Research article

On the oscillation of first-order differential equations with deviating arguments and oscillatory coefficients

  • Received: 26 October 2022 Revised: 08 December 2022 Accepted: 29 December 2022 Published: 09 January 2023
  • MSC : 34K11, 34K06

  • In this work, we study the oscillation problem of first-order differential equations with deviating arguments and oscillatory coefficients. We generalize and improve the work of Kwong [30] such that the delay (advanced) and the coefficient functions do not need to be monotone and nonnegative, respectively. This method essentially improves many known oscillation conditions. The significance and the substantial improvement of our results are shown by two illustrative examples.

    Citation: Emad R. Attia, George E. Chatzarakis. On the oscillation of first-order differential equations with deviating arguments and oscillatory coefficients[J]. AIMS Mathematics, 2023, 8(3): 6725-6736. doi: 10.3934/math.2023341

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  • In this work, we study the oscillation problem of first-order differential equations with deviating arguments and oscillatory coefficients. We generalize and improve the work of Kwong [30] such that the delay (advanced) and the coefficient functions do not need to be monotone and nonnegative, respectively. This method essentially improves many known oscillation conditions. The significance and the substantial improvement of our results are shown by two illustrative examples.



    Functional differential equations arise widely in many fields such as mathematical biology, economy, physics, or biology, see [16,19,28,40]. This explains the great interest in the qualitative properties of these kinds of equations. Oscillation phenomena appear in various models from real world applications; see, e.g., the papers [12,35,38] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. As a part of this approach, the oscillation theory of this type of equation has been extensively developed, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. In particular, the oscillation criteria of first-order differential equations with deviating arguments have numerous applications in the study of higher-order functional differential equations (e.g., one can study the oscillatory behavior of higher-order functional differential equations by relating oscillation of these equations to that of associated first-order functional differential equations); see, e.g., the papers [13,36,37].

    Recently, there has been great interest in studying the oscillation of all solutions of the first-order delay differential equation

    x(t)+b(t)x(τ(t))=0,tt0, (1.1)

    and its dual advanced equation

    x(t)c(t)x(σ(t))=0,tt0, (1.2)

    where b,c,τ,σC([t0,),[0,)) such that τ(t)t, limt τ(t)=, and σ(t)t. In most of these works, the delay (advanced) function is assumed to be nondecreasing, see [14,16,29,31,32,33,34,42,45] and the references therein. As shown in [8], the oscillation character of Eq (1.1) with nonmontone delay, is not an easy extension to the oscillation problem for the nondecreasing delay case. Many authors [1,3,5,6,7,8,9,10,11,15,20,25,27,39,43] have developed and generalized the methods used to study the oscillation of equations (1.1) and (1.2) with monotone delays and to study this property for the nonmonotone case. Only a few works, however, dealt with the oscillation of equations (1.1) and (1.2) with oscillatory coefficients. For example, [16,44] studied the oscillation of Eq (1.1) where the delay function τ(t) is assumed to be nondecreasing and constant (i.e., τ(t)=tα, α>0), respectively. Also, Kulenovic and Grammatikopoulos [29] studied the oscillation of a first-order nonlinear functional differential equation that contains both equations (1.1) and (1.2). The authors obtained liminf and limsup oscillation criteria for the case when the coefficient function does not need to be nonnegative. However, the delay (advanced) and the coefficient functions are assumed to be nondecreasing (for limsup conditions) and nonnegative on a sequence of intervals {(rn,sn)}n0 such that limn (snrn)= (for liminf conditions), respectively.

    Our aim in this work is to obtain oscillation criteria for equations (1.1) and (1.2) where b(t) and c(t) are continuous functions on [t0,). We relax the nonnegative restriction on the coefficient functions b(t) and c(t). To accomplish this goal, using the ideas of [27], we develop and enhance the work of Kwong [30]. This procedure leads to new sufficient oscillation criteria that improve and generalize those mentioned in [16,29,44].

    From now on, we assume that b(t) and c(t) are only continuous functions on [t0,).

    Let Λ(t) and Λi(t), tt0, iN be defined as follows (see [27]):

    Λ(t)=max{ut: τ(u)t},Λ1(t)=Λ(t),Λi(t)=Λ(t)Λi1(t), i=2,3, . (2.1)

    Also, we define the function g(t) and the sequence {Qn(v,u)}n=0, τ(v)uv, as follows:

    g(t)=suput τ(u),tt0 (2.2)

    and

    Q0(v,u)=1,Qn(v,u)=exp(vub(ζ)Qn1(ζ,τ(ζ))dζ),nN.

    The proofs of our main results are essentially based on the following lemma.

    Lemma 2.1. Let nN0, T>t0, TT and x(t) be a solution of Eq (1.1) such that x(t)>0 for all tT. If b(t)0 on [T,T1], T1Λn+2(T), then

    x(u)x(v)Qn(v,u),τ(v)uv,for  v[Λn+2(T),T1]. (2.3)

    Proof. It follows from Eq (1.1) that x(t)0 on [Λ1(T),T1]. Therefore,

    x(u)x(v)1=Q0(v,u),τ(v)uv, for v[Λ2(T),T1].

    Dividing Eq (1.1) by x(t) and integrating from u to v, τ(v)uv, we obtain

    x(u)x(v)=exp(vub(ζ)x(τ(ζ))x(ζ)dζ). (2.4)

    Since x(t)0 on [Λ1(T),T1], we get

    x(u)x(v)exp(vub(ζ)dζ)=exp(vub(ζ)Q0(ζ,τ(ζ))dζ)=Q1(v,u),τ(v)uv

    for v[Λ3(T),T1] and consequently, for uζv, we have

    x(τ(ζ))x(ζ)Q1(ζ,τ(ζ)),τ(v)uv for v[Λ4(T),T1].

    Substituting in (2.4), we get

    x(u)x(v)exp(vub(ζ)Q1(ζ,τ(ζ))dζ)=Q2(v,u) for v[Λ4(T),T1].

    Repeating this argument n times, we obtain

    x(u)x(v)exp(vub(ζ)Qn1(ζ,τ(ζ))dζ)=Qn(v,u) for t[Λn+2(T),T1].

    The proof of the lemma is complete.

    Let {Tk}k0 be a sequence of real numbers such that limk Tk= and

    b(t)0 for t[Tk,Λn+4(Tk)], for all  kN for some nN0. (2.5)

    Also, we define the sequence {βn}n1, βn>1 for all nN as follows:

    Qn(t,g(t))>βn,t[g(Λn+3(Tk)),Λn+3(Tk)] for all  kN0 for some nN. (2.6)

    Theorem 2.1. Let nN such that (2.5) and (2.6) are satisfied. If

    Λn+4(Tk)g(Λn+4(Tk))b(ζ)Qn+1(g(ζ),τ(ζ)) dζln(βn+1)+1βn+1for all   kN0,

    then every solution of Eq (1.1) is oscillatory.

    Proof. Assume, for the sake of contradiction, that x(t) is an eventually positive solution of Eq (1.1). Then there exists a sufficiently large T>t0 such that x(t)>0 for t>T. Suppose that Tk1{Tk}k0 such that Tk1>T. In view of (2.3), (2.5) and (2.6), it follows that

    x(g(Λn+4(Tk1)))x(Λn+4(Tk1))Qn+1(Λn+4(Tk1),g(Λn+4(Tk1)))>βn+1>1.

    Then there exists t(g(Λn+4(Tk1)),Λn+4(Tk1)) such that

    x(g(Λn+4(Tk1)))x(t)=βn+1. (2.7)

    Integrating Eq (1.1) from t to t, we get

    x(Λn+4(Tk1))x(t)+Λn+4(Tk1)tb(ζ)x(τ(ζ))dζ=0. (2.8)

    It is easy to see that

    x(τ(ζ))=x(g(ζ))exp(g(ζ)τ(ζ)b(ζ1)x(τ(ζ1))x(ζ1)dζ1). (2.9)

    Substituting in (2.8), we have

    x(Λn+4(Tk1))x(t)+Λn+4(Tk1)tb(ζ)exp(g(ζ)τ(ζ)b(ζ1)x(τ(ζ1))x(ζ1)dζ1)x(g(ζ))dζ=0.

    Since x(t)0 on [Λ1(Tk1),Λn+4(Tk1)], it follows that

    x(Λn+4(Tk1))x(t)+x(g(Λn+4(Tk1)))Λn+4(Tk1)tb(ζ)exp(g(ζ)τ(ζ)b(ζ1)x(τ(ζ1))x(ζ1)dζ1)dζ0.

    By (2.3) and ζ1[Λn+2(Tk1),Λn+3(Tk1)] for τ(ζ)<ζ1<g(ζ), g(Λn+4(Tk1))<ζ<Λn+4(Tk1), we get

    Λn+4(Tk1)tb(ζ)exp(g(ζ)τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζx(t)x(g(Λn+4(Tk1)))x(Λn+4(Tk1))x(g(Λn+4(Tk1)))<1βn+1. (2.10)

    Dividing Eq (1.1) by x(t) and integrating from g(Λn+4(Tk1)) to t, we obtain

    tg(Λn+4(Tk1))x(ζ)x(ζ)dζ=tg(Λn+4(Tk1))b(ζ) x(τ(ζ))x(ζ)dζ.

    Using (2.9), we get

    ln(x(g(Λn+4(Tk1)))x(t))=tg(Λn+4(Tk1))b(ζ) x(g(ζ)))x(ζ)exp(g(ζ)τ(ζ)b(ζ1)x(τ(ζ1))x(ζ1)dζ1)dζ.

    From this, (2.3) and (2.6), we get

    tg(Λn+4(Tk1))b(ζ)Qn+1(ζ,g(ζ)) exp(g(ζ)τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζln(x(g(Λn+4(Tk1)))x(t)).

    It follows from (2.6) and (2.7) that

    tg(Λn+4(Tk1))b(ζ) exp(g(ζ)τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζln(βn+1)βn+1.

    Combining this and (2.10) we get

    Λn+4(Tk1)g(Λn+4(Tk1))b(ζ) exp(g(ζ)τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζ<ln(βn+1)+1βn+1,

    that is

    Λn+4(Tk)g(Λn+4(Tk))b(ζ)Qn+1(g(ζ),τ(ζ)) dζ<ln(βn+1)+1βn+1.

    The proof of the theorem is complete.

    Theorem 2.2. Let nN0 such that (2.5) is satisfied. If

    Λn+4(Tk)g(Λn+4(Tk))b(ζ)Qn+1(g(Λn+4(Tk)),τ(ζ))dζ1for all kN0, (2.11)

    then every solution of Eq (1.1) is oscillatory.

    Proof. Let x(t) be an eventually positive solution of Eq (1.1). Then the exists T>t0 such that x(t)>0 for all tT. It is not difficult to prove that

    x(Λn+4(Tk1))x(g(Λn+4(Tk1)))+x(g(Λn+4(Tk1)))Λn+4(Tk1)g(Λn+4(Tk1))b(ζ)exp(g(Λn+4(Tk1))τ(ζ)b(ζ1)x(τ(ζ1))x(ζ1)dζ1)dζ=0,

    where Tk1{Tk} such that Tk1>T. Using (2.3), we get

    x(Λn+4(Tk1))+x(g(Λn+4(Tk1)))(Λn+4(Tk1)g(Λn+4(Tk1))b(ζ)exp(g(Λn+4(Tk1))τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζ1)0.

    Using the positivity of x(Λn+4(Tk1)) we have

    Λn+4(Tk1)g(Λn+4(Tk1))b(ζ)exp(g(Λn+4(Tk1))τ(ζ)b(ζ1)Qn(ζ1,τ(ζ1))dζ1)dζ<1.

    Then

    Λn+4(Tk)g(Λn+4(Tk))b(ζ)Qn+1(g(Λn+4(Tk)),τ(ζ))dζ<1,

    which contradicts (2.11). The proof of the theorem is complete.

    Remark 2.1.

    ● (1) It should be noted that the monotonicity of the delay function τ(t) is required in many previous works to study the oscillation of Eq (1.1) with oscillating coefficients; see, for example, [16,29,44]. In this work, the sequence {Λi(t)}i0 plays a central role in the derivation of our results. In fact the delay function τ(t) does not need to be monotone. Therefore, our results substantially improve and generalize [29, Theroems 6, 7] for Eq (1.1). Furthermore, using our approach, many previous oscillation studies for Eq (1.1) with monotone delays can be used to study the oscillation of Eq (1.1) with general delays (the delay does not need to be monotone) and oscillating coefficients.

    (2) There are numerous lower bounds for the quotient x(τ(t))x(t), where x(t) is a positive solution of Eq (1.1) with a nonnegative continuous function b(t), see [6,16,22,24,42]. For example, [22], Lemma 1] and Lemma 2.1 can be used instead of Lemma 2.1 to improve our results in the case where b(t) is a nonnegative continuous function. In this case, the adjusted version of the results improves Theorems 2.1 and 2.2. Even in the case where τ(t) is nondecreasing, the improvement is substantial.

    Similar results for the (dual) advanced differential equation (1.2) can be obtained easily. The details of the proofs are omitted since they are quite similar to Eq (1.1).

    We will use the following notation:

    h(t)=infut σ(u),tt0 (2.12)
    Ω1(t)=Ω(t),Ωi(t)=Ω(t)Ωi1(t), tt0, i=2,3,, (2.13)

    where

    Ω(t)=min{t0ut: σ(u)t}.

    Also, we define the sequence {Rn(u,v)}n=0, vuσ(v) as follows:

    R0(u,v)=1,Rn(u,v)=exp(uvb(ζ)Rn1(σ(ζ),ζ)dζ),nN.

    In order to obtain the oscillation criteria for Eq (1.2) we need the following conditions:

    Let the sequence {Tk}k0 be a sequence of real numbers such that limk Tk= and

    c(t)0 for t[Ωn+4(Tk),Tk] for all kN0 for some nN. (2.14)

    Also, we define the sequence {γn}n1, γn>1 for all nN as follows:

    Rn(h(t),t)>γn,t[Ωn+3(Tk),h(Ωn+3(Tk))] for all kN0 for some nN. (2.15)

    Theorem 2.3. Let nN such that (2.14) and (2.15) are satisfied. If

    h(Ωn+4   (Tk))Ωn+4   (Tk)c(ζ)Rn+1(σ(ζ),h(ζ))dζln(γn+1)+1γn+1   for all  kN0,

    then every solution of Eq (1.2) is oscillatory.

    Theorem 2.4. Let nN0 such that (2.14) is satisfied. If

    h(Ωn+4  (Tk))Ωn+4  (Tk)c(ζ)Rn+1(σ(ζ),h(Ωn+4(Tk)))dζ1for all   kN0,

    then every solution of Eq (1.2) is oscillatory.

    Example 3.1. Consider the delay differential equation

    x(t)+b(t)x(τ(t))=0,t1, (3.1)

    where bC([1,),R) such that

    b(t)=η>0 for t[3rk,3rk+645121] for all kN0,

    {rk}k0 is a sequence of positive integers such that rk+1>rk+215121 and limk rk=, and

    τ(t)={t1 if t[3l,3l+2],t+6l+3 if t[3l+2,3l+2.1],119t23l53 if t[3l+2.1,3l+3],lN0.

    In view of (2.1) and (2.2), it is easy to see that

    g(t)={t1 if t[3l,3l+2],3l+1 if t[3l+2,3l+2411],119t23l53 if t[3l++2411,3l+3],lN0

    and

    Λ(t)={t+1 if t[3l,3l+0.9],911t+611l+1511 if t[3l+0.9,3l+2],t+1 if t[3l+2,3l+3],lN0,

    respectively.

    Letting Tk=3rk, kN0, so Λ5(Tk)=3rk+645121, and hence

    b(t)=η for t[Tk,Λ5(Tk)] for all kN0. (3.2)

    It is obvious that g(Λ5(Tk))=3rk+4611 and

    t1.2τ(t)g(t)t1 for all t1.

    Therefore,

    Q2(t,g(t))=exp(tg(t)b(ζ)exp(ζτ(ζ)b(ζ1)dζ1)dζ)exp(tt1b(ζ)exp(ζζ1b(ζ1)dζ1)dζ)exp(ηexp(η))

    for t[3rk+4611,3rk+645121]. Denote β2=exp(ηexp(η))>1. Then

    Q2(t,g(t))>β2 for t[g(Λ5(Tk)),Λ5(Tk)] for all kN0. (3.3)

    Also,

    Λ5(Tk)g(Λ5(Tk))b(ζ)Q2(g(ζ),τ(ζ))dζ=3rk+6451213rk+4611b(ζ)exp(g(ζ)τ(ζ)Q1(ζ1,τ(ζ1))b(ζ1))dζ=117121η+20(exp(110ηexp(η))1)exp(η)11>0.707

    for all η0.61 and kN0 and

    (1+ln(β2)β2)<0.691 for all η0.61 and kN0.

    It is obvious that

    Λ5(Tk)g(Λ5(Tk))b(ζ)Q2(g(ζ),τ(ζ))dζ>(1+ln(β2)β2) for all η0.61 and kN0.

    In view of this, (3.2) and (3.3), all conditions of Theorem 2.1 with n=1 are satisfied for all η0.61. Therefore all solutions of Eq (3.1) are oscillatory for η0.61.

    However, if we assume that ak=3rk and bk=3rk+645121, then bkak=645121<. It follows that [29,Theorem 3] cannot be applied to Eq (3.1). Note also that since τ is not monotone, [29,Theorem 6] cannot be applied to this example.

    Example 3.2. Consider the advanced differential equation

    x(t)c(t)x(σ(t))=0,t0, (3.4)

    where cC([0,),R) such that

    c(t)={δ(1+sin(9πt)) for t[4rk689,4rk419],(αδ)(9t36rk+41)+δ for t[4rk419,4rk409],α for t[4rk409,4rk+103],kN0, (3.5)

    where α,δ0 and {rk}k0 is a sequence of positive integers such that rk+1>rk+4918 and limk rk=, and

    σ(t)={t+3 if t[4l,4l+2],t+8l+7 if t[4l+2,4l+3],3t8l5 if t[4l+3,4l+4],lN0.

    In view of (2.12) and (2.13), it follows that

    h(t)={t+3 if t[4l,4l+2],4l+5 if t[4l+2,4l+103],3t8l5 if t[4l+103,4l+4],lN0

    and

    Ω(t)={t3 if t[4l,4l+1],13t+83l1 if t[4l+1,4l+3],t3 if t[4l+3,4l+4],lN0,

    respectively.

    Clearly,

    t+1h(t)σ(t)t+3, for all t1.

    If we assume that Tk=4rk+103, kN0, then Ω4(Tk)=4rk689. It follows from (3.5) that

    c(t)0 for t[Ω4(Tk),Tk] for all  kN0.

    Thus

    h(Ω4(Tk))Ω4(Tk)c(ζ)R1(h(Ω4(Tk)),τ(ζ))dζ4rk4194rk689c(ζ)dζ=4rk4194rk689δ(1+sin(9πζ))dζ=δ(2+27π)9π1, for all δ9π2+27π and kN0.

    Therefore all conditions of Theorem 2.4 with n=0 are satisfied for all δ9π2+27π, and hence Eq (3.4) is oscillatory for δ9π2+27π.

    The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF-PSAU- 2022/01/22323).

    The authors declare that they have no competing of interests regarding the publication of this paper.



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