Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Special Issues

Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol

  • The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a 95 error band surrounding the estimated output signal.

    Citation: Lernik Asserian, Susan E. Luczak, I. G. Rosen. Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 20345-20377. doi: 10.3934/mbe.2023900

    Related Papers:

    [1] Deyue Zhang, Yukun Guo . Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory. Electronic Research Archive, 2021, 29(2): 2149-2165. doi: 10.3934/era.2020110
    [2] Xinlin Cao, Huaian Diao, Jinhong Li . Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29(1): 1753-1782. doi: 10.3934/era.2020090
    [3] Yan Chang, Yukun Guo . Simultaneous recovery of an obstacle and its excitation sources from near-field scattering data. Electronic Research Archive, 2022, 30(4): 1296-1321. doi: 10.3934/era.2022068
    [4] Yujie Wang, Enxi Zheng, Wenyan Wang . A hybrid method for the interior inverse scattering problem. Electronic Research Archive, 2023, 31(6): 3322-3342. doi: 10.3934/era.2023168
    [5] John Daugherty, Nate Kaduk, Elena Morgan, Dinh-Liem Nguyen, Peyton Snidanko, Trung Truong . On fast reconstruction of periodic structures with partial scattering data. Electronic Research Archive, 2024, 32(11): 6481-6502. doi: 10.3934/era.2024303
    [6] Yao Sun, Lijuan He, Bo Chen . Application of neural networks to inverse elastic scattering problems with near-field measurements. Electronic Research Archive, 2023, 31(11): 7000-7020. doi: 10.3934/era.2023355
    [7] Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu . A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28(2): 1123-1142. doi: 10.3934/era.2020062
    [8] Hyungyeong Jung, Sunghwan Moon . Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings. Electronic Research Archive, 2022, 30(12): 4436-4446. doi: 10.3934/era.2022225
    [9] Messoud Efendiev, Vitali Vougalter . Linear and nonlinear non-Fredholm operators and their applications. Electronic Research Archive, 2022, 30(2): 515-534. doi: 10.3934/era.2022027
    [10] Shiqi Ma . On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, 2021, 29(3): 2391-2415. doi: 10.3934/era.2020121
  • The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a 95 error band surrounding the estimated output signal.



    It is prudential to say that mathematical modeling with delay differential equations have drawn clear significance because of their potential applications in diverse fields, which includes biological sciences, physical sciences, gas and fluid mechanics, signal processing, robotics and traffic system, engineering, population dynamics, medicine and the like (see for example [9,16,17]). It is now realized that the oscillation and asymptotic solutions of various classes of differential equation are an important field of investigation and its theory is a lot richer than the qualitative theory of differential equations (see for example [8,10,22]). The problem of oscillatory and nonoscillatory of solutions of various classes of second/third order differential equations with delayed and mixed arguments has been widely investigated in the literature (see for example [2,4,5,6,7,11,12,18,23,24,25,26,27,28,29,30,31,32,33,34]). Various types of techniques appeared for investigations of such equations.

    The purpose of this work, we are concerned with third-order neutral differential equations with discrete and distributed delay

    (a2(t)[(a1(t)z(t))]λ)+q1(t)yλ(tσ1)+q2(t)yλ(t+σ1)=0,

    and

    (a2(t)[(a1(t)z(t))]λ)+dc˜q1(t,ξ)yλ(tξ)dξ+dc˜q2(t,ξ)yλ(t+ξ)dξ=0,

    where z(t)=y(t)+p1(t)y(tτ1)+p2(t)y(t+τ2), c<d and λ1. Now onwards, we assume that, ai(t),pi(t)C([t0,+)), ai(t)>0, pi(t)>0 for i=1,2 and 0pi(t)μi, μ1+μ2<1 where μi are constants, qiC([t0,+),R+), ~qi(t,ξ)C([t0,+)×[c,d], R+) for i=1,2, and not identically zero on [t,+)×[c,d], tt, constants τi0, for i=1,2, and the integral of (E2) is take in the sense of Riemann–Stieltjes.

    Let us recall that, a solution y(t)C([Ty,),R) of (E1) (or (E2)) is a non-trivial or y(t)0 with Tyt0, if the functions zC1([Ty,),R), a1zC2([Ty,),R) and a2[(a1z)]λC1([Ty,),R) for certain Tyt0 which satisfies (E1) (or (E2)). Our attention is restricted to those solutions of (E1) (or (E2)) which exist on half-line [Ty,) and the condition sup{|y(t)|:t>T}>0 satisfies for any Tty. A solution of (E1) (or (E2)), which is nontrivial (proper) for all large t, is called oscillatory if it has no last zero, otherwise, termed nonoscillatory.

    We define the operators,

    L[0]z=z,L[1]z=z,L[2]z=(a1L[1]z),L[3]z=a2[L[2]z]λ,L[4]z=(L[3]z).

    We shall consider the two cases,

    π1[t0,t]=tt0a1/λ2(s)ds,π2[t0,t]=tt0a11(s)ds.
    π1[t0,t]=,π2[t0,t]= as t, (1.1)

    and

    π1[t0,t]<,π2[t0,t]= as t. (1.2)

    Recently, Candan [24] investigated the oscillatory behavior of solutions of (E1) and (E2) by using the Riccati substitution techniques, he presented some new oscillation criteria for (E1) and (E2) by the assumption of condition (1.1). We notice that in [24], no criteria were found for (E1) (or (E2)) to be oscillatory for the assumption of condition (1.2). It would be interesting to improve and extend them in the condition (1.2).

    However, the corresponding result for (E1) (or (E2)) under (1.2) is still missing. In this work, we fill up this gap, also we strengthen and extend the main results of Candan [24] under the condition (1.1) and (1.2) respectively. We present several oscillatory criteria for (E1) and (E2), by applying three Riccati substitution techniques, integral averaging techniques and comparison principles. We present two examples in order to illustrate the main results at the end.

    In this section, we present some basic Lemmas for helping to prove the main results. We use throughout this paper the following notations for convenience and for shortening the equations:

    L[0]σz(t)=z(t+σ),L[1]σz(t)=z(t+σ),L[2]σz(t)=(a1(t+σ)z(t+σ)),L[3]σz(t)=a2(t+σ)[L[2]σz(t)]λ,L[4]σz(t)=(L[3]σz(t)),A(t)=tt0π1[t0,s]a1(s)ds.

    Lemma 2.1. Let λ1, assume u0. Then

    (u1+u2+u3)λ3λ1(uλ1+uλ2+uλ3). (2.1)

    Lemma 2.2. Let λ1, assume u0. Then

    (u1+u2+u3)λ(uλ1+uλ2+uλ3). (2.2)

    Lemma 2.3. If λ>0 and X,Y>0, then

    YvXvλ+1λλλ(1+λ)1+λY1+λXλ. (2.3)

    Lemma 2.4. Assume that (1.1) holds. Furthermore, assume that y is an eventually positive solution of (E1) (or (E2)). Then z for t1[t0,) satisfies, eventually of the following cases:

    (C1):L[0]z(t)>0,L[1]z(t)>0,andL[2]z(t)>0;(C2):L[0]z(t)>0,L[1]z(t)<0,andL[2]z(t)>0;

    and if (1.2) holds, then also

    (C3):L[0]z(t)>0,L[1]z(t)>0,andL[2]z(t)<0.

    Lemma 2.5. Assume that z satisfies (C1) for tt0. Then

    z(t)(L[3]z(t))1/λa1(t)π1[t0,t] (2.4)

    and

    z(t)(L[3]z(t))1/λA(t). (2.5)

    Proof. Since L[4]z(t)0, L[3]z(t) is nondecreasing. Then we have

    a1(t)z(t)a1(t)z(t)a1(t0)z(t0)=tt0a1/λ2(s)L[2]z(s)a1/λ2(s)dsa1/λ2(t)L[2]z(t)π1[t0,t].

    Again integrate, we get

    z(t)(L[3]z(t))1/λtt0π1[t0,s]a1(s)ds=(L[3]z(t))1/λA(t).

    Lemma 2.6 (See [24]). Assume that z is a solution of (E1) which satisfies (C2) in Lemma 2.4. Furthermore,

    t4a11(v)va1/λ2(u)(u(q1(s)+q2(s))ds)1/λdudv=. (2.6)

    Then, there is limtz(t)=0.

    Lemma 2.7 (See [24]). Assume that z is a solution of (E2) which satisfies (C2) in Lemma 2.4. Furthermore,

    t4a11(v)va1/λ2(u)(uba(~q1(s,ξ)+~q2(s,ξ))dξds)1/λdudv=. (2.7)

    Then, there is limtz(t)=0.

    In this section, we will establish several oscillation criteria for (E1). The following notations for convenience and for shortening the equations:

    P1(t)=min{q1(t),q1(tτ1),q1(t+τ2)},P2(t)=min{q2(t),q2(tτ1),q2(t+τ2)},P(t)=P1(t)+P2(t),B(t)=tt0sdua1/λ2(u)a1(s)ds.

    Let S0={(t,s):as<t<+}, S={(t,s):ast<+} the continuous function H(t,s), H:SR belongs to the class function

    (ⅰ) H(t,t)=0 for tt0 and H(t,s)>0 for (t,s)S0,

    (ⅱ) H(t,s)s0, (t,s)S0 and some locally integrable function h(t,s) such that

    sH(t,s)H(t,s)m(s)m(s)=h(t,s)(H(t,s))λλ+1m(s)for all (t,s)S0.

    Theorem 3.1. Let (1.1) hold and σ1τ1. If there exists an m(t)C1([t0,),R+) such that (2.6) and

    lim supt1H(t,t3)tt3[H(t,s)m(s)P(s)3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(sσ1)m(s)π1[t0,sσ1])λ]ds=, (3.1)

    then every solution y(t) of (E1) is either oscillatory or tends to 0.

    Proof. Suppose that (E1) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tσ1)>0 and y(t+σ1)>0 for tt1t0. Since y(t)>0 for all tt1, in view of (E1), we have

    L[4]z(t)=q1(t)yλ(tσ1)q2(t)yλ(t+σ1)0. (3.2)

    Assumption of (1.1), by Lemma 2.4 there exists two cases (C1) and (C2). If (C2) holds, then by Lemma 2.6, limtz(t)=0. If (C1) holds.

    L[4]z(t)+q1(t)yλ(tσ1)+q2(t)yλ(t+σ1)+μλ1L[4]τ1z(t)+μλ1q1(tτ1)yλ(tτ1σ1)+μλ1q2(tτ1)yλ(tτ1+σ1)+μλ2L[4]τ2z(t)+μλ2q1(t+τ2)yλ(t+τ2σ1)+μλ2q2(t+τ2)yλ(t+τ2+σ1)=0. (3.3)

    Furthermore, from Lemma 2.1, we get

    q1(t)yλ(tσ1)+μλ1q1(tτ1)yλ(tτ1σ1)+μλ1q1(t+τ2)yλ(t+τ2σ1)P1(t)3λ1zλ(tσ1). (3.4)

    Similarly, we get

    q2(t)yλ(t+σ1)+μλ2q2(tτ1)yλ(tτ1+σ1)+μλ2q2(t+τ2)yλ(t+τ2+σ1)P2(t)3λ1zλ(t+σ1). (3.5)

    Substituting (3.4), (3.5) into (3.3), we have

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P1(t)3λ1zλ(tσ1)+P2(t)3λ1zλ(t+σ1)0. (3.6)

    Using the fact of L[1]z(t)>0, we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P(t)3λ1zλ(tσ1)0. (3.7)

    Define

    w1(t)=m(t)L[3]z(t)zλ(tσ1). (3.8)

    We obtain w1(t)>0, then

    w1(t)=m(t)L[3]z(t)zλ(tσ1)+m(t)L[4]z(t)zλ(tσ1)λm(t)L[3]z(t)z(tσ1)zλ+1(tσ1). (3.9)

    By Lemma (2.5), one gets z(tσ1)a1/λ2(t)a1(tσ1)π1[t0,tσ1]L[2]z(t). Therefore

    w1(t)m(t)L[3]z(t)zλ(tσ1)+m(t)L[4]z(t)zλ(tσ1)λm(t)aλ+1λ2(t)π1[t0,tσ1]L[2]z(t)z(tσ1)zλ+1(tσ1)a1(tσ1). (3.10)

    Using (3.8) in (3.10), we obtain

    w1(t)(m(t))+m(t)w1(t)+m(t)L[4]z(t)zλ(tσ1)λ(w1(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1). (3.11)

    Next, define

    w2(t)=m(t)L[3]τ1z(t)zλ(tσ1). (3.12)

    We obtain w2(t)>0, then

    w2(t)=m(t)L[3]τ1z(t)zλ(tσ1)+m(t)L[4]τ1z(t)zλ(tσ1)λm(t)L[3]τ1z(t)z(tσ1)zλ+1(tσ1). (3.13)

    By Lemma (2.5), one gets z(tσ1)a1/λ2(tτ1)a1(tσ1)π1[t0,tσ1]L[2]τ1z(t) and using (3.12) in (3.13), we have

    w2(t)(m(t))+m(t)w2(t)+m(t)L[4]τ1z(t)zλ(tσ1)λ(w2(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1). (3.14)

    Finally, define

    w3(t)=m(t)L[3]τ2z(t)zλ(tσ1). (3.15)

    We obtain w3(t)>0, then

    w3(t)=m(t)L[3]τ2z(t)zλ(tσ1)+m(t)L[4]τ2z(t)zλ(tσ1)λm(t)L[3]τ2z(t)z(tσ1)zλ+1(tσ1). (3.16)

    By Lemma 2.5, one gets z(tσ1)a1/λ2(t+τ2)a1(tσ1)π1[t0,tσ1]L[2]τ2z(t) and using (3.15) in (3.16), we get

    w3(t)(m(t))+m(t)w3(t)+m(t)L[4]τ2z(t)zλ(tσ1)λ(w3(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1). (3.17)

    From (3.8), (3.10) and (3.15), we have

    w1(t)+μλ1w2(t)+μλ2w3(t)m(t)[L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)zλ(tσ1)]+[(m(t))+m(t)w1(t)λ(w1(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)]+μλ1[(m(t))+m(t)w2(t)λ(w2(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)]+μλ2[(m(t))+m(t)w3(t)λ(w3(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)]. (3.18)

    Using (3.7) in (3.18), we have

    w1(t)+μλ1w2(t)+μλ2w3(t)m(t)P(t)3λ1+[(m(t))+m(t)w1(t)λ(w1(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)]+μλ1[(m(t))+m(t)w2(t)λ(w2(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)]+μλ2[(m(t))+m(t)w3(t)λ(w3(t))λ+1λπ1[t0,tσ1](m(t))1/λa1(tσ1)], (3.19)

    that is,

    m(t)P(t)3λ1w1(t)μλ1w2(t)μλ2w3(t)+(m(t))+m(t)w1(t)λπ1[t0,tσ1](m(t))1/λa1(tσ1)(w1(t))λ+1λ+μλ1[(m(t))+m(t)w2(t)λπ1[t0,tσ1](m(t))1/λa1(tσ1)(w2(t))λ+1λ]+μλ2[(m(t))+m(t)w3(t)λπ1[t0,tσ1](m(t))1/λa1(tσ1)(w3(t))λ+1λ]. (3.20)

    Multiply H(t,s) and integrate (3.20) from t3 to t, one can get that

    tt3H(t,s)m(s)P(s)3λ1dstt3H(t,s)w1(s)dsμλ1tt3H(t,s)w2(s)dsμλ2tt3H(t,s)w3(s)ds+tt3H(t,s)(m(s))+m(s)w1(s)dstt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w1(s))λ+1λds+μλ1tt3H(t,s)(m(s))+m(s)w2(s)dsμλ1tt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w2(s))λ+1λds+μλ2tt3H(t,s)(m(s))+m(s)w3(s)dsμλ2tt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w3(s))λ+1λds. (3.21)

    Thus, we obtain

    tt3H(t,s)m(s)P(s)3λ1dsH(t,t3)w1(t3)+μλ1H(t,t3)w2(t3)+μλ2H(t,t3)w3(t3)tt3[sH(t,s)H(t,s)m(s)m(s)]w1(s)dstt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w1(s))λ+1λdsμλ1tt3[sH(t,s)H(t,s)m(s)m(s)]w2(s)dsμλ1tt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w2(s))λ+1λdsμλ2tt3[sH(t,s)H(t,s)m(s)m(s)]w3(s)dsμλ2tt3H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w3(s))λ+1λds. (3.22)

    Then

    tt3H(t,s)m(s)P(s)3λ1dsH(t,t3)w1(t3)+μλ1H(t,t3)w2(t3)+μλ2H(t,t3)w3(t3)+tt3[|h(t,s)|(H(t,s))λλ+1m(s)w1(s)H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w1(s))λ+1λ]ds+μλ1tt3[|h(t,s)|(H(t,s))λλ+1m(s)w2(s)H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w2(s))λ+1λ]ds+μλ2tt3[|h(t,s)|(H(t,s))λλ+1m(s)w3(s)H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1)(w3(s))λ+1λ]ds. (3.23)

    Setting Y=|h(t,s)|(H(t,s))λλ+1m(s), X=H(t,s)λπ1[t0,sσ1](m(s))1/λa1(sσ1) and u=wi(t) for i=1,2,3. By using the Lemma 2.3, we conclude that

    1H(t,t3)tt3[H(t,s)m(s)P(s)3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(sσ1)m(s)π1[t0,sσ1])λ]dsw1(t3)+μλ1w2(t3)+μλ2w3(t3) (3.24)

    which contradicts condition (3.20).

    Theorem 3.2. Let (1.1) hold and τ1σ1. If there exists an m(t)C1([t0,),R+) such that (2.6) and

    lim supt1H(t,t3)tt3[H(t,s)m(s)P(s)3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(sτ1)m(s)π1[t0,sτ1])λ]ds=, (3.25)

    then every solution y(t) of (E1) is either oscillatory or tends to 0.

    Proof. Suppose that (E1) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tσ1)>0 and y(t+σ1)>0 for tt1t0. Assumption of (1.1), by Lemma 2.4 there exists two cases (C1) and (C2). If (C2) holds, then by Lemma 2.6, limtz(t)=0. We only consider (C1), by using the fact that z(t)>0 and τ1σ1, we obtain that Using the fact of L[1]z(t)>0, we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P(t)3λ1zλ(tτ1)0. (3.26)

    Next, we categorize the functions as w1(t)=m(t)L[3]z(t)zλ(tτ1), w2(t)=m(t)L[3]τ1z(t)zλ(tτ1) and w3(t)=m(t)L[3]τ2z(t)zλ(tτ1) respectively. The rest of the proof is similar to that of Theorem 3.1, therefore, it is omitted.

    Theorem 3.3. Let (1.2) hold and σ1τ1. If there exists an m(t)C1([t0,),R+) such that (2.6),

    t3[m(s)P(s)3λ1(1+μλ1+μλ2)((m(s))+(λ+1))λ+1(a1(sσ1)m(s)π1[t0,sσ1])λ]ds=, (3.27)

    and

    t3[πλ(s+τ2)P(s)3λ1(s+σ1t2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+σ1)a1+1λ2(s)πλ(s+τ2)]ds=, (3.28)

    where (m(t))+=max{0,m(t)}, π(t)=t+σ1a1/λ2(s)ds, then every solution y(t) of (E1) is either oscillatory or tends to 0.

    Proof. Suppose that (E1) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tσ1)>0 and y(t+σ1)>0 for tt1t0. Since y(t)>0 for all tt1. Assumption of (1.2), by Lemma 2.4 there exists three cases (C1), (C2) and (C3). If case (C1) and (C2) holds, using the similar proof of ([24], Theorem 2.1) by using Lemma 2.1, we get the conclusion of Theorem 3.3.

    If case (C3) holds, z(tσ1)<0 for tt1. The facts that z(t)<0, c+d0 and (3.6), we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P(t)3λ1zλ(t+σ1)0. (3.29)

    Define

    w(t)=L[3]z(t)(a1(t+σ1)z(t+σ1))λ. (3.30)

    We obtain w(t)<0 for tt2. Noting that L[3]z(t) is decreasing, we obtain

    a2(s)[L[2]z(s)]λa2(t)[L[2]z(t)]λ (3.31)

    for stt2. Dividing (3.31) by a2(s) and integrating from t+σ1 to l(lt), we get

    a1(l)z(l)a1(t+σ1)z(t+σ1)+a1/λ2(t)[L[2]z(t)]lt+σ1a1/λ2(s)ds.

    letting l, we get

    1a1/λ2(t)[L[2]z(t)]a1(t+σ1)z(t+σ1)π(t), (3.32)

    for tt2. From (3.30), we have

    1w(t)πλ(t)0. (3.33)

    By (3.2) we have a1(t+σ1)z(t+σ1)a1(t)z(t). Differentiating (3.30) gives,

    w(t)(L[3]z(t))(a1(t+σ1)z(t+σ1))λλa2(t)[L[2]z(t)a1(t+σ1)z(t+σ1)]λ+1. (3.34)

    Using (3.30) in (3.34), we have

    w(t)L[4]z(t)(a1(t+σ1)z(t+σ1))λλw1+1λ(t)a1/λ2(t). (3.35)

    Again, we define

    w(t)=L[3]τ1z(t)(a1(t+σ1)z(t+σ1))λ. (3.36)

    We obtain w(t)<0 and w(t)w(t) for tt2. By (3.33), we obtain

    1w(t)πλ(t)0. (3.37)

    By (3.2) we have a1(t+σ1)z(t+σ1)a1(tτ1)z(tτ1). Differentiating (3.36) gives,

    w(t)(L[3]τ1z(t))(a1(t+σ1)z(t+σ1))λλa2(t)[L[2]τ1z(t)a1(t+σ1)z(t+σ1)]λ+1. (3.38)

    Using (3.36) in (3.38), we have

    w(t)L[4]τ1z(t)(a1(t+σ1)z(t+σ1))λλw1+1λ(t)a1/λ2(t). (3.39)

    Finally, we define a function

    w(t)=L[3]τ2z(t)(a1(t+τ2+σ1)z(t+τ2+σ1))λ. (3.40)

    We obtain w(t)<0 and w(t)=w(t+τ2) for tt2. By (3.33), we obtain

    1w(t)πλ(t+τ2)0. (3.41)

    By (3.2) we have a1(t+τ2+σ1)z(t+τ2+σ1)a1(t+τ2)z(t+τ2). Differentiating (3.40) gives,

    w(t)(L[3]τ2z(t))(a1(t+σ1)z(t+σ1))λλa2(t)[L[2]τ2z(t)a1(t+τ2+σ1)z(t+τ2+σ1)]λ+1. (3.42)

    Using (3.40) in (3.42), we have

    w(t)L[4]τ2z(t)(a1(t+σ1)z(t+σ1))λλw1+1λ(t)a1/λ2(t). (3.43)

    From (3.35), (3.39), (3.43) and (3.29) which implies

    w(t)+μλ1w(t)+μλ2w(t)P(t)3λ1zλ(t+σ1)(a1(t+σ1)z(t+σ1))λλw1+1λ(t)a1/λ2(t)μλ1λw1+1λ(t)a1/λ2(t)μλ2λw1+1λ(t)a1/λ2(t) (3.44)

    In case (C3), (a1(t)z(t))<0 we seen that

    z(t)a1(t)z(t)tt21a1(s)ds. (3.45)

    Using (3.45) in (3.44), we get

    w(t)+μλ1w(t)+μλ2w(t)P(t)3λ1(t+σ1t2dsa1(s))λλw1+1λ(t)a1/λ2(t)μλ1λw1+1λ(t)a1/λ2(t)μλ2λw1+1λ(t)a1/λ2(t). (3.46)

    Multiplying πλ(t+τ2) and integrating from t3(t3>t2) to t, yields

    πλ(t+τ2)w(t)πλ(t3+τ2)w(t3)+πλ(t+τ2)μλ1w(t)πλ(t3+τ2)μλ1w(t3)+πλ(t+τ2)μλ2w(t)πλ(t3+τ2)μλ2w(t3)λtt3[πλ1(s+τ2)(w(s))a1/λ2(s+τ2)πλ(s+τ2)(w(s))1+1λa1/λ2(s)]dsλμλ1tt3[πλ1(s+τ2)(w(s))a1/λ2(s+τ2)πλ(s+τ2)(w(s))1+1λa1/λ2(s)]dsλμλ2tt3[πλ1(s+τ2)(w(s))a1/λ2(s+τ2)πλ(s+τ2)(w(s))1+1λa1/λ2(s)]ds+tt3πλ(s+τ2)P(s)3λ1(s+σ1t2dua1(u))λds0. (3.47)

    Applying Lemma 2.3, we conclude that

    tt3[πλ(s+τ2)P(s)3λ1(s+σ1t2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+σ1)a1+1λ2(s)πλ(s+τ2)]ds[πλ(t+τ2)w(t)+μλ1πλ(t+τ2)w(t)+μλ2πλ(t+τ2)w(t)] (3.48)

    Using the fact of πλ(t+τ2)πλ(t) in (3.33), (3.37), (3.41) and (3.48) imply that

    tt3[πλ(s+τ2)P(s)3λ1(s+σ1t2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+σ1)a1+1λ2(s)πλ(s+τ2)]ds1+μλ1+μλ2. (3.49)

    a contradiction to (3.28).

    Finally, we establish new comparison theorems for (E1) under the case when (1.2) holds.

    Theorem 3.4. Let (1.2), (2.6) hold and σ1>τ1, σ1>τ2. If the first-order differential inequality

    ψ(t)+P1(t)3λ1Aλ(tσ1)1+μλ1+μλ2ψ(tσ1+τ1)0 (3.50)

    for tt0, has no positive nonincreasing solution and the first-order differential inequality

    ψ(t)P2(t)3λ1Bλ(t+σ1)1+μλ1+μλ2ψ(tτ2+σ1)0 (3.51)

    for tt0, has no positive nondecreasing solution. Then Eq. (E1) oscillatory.

    Proof. Suppose that (E1) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tσ1)>0 and y(t+σ1)>0 for tt1t0. Since y(t)>0 for all tt1. Assumption of (1.2), by Lemma 2.4, there exists three cases (C1), (C2) and (C3). If case (C2) hold, the proof is follows from Lemma 2.6.

    If case (C1) holds, we have L[2]z(t)>0, from (3.6), we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P1(t)3λ1zλ(tσ1)0. (3.52)

    By Lemma 2.5, one gets z(tσ1)(L[3]σ1z(t))1/λA(tσ1) and using in (3.52), we have

    (L[3]z(t)+μλ1L[3]τ1z(t)+μλ2L[3]τ2z(t))+P1(t)3λ1L[3]σ1z(t)Aλ(tσ1)0. (3.53)

    Now, set

    ψ(t)=L[3]z(t)+μλ1L[3]τ1z(t)+μλ2L[3]τ2z(t).

    Then ψ(t)>0 and the fact that L[3]z(t) is nonincreasing, we have

    ψ(t)L[3]τ1z(t)(1+μλ1+μλ2). (3.54)

    Using (3.54) and (3.53), we see that ψ(t) is a nonincreasing positive solution of the first order differential inequality

    ψ(t)+P1(t)3λ1Aλ(tσ1)1+μλ1+μλ2ψ(tσ1+τ1)0, (3.55)

    which is contradiction to (3.50).

    If case (C3) holds, we have L[2]z(t)<0, from (3.6), we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+P2(t)3λ1zλ(t+σ1)0. (3.56)

    Since L[3]z(t) is nondecreasing. Then we get

    L[3]z(s)L[3]z(t) for all stt1t0.

    Integrating above inequality from t to l, we get

    a1(l)z(l)a1(t)z(t)+lta1/λ2(t)L[2]z(t)a1/λ2(s)dsa1(t)z(t)+(L[3]z(s))1/λltdsa1/λ2(s).

    Letting l, we get

    a1(t)z(t)(L[3]z(s))1/λtdsa1/λ2(s).

    Again integrating, we get

    z(t)(L[3]z(t))1/λtt0tdua1/λ2(u)a1(s)ds=(L[3]z(t))1/λB(t). (3.57)

    From 3.57, one gets z(t+σ1)(L[3]σ1z(t))1/λB(t+σ1) and using in (3.56), we have

    (L[3]z(t)+μλ1L[3]τ1z(t)+μλ2L[3]τ2z(t))P2(t)3λ1L[3]σ1z(t)Bλ(t+σ1)0. (3.58)

    Now, set

    ψ(t)=L[3]z(t)+μλ1L[3]τ1z(t)+μλ2L[3]τ2z(t).

    Then ψ(t)>0, ψ(t)0 and the fact that L[3]z(t) is nondecreasing, we have

    ψ(t)L[3]τ2z(t)(1+μλ1+μλ2). (3.59)

    Using (3.59) and (3.58), we see that ψ(t) is a nonincreasing positive solution of the first order differential inequality

    ψ(t)P2(t)3λ1Bλ(t+σ1)1+μλ1+μλ2ψ(tτ2+σ1)0 (3.60)

    which is contradiction to (3.51).

    Corollary 3.5. Let (1.2), (2.6) hold and σ1>τ1, σ1>τ2. If

    lim inftttσ1+τ1P1(s)Aλ(sσ1)ds>3λ1e(1+μλ1+μλ2) (3.61)

    and

    lim inftttτ2+σ1P2(s)Bλ(s+σ1)ds>3λ1e(1+μλ1+μλ2) (3.62)

    hold, then Eq. (E1) oscillatory.

    Proof. The proof follows from Theorem 3.4 and ([10], Theorem 2.1.1), and the details are omitted.

    Example 3.6. Consider the third order differential equation

    ((((y(t)+e23y(t2)+e3y(t+1))))3/2)+3e34(53)3/2y3/2(t2)+3e34(53)3/2y3/2(t+2)=0. (3.63)

    Compared with (E1), we can see that a1(t)=a2(t)=1, p1(t)=e23, p2(t)=e13, q1(t)=3e34(53)3/2, q2(t)=3e34(53)3/2, λ=3/2, τ1=2, τ2=1 and σ1=2. By taking m(t)=1, H(t,s)=(ts)2, we obtain h(t,s)=(3st)(ts)1/5. It is easy to verify that all conditions of Theorem 3.1 are satisfied. Therefore, all the solutions of (3.63) is either oscillates or tends to 0 and y(t)=et is a such solution of (3.63).

    Example 3.7. Consider the third order differential equation

    [t2(y(t)+k1y(tτ1)+k2y(t+τ2))]+k3ty(tσ1)+k4y(t+σ1)=0,t1. (3.64)

    Compared with (E1), we can see that a1(t)=1, a2(t)=t2, p1(t)=k1, p2(t)=k2, q1(t)=k3t, q2(t)=k4, λ=1 and k1, k2, k3, k4 are nonnegative constants. It is easy to verify that all conditions of Corollary 3.5 are satisfied and hence all solutions of equation (3.64) are oscillatory.

    In this section, we will establish several oscillation criteria for (E2). For convenience, we define,

    Q1(t,ξ)=min{˜q1(t,ξ),˜q1(tτ1,ξ),˜q1(t+τ2,ξ)},Q2(t,ξ)=min{˜q2(t,ξ),˜q2(tτ1,ξ),˜q2(t+τ2,ξ)},Q(t,ξ)=Q1(t,ξ)+Q2(t,ξ).

    Theorem 4.1. Let (1.1) holds and c+d0, bτ1. If there exists an m(t)C1([t0,),R+) such that (2.7) and

    lim supt1H(t,t3)tt3[H(t,s)m(s)dcQ(s,ξ)dξ3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(sd)m(s)π1[t0,sd])λ]ds=, (4.1)

    then every solution y(t) of (E2) is either oscillatory or tends to 0.

    Proof. Suppose that (E2) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tξ)>0 and y(t+ξ)>0 for tt1t0 and ξ[c,d]. Since y(t)>0 for all tt1, in view of (E2), we have

    L[4]z(t)=dc˜q1(t,ξ)yλ(tξ)dξdc˜q2(t,ξ)yλ(t+ξ)dξ0. (4.2)

    Assumption of (1.1), by Lemma 2.4 there exists two cases (C1) and (C2). If (C2) holds, then by Lemma 2.7, limtz(t)=0. If (C1) holds.

    L[4]z(t)+dc˜q1(t,ξ)yλ(tξ)dξ+dc˜q2(t,ξ)yλ(t+ξ)dξ+μλ1L[4]τ1z(t)+μλ1dc˜q1(tτ1,ξ)yλ(tτ1ξ)dξ+μλ1dc˜q2(tτ1,ξ)yλ(tτ1+ξ)dξ+μλ2L[4]τ2z(t)+μλ2dc˜q1(t+τ2,ξ)yλ(t+τ2ξ)dξ+μλ2dc˜q2(t+τ2,ξ)yλ(t+τ2+ξ)dξ=0. (4.3)

    Furthermore, from Lemma 2.1, we have

    ˜q1(t,ξ)yλ(tξ)+μλ1˜q1(tτ1,ξ)yλ(tτ1ξ)+μλ1˜q1(t+τ2,ξ)yλ(t+τ2ξ)Q1(t,ξ)3λ1zλ(tξ). (4.4)

    Similarly, we get

    ˜q2(t,ξ)yλ(t+ξ)+μλ2˜q2(tτ1,ξ)yλ(tτ1+ξ)+μλ2˜q2(t+τ2,ξ)yλ(t+τ2+ξ)Q2(t,ξ)3λ1zλ(t+ξ). (4.5)

    Substituting (4.4), (4.5) into (4.3), we have

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+dcQ1(t,ξ)dξ3λ1zλ(tξ)+dcQ2(t,ξ)dξ3λ1zλ(t+ξ)0. (4.6)

    Using the fact of L[1]z(t)>0 and c+d0, we have

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+dcQ(t,ξ)dξ3λ1zλ(td)0. (4.7)

    Define a function

    w1(t)=m(t)L[3]z(t)zλ(td). (4.8)

    We obtain w1(t)>0, then

    w1(t)=m(t)L[3]z(t)zλ(td)+m(t)L[4]z(t)zλ(td)λm(t)L[3]z(t)z(td)zλ+1(td). (4.9)

    By Lemma (2.5), one gets z(td)a1/λ2(t)a1(td)π1[t0,td]L[2]z(t). Therefore

    w1(t)m(t)L[3]z(t)zλ(td)+m(t)L[4]z(t)zλ(td)λm(t)aλ+1λ2(t)π1[t0,td]L[2]z(t)z(td)zλ+1(td)a1(td). (4.10)

    Using (4.8) in (4.10), we have

    w1(t)(m(t))+m(t)w1(t)+m(t)L[4]z(t)zλ(td)λ(w1(t))λ+1λπ1[t0,td](m(t))1/λa1(td). (4.11)

    Next, define

    w2(t)=m(t)L[3]τ1z(t)zλ(td). (4.12)

    We obtain w2(t)>0, then

    w2(t)=m(t)L[3]τ1z(t)zλ(td)+m(t)L[4]τ1z(t)zλ(td)λm(t)L[3]τ1z(t)z(td)zλ+1(td). (4.13)

    By Lemma (2.5), one gets z(td)a1/λ2(tτ1)a1(td)π1[t0,td]L[2]τ1z(t) and using (4.12) in (4.13), we have

    w2(t)(m(t))+m(t)w2(t)+m(t)L[4]τ1z(t)zλ(td)λ(w2(t))λ+1λπ1[t0,td](m(t))1/λa1(td). (4.14)

    Finally, define

    w3(t)=m(t)L[3]τ2z(t)zλ(td). (4.15)

    We obtain w3(t)>0, then

    w3(t)=m(t)L[3]τ2z(t)zλ(td)+m(t)L[4]τ2z(t)zλ(td)λm(t)L[3]τ2z(t)z(td)zλ+1(td). (4.16)

    By Lemma 2.5, one gets z(td)a1/λ2(t+τ2)a1(td)π1[t0,td]L[2]τ2z(t) and using (4.15) in (4.16), we have

    w3(t)(m(t))+m(t)w3(t)+m(t)L[4]τ2z(t)zλ(td)λ(w3(t))λ+1λπ1[t0,td](m(t))1/λa1(td). (4.17)

    From (4.8), (4.10) and (4.15), we have

    w1(t)+μλ1w2(t)+μλ2w3(t)m(t)[L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)zλ(td)]+[(m(t))+m(t)w1(t)λ(w1(t))λ+1λπ1[t0,td](m(t))1/λa1(td)]+μλ1[(m(t))+m(t)w2(t)λ(w2(t))λ+1λπ1[t0,td](m(t))1/λa1(td)]+μλ2[(m(t))+m(t)w3(t)λ(w3(t))λ+1λπ1[t0,td](m(t))1/λa1(td)]. (4.18)

    Using (4.7) in (4.18), we have

    w1(t)+μλ1w2(t)+μλ2w3(t)m(t)dcQ(t,ξ)dξ3λ1+[(m(t))+m(t)w1(t)λ(w1(t))λ+1λπ1[t0,td](m(t))1/λa1(td)]+μλ1[(m(t))+m(t)w2(t)λ(w2(t))λ+1λπ1[t0,td](m(t))1/λa1(td)]+μλ2[(m(t))+m(t)w3(t)λ(w3(t))λ+1λπ1[t0,td](m(t))1/λa1(td)], (4.19)

    that is,

    m(t)dcQ(t,ξ)dξ3λ1w1(t)μλ1w2(t)μλ2w3(t)+(m(t))+m(t)w1(t)λπ1[t0,td](m(t))1/λa1(td)(w1(t))λ+1λ+μλ1[(m(t))+m(t)w2(t)λπ1[t0,td](m(t))1/λa1(td)(w2(t))λ+1λ]+μλ2[(m(t))+m(t)w3(t)λπ1[t0,td](m(t))1/λa1(td)(w3(t))λ+1λ]. (4.20)

    Multiply both sides H(t,s) and integrate (4.51) from t3 to t, one can get that

    tt3H(t,s)m(s)dcQ(s,ξ)dξ3λ1dstt3H(t,s)w1(s)dsμλ1tt3H(t,s)w2(s)dsμλ2tt3H(t,s)w3(s)ds+tt3H(t,s)(m(s))+m(s)w1(s)dstt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w1(s))λ+1λds+μλ1tt3H(t,s)(m(s))+m(s)w2(s)dsμλ1tt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w2(s))λ+1λds+μλ2tt3H(t,s)(m(s))+m(s)w3(s)dsμλ2tt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w3(s))λ+1λds. (4.21)

    Thus, we obtain

    tt3H(t,s)m(s)dcQ(s,ξ)dξ3λ1dsH(t,t3)w1(t3)+μλ1H(t,t3)w2(t3)+μλ2H(t,t3)w3(t3)tt3[sH(t,s)H(t,s)m(s)m(s)]w1(s)dstt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w1(s))λ+1λdsμλ1tt3[sH(t,s)H(t,s)m(s)m(s)]w2(s)dsμλ1tt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w2(s))λ+1λdsμλ2tt3[sH(t,s)H(t,s)m(s)m(s)]w3(s)dsμλ2tt3H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w3(s))λ+1λds. (4.22)

    Then

    tt3H(t,s)m(s)dcQ(s,ξ)dξ3λ1dsH(t,t3)w1(t3)+μλ1H(t,t3)w2(t3)+μλ2H(t,t3)w3(t3)+tt3[|h(t,s)|(H(t,s))λλ+1m(s)w1(s)H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w1(s))λ+1λ]ds+μλ1tt3[|h(t,s)|(H(t,s))λλ+1m(s)w2(s)H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w2(s))λ+1λ]ds+μλ2tt3[|h(t,s)|(H(t,s))λλ+1m(s)w3(s)H(t,s)λπ1[t0,sd](m(s))1/λa1(sd)(w3(s))λ+1λ]ds. (4.23)

    Setting Y=|h(t,s)|(H(t,s))λλ+1m(s), X=H(t,s)λπ1[t0,sd](m(s))1/λa1(sd) and u=wi(t) for i=1,2,3. By using the Lemma 2.3, we conclude that

    1H(t,t3)tt3[H(t,s)m(s)dcQ(s,ξ)dξ3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(sd)m(s)π1[t0,sd])λ]dsw1(t3)+μλ1w2(t3)+μλ2w3(t3) (4.24)

    which contradicts condition (4.51).

    Theorem 4.2. Let (1.1) holds and c+d0, cτ1. If there exists an m(t)C1([t0,),R+) such that (2.7) and

    lim supt1H(t,t3)tt3[H(t,s)m(s)dcQ(s,ξ)dξ3λ11+μλ1+μλ2(λ+1)λ+1(|h(t,s)|a1(s+c)m(s)π1[t0,s+c])λ]ds=, (4.25)

    then every solution y(t) of (E2) is either oscillatory or tends to 0.

    Proof. Suppose that (E2) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tξ)>0 and y(t+ξ)>0 for tt1t0 and ξ[c,d]. Assumption of (1.1), by Lemma 2.4 there exists two cases (C1) and (C2). If (C2) holds, then by Lemma 2.7, limtz(t)=0. We only consider (C1), by using the fact that z(t)>0 and cτ1, we obtain that Using the fact of L[1]z(t)>0, we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+dcQ(t,ξ)dξ3λ1zλ(t+c)0. (4.26)

    Next, we categorize the functions as w1(t)=m(t)L[3]z(t)zλ(t+c), w2(t)=m(t)L[3]τ1z(t)zλ(t+c) and w3(t)=m(t)L[3]τ2z(t)zλ(t+c) respectively. The rest of the proof is similar to that of Theorem 4.1, therefore, it is omitted.

    Theorem 4.3. Let (1.2) holds and bτ1 (or bτ1). If there exists an m(t)C1([t0,),R+) such that (2.7),

    t3[m(s)dcQ(s,ξ)dξ3λ1(1+μλ1+μλ2)((m(s))+(λ+1))λ+1(a1(sd)m(s)π1[t0,sd])λ]ds=, (4.27)

    and

    t3[πλ(s+τ2)dcQ(s,ξ)dξ3λ1(s+dt2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+d)a1+1λ2(s)βλ(s+τ2)]ds=, (4.28)

    where β(t)=t+da1/λ2(s)ds, then every solution y(t) of (E2) is either oscillatory or tends to 0.

    Proof. Suppose that (E1) has a nonoscillatory solution y. Without loss of generality, we may take y(t)>0, y(tτ1)>0, y(t+τ2)>0, y(tξ)>0 and y(t+ξ)>0 for tt1t0 and ξ[c,d]. Since y(t)>0 for all tt1. Assumption of (1.2), by Lemma 2.4 there exists three cases (C1), (C2) and (C3). If case (C1) and (C2) holds, using the similar proof of ([24], Theorem 2.3) by using Lemma 2.1, we get the conclusion of Theorem 4.3

    If case (C3) holds, z(td)<0 for tt1. The facts that z(t)<0, c+d0 and (4.6), we obtain

    L[4]z(t)+μλ1L[4]τ1z(t)+μλ2L[4]τ2z(t)+dcQ(t,ξ)dξ3λ1zλ(t+d)0. (4.29)

    Define

    w(t)=L[3]z(t)(a1(t+d)z(t+d))λ. (4.30)

    We obtain w(t)<0 for tt2. Noting that L[3]z(t) is decreasing, we obtain

    a2(s)[L[2]z(s)]λa2(t)[L[2]z(t)]λ (4.31)

    for stt2. Dividing (4.31) by a2(s) and integrating from t+d to l(lt), we get

    a1(l)z(l)a1(t+d)z(t+d)+a1/λ2(t)[L[2]z(t)]lt+da1/λ2(s)ds.

    letting l, we get

    1a1/λ2(t)[L[2]z(t)]a1(t+d)z(t+d)π(t),tt2. (4.32)

    From (4.30), we have

    1w(t)βλ(t)0. (4.33)

    By (4.2) we have a1(t+d)z(t+d)a1(t)z(t). Differentiating (4.30) gives,

    w(t)(L[3]z(t))(a1(t+d)z(t+d))λλa2(t)[L[2]z(t)a1(t+d)z(t+d)]λ+1. (4.34)

    Using (4.30) in (4.34), we have

    w(t)L[4]z(t)(a1(t+d)z(t+d))λλw1+1λ(t)a1/λ2(t). (4.35)

    Next, we define

    w(t)=L[3]τ1z(t)(a1(t+d)z(t+d))λ. (4.36)

    We obtain w(t)<0 and w(t)w(t) for tt2. By (4.33), we obtain

    1w(t)βλ(t)0. (4.37)

    By (3.2) we have a1(t+d)z(t+d)a1(tτ1)z(tτ1). Differentiating (4.36) gives,

    w(t)(L[3]τ1z(t))(a1(t+d)z(t+d))λλa2(t)[L[2]τ1z(t)a1(t+d)z(t+d)]λ+1. (4.38)

    Using (4.36) in (4.38), we have

    w(t)L[4]τ1z(t)(a1(t+d)z(t+d))λλw1+1λ(t)a1/λ2(t). (4.39)

    Finally, We define a function

    w(t)=L[3]τ2z(t)(a1(t+τ2+d)z(t+τ2+d))λ. (4.40)

    We obtain w(t)<0 and w(t)=w(t+τ2) for tt2. By (4.33), we obtain

    1w(t)βλ(t+τ2)0. (4.41)

    By (4.2) we have a1(t+τ2+d)z(t+τ2+d)a1(t+τ2)z(t+τ2). Differentiating (4.40) gives,

    w(t)(L[3]τ2z(t))(a1(t+d)z(t+d))λλa2(t)[L[2]τ2z(t)a1(t+τ2+d)z(t+τ2+d)]λ+1. (4.42)

    Using (4.40) in (4.42), we have

    w(t)L[4]τ2z(t)(a1(t+d)z(t+d))λλw1+1λ(t)a1/λ2(t). (4.43)

    From (4.35), (4.39), (4.43) and (4.29) which implies

    w(t)+μλ1w(t)+μλ2w(t)dcQ(t,ξ)dξ3λ1zλ(t+d)(a1(t+d)z(t+d))λλw1+1λ(t)a1/λ2(t)μλ1λw1+1λ(t)a1/λ2(t)μλ2λw1+1λ(t)a1/λ2(t) (4.44)

    In case (C3), (a1(t)z(t))<0 we seen that

    z(t)a1(t)z(t)tt2dsa1(s). (4.45)

    Using (4.45) in (4.44), we get

    w(t)+μλ1w(t)+μλ2w(t)dcQ(t,ξ)dξ3λ1(t+dt2dsa1(s))λλw1+1λ(t)a1/λ2(t)μλ1λw1+1λ(t)a1/λ2(t)μλ2λw1+1λ(t)a1/λ2(t) (4.46)

    Multiplying βλ(t+τ2) and integrating from t3(t3>t2) to t, yields

    βλ(t+τ2)w(t)βλ(t3+τ2)w(t3)+βλ(t+τ2)μλ1w(t)βλ(t3+τ2)μλ1w(t3)+βλ(t+τ2)μλ2w(t)βλ(t3+τ2)μλ2w(t3)λtt3[βλ1(s+τ2)(w(s))a1/λ2(s+τ2)βλ(s+τ2)(w(s))1+1λa1/λ2(s)]dsλμλ1tt3[βλ1(s+τ2)(w(s))a1/λ2(s+τ2)βλ(s+τ2)(w(s))1+1λa1/λ2(s)]dsλμλ2tt3[βλ1(s+τ2)(w(s))a1/λ2(s+τ2)βλ(s+τ2)(w(s))1+1λa1/λ2(s)]ds+tt3βλ(s+τ2)dcQ(s,ξ)dξ3λ1(s+dt2dua1(u))λds0. (4.47)

    Applying Lemma 2.3, we conclude that

    tt3[βλ(s+τ2)dcQ(s,ξ)dξ3λ1(s+dt2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+d)a1+1λ2(s) βλ(s+τ2)]ds[βλ(t+τ2)w(t)+μλ1βλ(t+τ2)w(t)+μλ2βλ(t+τ2)w(t)] (4.48)

    Using the fact of βλ(t+τ2)βλ(t) in (4.33), (4.37), (4.41) and (4.48) imply that

    tt3[βλ(s+τ2)dcQ(s,ξ)dξ3λ1(s+dt2dua1(u))λ(λ1+λ)1+λ(1+μλ1)a2(s)+μλ2a2(s+τ2+d)a1+1λ2(s)βλ(s+τ2)]ds1+μλ1+μλ2. (4.49)

    a contradiction to (4.28).

    Example 4.4. Consider a third-order differential equation

    (12(y(t)+(1/3)y(tπ/4)+(2/3)y(t+π/2)))+π0y(tξ)dξ+32π0y(t+ξ)dξ=0, (4.50)

    Compared with (E2), we can see that c=0, d=π, a1(t)=1/2, a2(t)=1, p1(t)=13, p2(t)=23, ˜q1(t,ξ)=˜q2(t,ξ)=1, λ=1, τ1=π/4 and τ2=π/2. By taking m(t)=1, we obtain

    12t4vu2πdsdudv=

    and we take H(t,s)=(ts)2 then h(t,s)=(3st)(ts)1/5 and 0<μ1+μ2<1, we see that

    lim supt1(tt3)2tt3[2π(ts)21+μ1+μ28((3st)(ts)1/5sπt0)λ]ds=. (4.51)

    Since all the conditions of Theorem 4.1 hold, (4.50) is either oscillates or tends to 0.

    In this paper, we have used Riccati substitution techniques, integral averaging technique and some new oscillation and asymptotic theorems for (E1) and (E2) under the conditions (1.1) and (1.2) have been established. Additionally, we established new comparison theorem that permit to study properties of (E1) regardless under the conditions (1.2). The results obtained indicated that it improved theorems reported by Candan [24]. Similar results can be presented under the assumption that λ1. In this case, using Lemma 2.2, one has to simply replace 3λ1 by 1 and proceed as above. In literature, very few works has been paid in the research activities related to qualitative behavior of solutions of various types of stochastic differential equations, see the recent works [1,3,13,14,15,19,20,21]. The results of this paper could be extended to the stochastic differential equations with time delay in further research.

    The authors would like to thank the anonymous reviewers for their valuable suggestions on improving the content of this article.

    The authors declare there are no conflicts of interest.



    [1] D. A. Labianca, The chemical basis of the Breathalyzer: A critical analysis, J. Chem. Educ., 67 (1990), 259–261. https://doi.org/10.1021/ed067p259 doi: 10.1021/ed067p259
    [2] J. T. Sakai, S. K. Mikulich-Gilbertson, R. J. Long, T. J. Crowley, Validity of transdermal alcohol monitoring: fixed and self-regulated dosing, Alcohol.: Clin. Exp. Res., 30 (2006), 26–33. https://doi.org/10.1111/j.1530-0277.2006.00004.x doi: 10.1111/j.1530-0277.2006.00004.x
    [3] R. M. Swift, Transdermal alcohol measurement for estimation of blood alcohol concentration, Alcohol.: Clin. Exp. Res., 24 (2000), 422–423.
    [4] P. R. Marques, A. S. McKnight, Field and laboratory alcohol detection with 2 types of transdermal devices, Alcohol.: Clin. Exp. Res., 33 (2009), 703–711. https://doi.org/10.1111/j.1530-0277.2008.00887.x doi: 10.1111/j.1530-0277.2008.00887.x
    [5] H. T. Banks, K. Ito, Approximation in LQR problems for infinite dimensional systems with unbounded input operators, J. Math. Syst. Estim. Control, 7 (1997), 1–34.
    [6] H. T. Banks, K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, (1989).
    [7] Z. Dai, I. G. Rosen, C. Wang, N. P. Barnett, S. E. Luczak, Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution-based data analysis software for transdermal alcohol biosensors, Math. Biosci. Eng., 13 (2016), 911–934. https://doi.org/10.3934/mbe.2016023 doi: 10.3934/mbe.2016023
    [8] M. A. Dumett, I. G. Rosen, J. Sabat, A. Shaman, L. Tempelman, C. Wang, et al., Deconvolving an estimate of breath measured blood alcohol concentration from biosensor collected transdermal ethanol data, Appl. Math. Comput., 196 (2008), 724–743.
    [9] I. G. Rosen, S. E. Luczak, J. Weiss, Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data, Appl. Math. Comput., 231 (2014), 357–376.
    [10] W. F. Smith, J. Hashemi, F. Presuel-Moreno, Foundations of Materials Science and Engineering, 3rd edition, McGraw-Hill, New York, (2004).
    [11] M. Allayioti, C. Oszkinat, E. Saldich, L. Goldstein, S. E. Luczak, C. Wang, et al., Parametric and non-parametric estimation of a random diffusion equation-based population model for deconvolving blood/breath alcohol concentration from transdermal alcohol biosensor data with uncertainty quantification, in American Control Conference (ACC), (2023). https://doi.org/10.23919/ACC55779.2023.10156287
    [12] K. Hawekotte, S. E. Luczak, I. G. Rosen, A Bayesian approach to quantifying uncertainty in transport model parameters for, and breath alcohol concentration deconvolved from, biosensor measured transdermal alcohol level, Math. Biosci. Eng., 18 (2021), 6739–6770.
    [13] H. Liu, L. Goldstein, S. E. Luczak, I. G. Rosen, Confidence bands for evolution systems described by parameter-dependent analytic semigroups, in SIAM Conference on Control and its Applications, (2023). https://doi.org/10.1137/1.9781611977745.17
    [14] H. Liu, L. Goldstein, S. E. Luczak, I. G. Rosen, Delta-method induced confidence bands for a parameter-dependent evolution system with application to transdermal alcohol concentration monitoring, in Conference on Decision and Control, (2023).
    [15] M. Sirlanci, S. E. Luczak, C. E. Fairbairn, D. Kang, R. Pan, X. Yu, et al., Estimating the distribution of random parameters in a diffusion equation forward model for a transdermal alcohol biosensor, Automatica, 106 (2019), 101–109. https://doi.org/10.1016/j.automatica.2019.04.026 doi: 10.1016/j.automatica.2019.04.026
    [16] M. Sirlanci, S. E. Luczak, I. G. Rosen, Approximation and convergence in the estimation of random parameters in linear holomorphic semigroups generated by regularly dissipative operators, in American Control Conference (ACC), (2017), 3171–3176. https://doi.org/10.23919/ACC.2017.7963435
    [17] M. Sirlanci, S. E. Luczak, I. G. Rosen, Estimation of the distribution of random parameters in discrete time abstract parabolic systems with unbounded input and output: approximation and convergence, Commun. Appl. Anal., 23 (2019), 287–329. https://doi.org/10.12732/caa.v23i2.4 doi: 10.12732/caa.v23i2.4
    [18] C. Oszkinat, T. Shao, C. Wang, I. G. Rosen, A. D. Rosen, E. Saldich, et al., Estimation and uncertainty quantification via forward and inverse filtering for a covariate-dependent, physics-informed, hidden Markov model, Inverse Probl., 38 (2022). https://doi.org/10.1088/1361-6420/ac5ac7 doi: 10.1088/1361-6420/ac5ac7
    [19] C. Oszkinat, S. E. Luczak, I. G. Rosen, Uncertainty quantification in estimating blood alcohol concentration from transdermal alcohol level with physics-informed neural networks, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 8094–8101. https://doi.org/10.1109/tnnls.2022.3140726 doi: 10.1109/tnnls.2022.3140726
    [20] C. Oszkinat, S. E. Luczak, I. G. Rosen, An abstract parabolic system-based physics-informed long short-term memory network for estimating breath alcohol concentration from transdermal alcohol biosensor data, Neural Comput. Appl., 34 (2022), 1–19. https://doi.org/10.1007/s00521-022-07505-w doi: 10.1007/s00521-022-07505-w
    [21] H. T. Banks, W. C. Thompson, Least Squares Estimation of Probability Measures in the Prohorov Metric Framework, Technical report, (2012).
    [22] H. T. Banks, K. B. Flores, I. G. Rosen, E. M. Rutter, M. Sirlanci, W. C. Thompson, The Prohorov metric framework and aggregate data inverse problems for random PDEs, Commun. Appl. Anal., 22 (2018), 415–446.
    [23] M. Davidian, D. Giltinan, Nonlinear Models for Repeated Measurement Data, Chapman and Hall, New York, (1995).
    [24] M. Davidian, D. M. Giltinan, Nonlinear models for repeated measurement data: An overview and update, Agric. Biol. Environ. Stat., 8 (2003), 387–419. https://doi.org/10.1198/1085711032697 doi: 10.1198/1085711032697
    [25] E. Demidenko, Mixed Models, Theory and Applications, 2nd edition, John Wiley and Sons, Hoboken, (2013).
    [26] M. Lovern, M. Sargentini-Maier, C. Otoul, J. Watelet, Population pharmacokinetic and pharmacodynamic analysis in allergic diseases, Drug Metab. Rev., 41 (2009), 475–485. https://doi.org/10.1080/10837450902891543 doi: 10.1080/10837450902891543
    [27] R. Tatarinova, M. Neely, J. Bartroff, M. van Guilder, W. Yamada, D. Bayard, et al., Two general methods for population pharmacokinetic modeling: non-parametric adaptive grid and non-parametric Bayesian, J. Pharmacokinet. Pharmacodyn., 40 (2013), 189–199. https://doi.org/10.1007/s10928-013-9302-8 doi: 10.1007/s10928-013-9302-8
    [28] J. Li, S. E. Luczak, I. G. Rosen, Comparing a distributed parameter model-based system identification technique with more conventional methods for inverse problems, J. Inverse Ill-Posed Probl., 27 (2019), 703–717. https://doi.org/10.1515/jiip-2018-0006 doi: 10.1515/jiip-2018-0006
    [29] M. Sirlanci, I. G. Rosen, S. E. Luczak, C. E. Fairbairn, K. Bresin, D. Kang, Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data, Inverse Probl., 34 (2018), 125006. https://doi.org/10.1088/1361-6420/aae791 doi: 10.1088/1361-6420/aae791
    [30] M. Yao, S. E. Luczak, I. G. Rosen, Tracking and blind deconvolution of blood alcohol concentration from transdermal alcohol biosensor data: A population model-based LQG approach in Hilbert space, Automatica, 147 (2023). https://doi.org/10.1016/j.automatica.2022.110699 doi: 10.1016/j.automatica.2022.110699
    [31] D. M. Dougherty, N. E. Charles, A. Acheson, S. John, R. M. Furr, N. Hill-Kapturczak, Comparing the detection of transdermal and breath alcohol concentrations during periods of alcohol consumption ranging from moderate drinking to binge drinking, Exp. Clin. Psychopharmacol., 20 (2012), 373–81. https://doi.org/10.1037/a0029021 doi: 10.1037/a0029021
    [32] D. M. Dougherty, T. E. Karns, J. Mullen, Y. Liang, S. L. Lake, J. D. Roache, et al., Transdermal alcohol concentration data collected during a contingency management program to reduce at-risk drinking, Drug Alcohol Depend., 148 (2015), 77–84. https://doi.org/10.1016/j.drugalcdep.2014.12.021 doi: 10.1016/j.drugalcdep.2014.12.021
    [33] C. E. Fairbairn, D. Kang, N. Bosch, Using machine learning for real-time BAC estimation from a new-generation transdermal biosensor in the laboratory, Drug Alcohol Depend., 216 (2021), 108205. https://doi.org/10.1016/j.drugalcdep.2020.108205 doi: 10.1016/j.drugalcdep.2020.108205
    [34] B. Lindsay, The geometry of mixture likelihoods: a general theory, Ann. Stat., 11 (1983), 86–94. https://doi.org/10.1214/aos/1176346059 doi: 10.1214/aos/1176346059
    [35] A. Mallet, A maximum likelihood estimation method for random coefficient regression models, Biometrika, 73 (1986), 645–656. https://doi.org/10.2307/2336529 doi: 10.2307/2336529
    [36] J. Kiefer, J. Wolfowitz, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters, Ann. Math. Stat., 27 (1956), 887–906. https://doi.org/10.1214/aoms/1177728066 doi: 10.1214/aoms/1177728066
    [37] H. Tanabe, Equations of Evolution (Monographs and Studies in Mathematics), Pitman Publishing, (1979).
    [38] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Berlin, Heidelberg, (1971).
    [39] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, (1983).
    [40] H. T. Banks, K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), 684–698. https://doi.org/10.1137/0322043 doi: 10.1137/0322043
    [41] H. T. Banks, K. Ito, A Unified Framework for Approximation in Inverse Problems for Distributed Parameter Systems, NASA. Hampton, VA. Technical Reports NASA-CR-181621, (1988).
    [42] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Elsevier, (2003).
    [43] M. H. Schultz, Spline Analysis, Prentice-Hall, (1973).
    [44] S. E. Luczak, I. G. Rosen, T. L. Wall, Development of a real-time repeated-measures assessment protocol to capture change over the course of drinking episodes, Alcohol Alcohol., 50 (2015), 1–8. https://doi.org/10.1093/alcalc/agu100 doi: 10.1093/alcalc/agu100
    [45] E. B. Saldich, C. Wang, I. G. Rosen, L. Goldstein, J. Bartroff, R. M. Swift, et al., Obtaining high-resolution multi-biosensor data for modeling transdermal alcohol concentration data, Alcohol.: Clin. Exp. Res., 44 (2020). https://doi.org/10.1111/acer.14358 doi: 10.1111/acer.14358
    [46] A. Kryshchenko, M. Sirlanci, B. Vader, Nonparametric estimation of blood alcohol concentration from transdermal alcohol measurements using alcohol biosensor devices, Adv. Data Sci. Adapt. Anal., 26 (2021), 329–360. https://doi.org/10.1007/978-3-030-79891-8_13 doi: 10.1007/978-3-030-79891-8_13
  • This article has been cited by:

    1. Marappan Sathish Kumar, Omar Bazighifan, Alanoud Almutairi, Dimplekumar N. Chalishajar, Philos-Type Oscillation Results for Third-Order Differential Equation with Mixed Neutral Terms, 2021, 9, 2227-7390, 1021, 10.3390/math9091021
    2. Sathish Kumar Marappan, Alanoud Almutairi, Loredana Florentina Iambor, Omar Bazighifan, Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms, 2023, 15, 2073-8994, 553, 10.3390/sym15020553
    3. M. Sathish Kumar, V. Ganesan, Oscillatory behavior of solutions of certain third-order neutral differential equation with continuously distributed delay, 2021, 1850, 1742-6588, 012091, 10.1088/1742-6596/1850/1/012091
    4. M. Sathish Kumar, Omar Bazighifan, Khalifa Al-Shaqsi, Fongchan Wannalookkhee, Kamsing Nonlaopon, Symmetry and Its Role in Oscillation of Solutions of Third-Order Differential Equations, 2021, 13, 2073-8994, 1485, 10.3390/sym13081485
    5. Nagamanickam Nagajothi, Vadivel Sadhasivam, Omar Bazighifan, Rami Ahmad El-Nabulsi, Existence of the Class of Nonlinear Hybrid Fractional Langevin Quantum Differential Equation with Dirichlet Boundary Conditions, 2021, 5, 2504-3110, 156, 10.3390/fractalfract5040156
    6. R. Elayaraja, M. Sathish Kumar, V. Ganesan, Nonexistence of Kneser solution for third order nonlinear neutral delay differential equations, 2021, 1850, 1742-6588, 012054, 10.1088/1742-6596/1850/1/012054
    7. M. Sathish Kumar, R. Elayaraja, V. Ganesan, Omar Bazighifan, Khalifa Al-Shaqsi, Kamsing Nonlaopon, Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order, 2021, 5, 2504-3110, 95, 10.3390/fractalfract5030095
    8. R. Elayaraja, V. Ganesan, Omar Bazighifan, Clemente Cesarano, Oscillation and Asymptotic Properties of Differential Equations of Third-Order, 2021, 10, 2075-1680, 192, 10.3390/axioms10030192
    9. Waed Muhsin, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, Elmetwally M. Elabbasy, Delay Differential Equations with Several Sublinear Neutral Terms: Investigation of Oscillatory Behavior, 2023, 15, 2073-8994, 2105, 10.3390/sym15122105
    10. Rami Ahmad El-Nabulsi, Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coefficients, 2024, 189, 09600779, 115737, 10.1016/j.chaos.2024.115737
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1512) PDF downloads(44) Cited by(0)

Figures and Tables

Figures(5)  /  Tables(11)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog