R) belongs to the class Y (φY), if

    (i) φ(u,u,w)=0, φ(u,w,w)=0, and φ(u,s,w)0 for w<s<u.

    (ii) φ has the partial derivative φ/v on K and φ/v is locally integrable with respect to v in K.

    For any function fC1(I,R), we define the following operator:

    T[f;u,w]:=uwf(v)φ(u,v,w)dv,

    for wvu. Moreover, we define the function μ(u,v,w) by

    μ(u,v,w):=1φ(u,v,w)φ(u,v,w)s,

    for all w<v<u. We notice that T[;u,w] is linear. Using integration by parts, we find that

    T[f;u,w]=T[(fμ);u,w]. (1.5)

    Here, we investigate the improved monotonic features of non-oscillatory solutions and then establish new oscillation conditions for (1.1). For simplicity, the class of all positive non-oscillatory solutions of (1.1) is denoted by the symbol S. Also, we assume that γ=1α(α/(α+1))α+1,

    At(t):=tt1[a(s)]1/αds

    for tt, and

    Gq(t;m):=q(t)F(mr=0(2rl=0p(h[l](g(t))))[1p(h[2r](g(t)))1][At1(h[2r](g(t)))At1(g(t))]κ).

    First, we infer the monotonic behavior of positive solutions in the next lemma.

    Lemma 2. If xS, then the corresponding function z of x conforms to z>0, z>0, and (aψ(x)[z]α)<0, eventually.

    Proof. Suppose that xS. From (A2), there exists a t1I whereby (xh)(t) and (xg)(t) are positive for tt1. Therefore, z(t)>0. Using (A4), we have that F(x(g(t)))>0. Hence, (1.1) becomes

    (aψ(x)[z]α)(t)=q(t)F(x(g(t)))<0.

    Thus, we find that aψ(x)[z]α has a constant sign. This is the same as stating that z>0 or z<0 for tt2, where t2 is large enough. But, when z(t)<0, this case contradicts (1.2), as displayed next:

    Let z(t)<0 for tt2. Then

    (aψ(x)[z]α)(t)(aψ(x)[z]α)(t2):=L<0.

    Hence,

    zL1/αa1/α(t)[ψ(x)]1/α,

    which with (A3) gives

    zL1/αM1/αa1/α.

    Thus,

    z(t)z(t2)L1/αM1/αtt2a1/α(s)ds.

    But condition (1.2) results in z(t) as t, a contradiction.

    Consequently, the proof ends.

    Lemma 3. Assume that xS. Then

    z(t)M1/α[a(t)ψ(x(t))]1/αz(t)At1(t).

    Moreover,

    z(t)[At1(t)]κ

    is nonincreasing, for tt1.

    Proof. Let xS. From (A2), there is a t1I such that (xh)(t)>0 and (xg)(t)>0 for tt1. Using Lemma 2, we obtain

    z(t)=z(t1)+tt11[a(ξ)ψ(x(ξ))]1/α[a(ξ)ψ(x(ξ))]1/αz(ξ)dξtt11[Ma(ξ)]1/α[a(ξ)ψ(x(ξ))]1/αz(ξ)dξ1M1/α[a(t)ψ(x(t))]1/αz(t)tt11[a(ξ)]1/αdξ=1M1/α[a(t)ψ(x(t))]1/αz(t)At1(t).

    Hence, for any tt1,

    0zM1/α[a(t)ψ(x)]1/αAt1zzM1/αm1/α[a]1/αAt1z. (2.1)

    Since At1(t)0, At1(t1)=0 and At1()=, there is a t2t1 such that At1(t2)=1. Now, from (2.1), we obtain

    0ddt(z(t)exp[κtt2[a(s)]1/αAt1(s)ds])=ddt(z(t)exp[κlnAt1(t)])=ddt(z(t)[At1(t)]κ).

    Consequently, this proof ends.

    Next, we improve the oscillation results by deriving a new substitution for x by z.

    Lemma 4. Assume that xS. Then (1.1) can be articulated as

    (a(t)ψ(x(t))[z(t)]α)+Gq(t;m)F(z(g(t)))0, (2.2)

    for t>t1 and any integer m0.

    Proof. Let xS. From (A2), there is a t1I such that (xh)(t)>0 and (xg)(t)>0 for tt1. Using Lemma 1, we obtain

    x>mr=0(2rl=0p(h[l]))[z(h[2r])p(h[2r])z(h[2r+1])] (2.3)

    From Lemmas 2–3, we find that

    z>0  and  (z[At1]κ)0.

    Then,

    z(h[2r])z(h[2r+1])

    and

    z(h[2r])[At1(h[2r])At1]κz.

    Using previous relationships in (2.3), we arrive at

    x>zmr=0(2rl=0p(h[l]))[1p(h[2r])1][At1(h[2r])At1]κ,

    which with (1.1) gives

    0=(aψ(x)[z]α)+qF(z(g)mr=0(2rl=0p(h[l](g)))[1p(h[2r](g))1][At1(h[2r](g))At1(g)]κ)(aψ(x)[z]α)+qF(z(g))F(mr=0(2rl=0p(h[l](g)))[1p(h[2r](g))1][At1(h[2r](g))At1(g)]κ)=(aψ(x)[z]α)+Gq(t;m)F(z(g)),

    where assumption (A4) was used.

    Therefore, the proof ends.

    The following theorem transforms the studied equation into the form of a Riccati inequality, or what is known as the Riccati technique.

    Theorem 2.1. Assume that xS, and there is a constant L>0 such that

    F(υ)L[F(υ)]11/α, (2.4)

    for u0. If we define the function ω as

    ω:=υaψ(x)[z]αF(z(g)), (2.5)

    then ω satisfies

    ω(t)υ(t)υ(t)ω(t)υ(t)Gq(t;m)Lg(t)(Ma(g(t))υ(t))1/α[ω(t)]1+1/α, (2.6)

    where υC1(I,R+).

    Proof. Assume that xS. By differentiating ω, we find

    ω=υυω+υ[(a(ψx)[z]α)(Fzg)a(ψx)[z]α(Fzg)2(Fzg)],

    which with (2.2) yields

    ωυυω+υ[Gqa(ψx)[z]α(Fzg)2(Fzg)(zg)g]. (2.7)

    Since (a(t)ψ(x(t))[z(t)]α)<0 and g(t)t, we have

    a(t)ψ(x(t))[z(t)]αa(g(t))ψ(x(g(t)))[z(g(t))]αMa(g(t))[z(g(t))]α,

    and then

    z(g(t))(a(t)ψ(x(t))[z(t)]α)1/αM1/α[a(g(t))]1/α. (2.8)

    Combining (2.7) and (2.8), we arrive at

    ωυυω+υ[Gq[a(ψx)[z]α]1+1/α(Fzg)2(Fzg)gM1/α(ag)1/α]=υυω+υ[Gqω1+1/αυ1+1/α(Fzg)(Fzg)11/αgM1/α(ag)1/α]. (2.9)

    Using (2.4), we find

    ωυυωυGqLgM1/α(ag)1/αυ1/αω1+1/α.

    Therefore, the proof ends.

    Here, we employ the previous results to derive a novel criterion that tests whether all solutions are oscillatory.

    Theorem 3. Assume that φY, υC1(I,R+), and there is a constant L>0 such that (2.4) holds. If

    limsuptT[υGqγMLα(υυ+μ)α+1(ag)υ(g)α;t,l]>0, (2.10)

    then (1.1) oscillates.

    Proof. Assuming the opposite of what is required means that there is a non-oscillatory solution to the studied equation, and this necessarily leads to guaranteeing the existence of a solution that eventually becomes positive for this equation. Let it be x. From Theorem 2.1, if we define the function ω as in (2.5), then ω satisfies (2.6).

    Next, applying the operator T[;t,l] to (2.6), we obtain

    T[ω;t,l]T[(υυωLg(M(ag)υ)1/αω1+1/α);t,l]T[υGq;t,l].

    Using the property (1.5), we obtain

    T[υGq;t,l]T[(υυ+μ)ωLg(M(ag)υ)1/αω1+1/α;t,l]. (2.11)

    By simple calculation, we find that the function H(ω)=c1ωc2ω1+1/α, where c1, c2>0, has the maximum

    H(ω)H(ωmax)=γcα+11cα2, (2.12)

    at ωmax=(αc1/((α+1)c2))α. Using (2.12), (2.11) becomes

    T[υGq;t,l]T[γMLα(υυ+μ)α+1(ag)υ(g)α;t,l]. (2.13)

    Taking the super limit for (2.13), we have

    limsuptT[υGqγMLα(υυ+μ)α+1(ag)υ(g)α;t,l]0.

    This contradicts assumption (2.10).

    Therefore, the proof ends.

    Corollary 1. Assume that there is a constant L>0 such that (2.4) holds. If there are υ, ηC1(I,R+) and ϱ, k>max{1/2,α} such that

    limsupttlη(s)(ts)ϱ(sl)k[υ(s)Gq(s;m)γMLαH(t,s,l)]ds>0, (2.14)

    for lt0, then all solutions of (1.1) are oscillatory, where

    H(t,s,l):=[υ(s)υ(s)+η(s)η(s)+kt(ϱ+k)s+ϱl(ts)(sl)]α+1(a(g(s)))υ(s)(g(s))α. (2.15)

    Proof. By choosing

    φ(u,v,w)=η(v)(uv)ϱ(vw)k,

    we find

    μ(u,v,w)=η(v)η(v)+ku(ϱ+k)v+ϱw(uv)(vw).

    Using Theorem 3, condition (2.10) reduces to (2.14).

    Therefore, the proof ends.

    Corollary 2. Assume that there is a constant L>0 such that (2.4) holds. If g(t)=λt, λ(0,1], αZ+, a(t)1, and there is ϱ, k>α such that

    limsuptT[Gq;t,l][At1(t)At1(l)]ϱ+kα>1λαγMLαα+1i=0(α+1i)(ϱ)ikα+1iβ(k+iα,ϱi+1), (2.16)

    for lt0, then all solutions of (1.1) are oscillatory, where β(,) is the beta function.

    Proof. Assume that

    υ(t)1 and φ(u,v,w)=(At1(u)At1(v))ϱ(At1(v)At1(w))k. (2.17)

    Then, we find that

    T[(μ)α+1(ag)(g)α;t,l]=1λαtl(At1(t)At1(s))ϱ(At1(s)At1(l))k[ϱAt1(t)(ϱ+k)At1(s)+kAt1(l)a(s)(At1(t)At1(s))(At1(s)At1(l))]α+1(a(λs))ds=1λαtl(At1(t)At1(s))ϱα1(At1(s)At1(l))k+α+1[ϱAt1(t)(ϱ+k)At1(s)+kAt1(l)]α+1[a(s)]α+1(a(λs))ds.

    Since a(t)0, we have that a(λs)/a(s)1. Hence,

    T[(μ)α+1(ag)(g)α;t,l]1λαtl(At1(t)At1(s))ϱα1(At1(s)At1(l))k+α+1[ϱAt1(t)(ϱ+k)At1(s)+kAt1(l)]α+1[a(s)]αds1λαtl(At1(t)At1(s))ϱα1(At1(s)At1(l))k+α+1[ϱAt1(t)(ϱ+k)At1(s)+kAt1(l)]α+1[a(s)]1/αds=1λαtl(At1(t)At1(s))ϱα1(At1(s)At1(l))k+α+1[ϱAt1(t)(ϱ+k)At1(s)+kAt1(l)]α+1dAt1(s).

    Let w:=At1(s)At1(l). Thus,

    T[(μ)α+1(ag)(g)α;t,l]1λαϑ0(ϑw)ϱα1wkα1[k(ϑw)ϱw]α+1dw=1λαϑ0[(ϑw)ϱα1wkα1α+1i=0(α+1i)(ϱ)ikα+1i(ϑw)α+1iwi]dw=1λαα+1i=0(α+1i)(ϱ)ikα+1iϑ0(ϑw)ϱiwk+iα1dw=1λαα+1i=0(α+1i)(ϱ)ikα+1iϑϱ+kαϑ0(1wϑ)ϱi(wϑ)k+iα11ϑdw=1λαϑϱ+kαα+1i=0(α+1i)(ϱ)ikα+1i10(1y)ϱi(y)k+iα1dy=1λαϑϱ+kαα+1i=0(α+1i)(ϱ)ikα+1iβ(k+iα,ϱi+1). (2.18)

    Now, as in the proof of Theorem 3, if we assume the contrary, then we arrive at (2.13). Under assumptions in (2.17), inequality (2.13) reduces to

    T[Gq;t,l]γMLαT[(μ)α+1(ag)(g)α;t,l],

    which with (2.18) gives

    T[Gq;t,l]1λαγMLαϑϱ+kαα+1i=0(α+1i)(ϱ)ikα+1iβ(k+iα,ϱi+1).

    Thus, we obtain

    limsupt1ϑϱ+kαT[Gq;t,l]1λαγMLαα+1i=0(α+1i)(ϱ)ikα+1iβ(k+iα,ϱi+1).

    This contradicts assumption (2.16).

    Therefore, the proof ends.

    In the following corollaries, we present an oscillation criterion for a special case of the studied equation, which is

    ddt(a(t)(ψx)(t)[ddtz(t)]α)+q(t)(xαg)=0.

    It is easy to notice that the function F(u)=uα satisfies (A4) and (2.4) with L=α.

    Corollary 3. If there are υ, ηC1(I,R+) and ϱ, k>max{1/2,α} such that

    limsupttlη(s)(ts)ϱ(sl)k[υ(s)Gq(s;m)γMααH(t,s,l)]ds>0, (2.19)

    for lt0, then all solutions of (1.1) are oscillatory, where H(t,s,l) is defined as in (2.15).

    Corollary 4. If g(t)=λt, λ(0,1], α=1, a(t)1, and there is ϱ>1/2 such that

    limsupt1[At1(t)]2ϱ+1T[Gq;t,l]>Mλϱ4ϱ21, (2.20)

    for lt0, then all solutions of (1.1) are oscillatory.

    Example 1. Consider the delay equation

    ddt(11+sin2(x(t))ddt[x(t)+12x(h0t)])+q0t2x(g0t)=0, (2.21)

    where h0, g0(0,1], and q0>0. We note that α=1, a(t)=1, p(t)=1/2, h(t)=h0t, g(t)=g0t, q(t)=q0/t2, F(u)=u, and ψ(u)=1/(1+sin2u). It is easy to verify that

    m=12ψ(t)1=M,

    κ=2, At0()=, h[r](t)=hr0t for r=0,1,...,

    Gq(t;m)=q0t2(mr=0h4r0(2rl=0(12)))=q02t2mr=0(h202)2r.

    Now, we have

    1[At1(t)]2ϱ+1T[Gq;t,l]=1[tt0]2ϱ+1tlGq(s;m)φ(t,s,l)ds=q02[tt0]2ϱ+1mr=0(h202)2rtl1s2(ts)2ϱ(sl)2ds.

    Hence, condition (2.20) becomes

    q02(1+2ϱ)mr=0(h202)2r>1g0ϱ4ϱ21.

    Then, by using Corollary 4, all solutions of (2.21) are oscillatory if

    q0mr=0(h202)2r>1g0. (2.22)

    Remark 1. Using Theorem 1.1 with H(t,s)=(ts)2 and ρ(t)=t2, we obtain that all solutions of (2.21) are oscillatory if

    q0>2g0. (2.23)

    Applying to the special case when g0=0.5 and h0=0.9, we notice that criteria (2.22) and (2.23) lead to q0>1.672 and q0>4, respectively. Therefore, our results improve results in [18]. For example, we find that our results guarantee that all solutions of equation

    ddt(11+sin2(x(t))ddt[x(t)+12x(t2)])+2t2x(t2)=0

    oscillate while the criteria of [18] do not apply (2=q02g0=4).

    Figure 1 shows one of the numerical solutions to Eq (2.21).

    Figure 1.  The numerical solution of (2.21) when h0=g0=1 and q0=2.

    Example 2. Consider the delay equation

    ddt(1+ex2(t)etddt[x(t)+p0x(th0)])+q0etx(tg0)=0, (2.24)

    where h0, g0, and q0 are positive. We note that α=1, a(t)=et, p(t)=p0, h(t)=th0, g(t)=tg0, q(t)=q0et, F(u)=u, and ψ(u)=1+eu2. It is easy to verify that m=1, M=2, κ=2,

    Gq(t;m)=q0et[1p0]mr=0p2r0e4rh0.

    By choosing η(t)=υ(t)=et and ϱ=k=2, we obtain

    H(t,s,l)=eg0[2+2t4s+2l(ts)(sl)]2.

    Thus,

    limsupttles(ts)2(sl)2×[q0[1p0]mr=0p2r0e4rh012eg0[2+2t4s+2l(ts)(sl)]2]ds=(q0[1p0]mr=0p2r0e4rh0eg0)(+),

    which is fulfilled if

    q0[1p0]mr=0p2r0e4rh0>eg0. (2.25)

    Then, by using Corollary 3, all solutions of (2.24) are oscillatory if (2.25) holds.

    Remark 2. Recent results in papers [26,27,28,29,30] provided many improved criteria that test the oscillatory characteristics of second-order neutral DEs. However, these results fail to apply to Eqs (2.21) and (2.24), because these results only apply in the case of ψ(u)=u. Figure 2 shows one of the numerical solutions to Eq (2.24).

    Figure 2.  The numerical solution of (2.24) when h0=g0=1, p0=1/2, and q0=e5.

    The investigation into the oscillatory behavior of FDEs is affected by the accuracy of the relationships and inequalities used. In this article, we studied the oscillations of solutions of the class of FDEs of the neutral type (1.1). As an extension of the results in [21], we have derived a novel relation between x and z. We used the Riccati approach to couple the studied equation with an inequality of the Riccati type. Then, we presented Kamenev-type criteria that ensure the oscillation of (1.1).

    Our results—as shown in Remark 2—have the advantage of being applied to a more general class of second-order FDEs of the neutral type compared to the results in [26,27,28]. Our results also presented more sharp conditions in the oscillation test than the results that dealt with the same equation (see Remark 1).

    It would be of interest to formally extend our findings to the noncanonical case (At()<). Also, an interesting point, as a future work, is to obtain an improved relation between x and z without the need for the constraint ψ(u)m>0, which excludes a large class of bounded functions such as sin2u, eu2 and 11+u2.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    The authors declare there is no conflicts of interest.



    Conflict of interest



    The authors declare no conflict of interest.

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