# On Nonoscillation of Advanced Differential Equations with Several Terms

Authors: L. Berezansky; E. Braverman
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Existence of positive solutions for advanced equations with several termsx˙(t)+∑k=1mak(t)x(hk(t))=0,hk(t)≥t is investigated in the following three cases: (a) all coefficients ak are positive; (b) all coefficients ak are negative; (c) there is an equal number of positive and negative coefficients. Results on asymptotics of nonoscillatory solutions are also presented.


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## Body

## 1. Introduction

This paper deals with nonoscillation properties of scalar advanced differential equations. Advanced differential equations appear in several applications, especially as mathematical models in economics; an advanced term may, for example, reflect the dependency on anticipated capital stock [1, 2].It is not quite clear how to formulate an initial value problem for such equations, and existence and uniqueness of solutions becomes a complicated issue. To study oscillation, we need to assume that there exists a solution of such equation on the halfline. In the beginning of 1980s, sufficient oscillation conditions for first-order linear advanced equations with constant coefficients and deviations of arguments were obtained in [3] and for nonlinear equations in [4]. Later oscillation properties were studied for other advanced and mixed differential equations (see the monograph [5], the papers [6–12] and references therein). Overall, these publications mostly deal with sufficient oscillation conditions; there are only few results [7, 9, 12] on existence of positive solutions for equations with several advanced terms and variable coefficients, and the general nonoscillation theory is not complete even for first-order linear equations with variable advanced arguments and variable coefficients of the same sign. The present paper partially fills up this gap. We obtain several nonoscillation results for advanced equations using the generalized characteristic inequality [13]. The main method of this paper is based on fixed point theory; thus, we also state the existence of a solution in certain cases.In the linear case, the best studied models with advanced arguments were the equations of the types(1.1)ẋ(t)-a(t)x(h(t))+b(t)x(t)=0,ẋ(t)-a(t)x(t)+b(t)x(g(t))=0,
where a(t)≥0, b(t)≥0, h(t)≥t, and g(t)≥t.Let us note that oscillation of higher order linear and nonlinear equations with advanced and mixed arguments was also extensively investigated, starting with [14]; see also the recent papers [15–19] and references therein.For equations with an advanced argument, the results obtained in [20, 21] can be reformulated as Theorems A–C below.Theorem A (see [20]).
Ifa,  b, and  h are equicontinuous on  [0,∞),  a(t)≥0,  b(t)≥0,  h(t)≥t, and  limsupt→∞[h(t)-t]<∞, then the advanced equation
(1.2)ẋ(t)+a(t)x(h(t))+b(t)x(t)=0
has a nonoscillatory solution.In the present paper, we extend Theorem A to the case of several deviating arguments and coefficients (Theorem2.10).Theorem B (see [20]).
Ifa,  b, and  h are equicontinuous on  [0,∞),  a(t)≥0,  b(t)≥0,  h(t)≥t,  limsupt→∞[h(t)-t]<∞, and
(1.3)limsup⁡t→∞∫th(t)a(s)exp⁡{∫sh(s)b(τ)dτ}ds<1e,
then the advanced equation
(1.4)ẋ(t)-a(t)x(h(t))-b(t)x(t)=0
has a nonoscillatory solution.Corollary2.3 of the present paper extends Theorem B to the case of several coefficients ak≥0 and advanced arguments hk (generally, b(t)≡0); if (1.5)∫tmax⁡khk(t)∑i=1mai(s)ds≤1e,
then the equation (1.6)ẋ(t)+∑k=1mak(t)x(hk(t))=0has an eventually positive solution. To the best of our knowledge, only the opposite inequality (withminkhk(t) rather than maxkhk(t) in the upper bound) was known as a sufficient oscillation condition. Coefficients and advanced arguments are also assumed to be of a more general type than in [20]. Comparison to equations with constant arguments deviations, and coefficients (Corollary 2.8) is also outlined.For advanced equations with coefficients of different sign, the following result is known.Theorem C (see [21]).
If0≤a(t)≤b(t) and  h(t)≥t, then there exists a nonoscillatory solution of the equation
(1.7)ẋ(t)-a(t)x(h(t))+b(t)x(t)=0.This result is generalized in Theorem2.13 to the case of several positive and negative terms and several advanced arguments; moreover, positive terms can also be advanced as far as the advance is not greater than in the corresponding negative terms.We also study advanced equations with positive and negative coefficients in the case when positive terms dominate rather than negative ones; some sufficient nonoscillation conditions are presented in Theorem2.15; these results are later applied to the equation with constant advances and coefficients. Let us note that analysis of nonoscillation properties of the mixed equation with a positive advanced term (1.8)ẋ(t)+a(t)x(h(t))-b(t)x(g(t))=0,h(t)≥t,g(t)≤t,a(t)≥0,b(t)≥0was also more complicated compared to other cases of mixed equations with positive and negative coefficients [21].In nonoscillation theory, results on asymptotic properties of nonoscillatory solutions are rather important; for example, for equations with several delays and positive coefficients, all nonoscillatory solutions tend to zero if the integral of the sum of coefficients diverges; under the same condition for negative coefficients, all solutions tend to infinity. In Theorems2.6 and 2.11, the asymptotic properties of nonoscillatory solutions for advanced equations with coefficients of the same sign are studied.The paper is organized as follows. Section2 contains main results on the existence of nonoscillatory solutions to advanced equations and on asymptotics of these solutions: first for equations with coefficients of the same sign, then for equations with both positive and negative coefficients. Section 3 involves some comments and open problems.


## 2. Main Results

Consider first the equation(2.1)ẋ(t)-∑k=1mak(t)x(hk(t))=0,
under the following conditions:(a1)
ak(t)≥0, k=1,…,m, are Lebesgue measurable functions locally essentially bounded for t≥0,(a2)
hk:[0,∞)→ℝ are Lebesgue measurable functions, hk(t)≥t, k=1,…,m.Definition 2.1.
A locally absolutely continuous functionx:[t0,∞)→R is called a solution of problem (2.1) if it satisfies (2.1) for almost all t∈[t0,∞).The same definition will be used for all further advanced equations.Theorem 2.2.
Suppose that the inequality(2.2)u(t)≥∑k=1mak(t)exp⁡{∫thk(t)u(s)ds},t≥t0
has a nonnegative solution which is integrable on each interval [t0,b], then (2.1) has a positive solution for t≥t0.Proof.
Letu0(t) be a nonnegative solution of inequality (2.2). Denote
(2.3)un+1(t)=∑k=1mak(t)exp⁡{∫thk(t)un(s)ds},n=0,1,…,
then
(2.4)u1(t)=∑k=1mak(t)exp⁡{∫thk(t)u0(s)ds}≤u0(t).
By induction we have 0≤un+1(t)≤un(t)≤u0(t). Hence, there exists a pointwise limit u(t)=limn→∞un(t). By the Lebesgue convergence theorem, we have
(2.5)u(t)=∑k=1mak(t)exp⁡{∫thk(t)u(s)ds}.
Then, the function
(2.6)x(t)=x(t0)exp⁡{∫t0tu(s)ds}foranyx(t0)>0
is a positive solution of (2.1).Corollary 2.3.
If(2.7)∫tmax⁡khk(t)∑i=1mai(s)ds≤1e,t≥t0,
then (2.1) has a positive solution for t≥t0.Proof.
Letu0(t)=e∑k=1mak(t), then u0 satisfies (2.2) at any point t where ∑k=1mak(t)=0. In the case when ∑k=1mak(t)≠0, inequality (2.7) implies
(2.8)u0(t)∑k=1mak(t)exp⁡{∫thk(t)u0(s)ds}≥u0(t)∑k=1mak(t)exp⁡{∫tmax⁡khk(t)u0(s)ds}=e∑k=1mak(t)∑k=1mak(t)exp⁡{e∫tmax⁡khk(t)∑i=1mai(s)ds}≥e∑k=1mak(t)∑k=1mak(t)e=1.
Hence, u0(t) is a positive solution of inequality (2.2). By Theorem 2.2, (2.1) has a positive solution for t≥t0.Corollary 2.4.
If there existsσ>0 such that hk(t)-t≤σ and ∫0∞∑k=1mak(s)ds<∞, then (2.1) has an eventually positive solution.Corollary 2.5.
If there existsσ>0 such that hk(t)-t≤σ and limt→∞ak(t)=0, then (2.1) has an eventually positive solution.Proof.
Under the conditions of either Corollary2.4 or Corollary 2.5, obviously there exists t0≥0 such that  (2.7) is satisfied.Theorem 2.6.
Let∫∞∑k=1mak(s)ds=∞ and  x be an eventually positive solution of (2.1), then  limt→∞x(t)=∞.Proof.
Suppose thatx(t)>0 for t≥t1, then ẋ(t)≥0 for t≥t1 and
(2.9)ẋ(t)≥∑k=1mak(t)x(t1),t≥t1,
which implies
(2.10)x(t)≥x(t1)∫t1t∑k=1mak(s)ds.
Thus, limt→∞x(t)=∞.Consider together with (2.1) the following equation:(2.11)ẋ(t)-∑k=1mbk(t)x(gk(t))=0,
for t≥t0. We assume that for (2.11) conditions (a1)-(a2) also hold.Theorem 2.7.
Suppose thatt≤gk(t)≤hk(t),  0≤bk(t)≤ak(t),  t≥t0, and the conditions of Theorem 2.2 hold, then (2.11) has a positive solution for  t≥t0.Proof.
Letu0(t)≥0 be a solution of inequality (2.2) for t≥t0, then this function is also a solution of this inequality if ak(t) and hk(t) are replaced by bk(t) and gk(t). The reference to Theorem 2.2 completes the proof.Corollary 2.8.
Suppose that there existak>0 and σk>0 such that 0≤ak(t)≤ak,  t≤hk(t)≤t+σk, t≥t0, and the inequality
(2.12)λ≥∑k=1makeλσk
has a solution λ≥0, then (2.1) has a positive solution for t≥t0.Proof.
Consider the equation with constant parameters(2.13)ẋ(t)-∑k=1makx(t+σk)=0.
Since the function u(t)≡λ is a solution of inequality (2.2) corresponding to (2.13), by Theorem 2.2, (2.13) has a positive solution. Theorem 2.7 implies this corollary.Corollary 2.9.
Suppose that0≤ak(t)≤ak, t≤hk(t)≤t+σ for t≥t0, and
(2.14)∑k=1mak≤1eσ,
then (2.1) has a positive solution for t≥t0.Proof.
Since∑k=1mak≤1/eσ, the number λ=1/σ is a positive solution of the inequality
(2.15)λ≥(∑k=1mak)eλσ,
which completes the proof.Consider now the equation with positive coefficients(2.16)ẋ(t)+∑k=1mak(t)x(hk(t))=0.Theorem 2.10.
Suppose thatak(t)≥0 are continuous functions bounded on  [t0,∞) and  hk are equicontinuous functions on  [t0,∞) satisfying  0≤hk(t)-t≤δ, then (2.16) has a nonoscillatory solution.Proof.
In the spaceC[t0,∞) of continuous functions on [t0,∞), consider the set
(2.17)M={u∣0≤u≤∑k=1mak(t)},
and the operator
(2.18)(Hu)(t)=∑k=1mak(t)exp⁡{-∫thk(t)u(s)ds}.
If u∈M, then Hu∈M.
For the integral operator(2.19)(Tu)(t)∶=∫thk(t)u(s)ds,
we will demonstrate that TM is a compact set in the space C[t0,∞). If u∈M, then
(2.20)‖(Tu)(t)‖C[t0,∞)≤sup⁡t≥t0∫tt+δ|u(s)|ds≤sup⁡t≥t0∑k=1mak(t)δ<∞.
Hence, the functions in the set TM are uniformly bounded in the space C[t0,∞).
Functionshk are equicontinuous on [t0,∞), so for any ɛ>0, there exists a σ0>0 such that for |t-s|<σ0, the inequality
(2.21)|hk(t)-hk(s)|<ɛ2(sup⁡t≥t0∑k=1mak(t))-1,k=1,…,m
holds. From the relation
(2.22)∫t0hk(t0)-∫thk(t)=∫t0t+∫thk(t0)-∫thk(t0)-∫hk(t0)hk(t)=∫t0t-∫hk(t0)hk(t),
we have for |t-t0|<min{σ0,ɛ/2supt≥t0∑k=1mak(t)} and u∈M the estimate
(2.23)|(Tu)(t)-(Tu)(t0)|=|∫thk(t)u(s)-∫t0hk(t0)u(s)ds|≤∫t0t|u(s)|ds+∫hk(t0)hk(t)|u(s)|ds≤|t-t0|sup⁡t≥t0∑k=1mak(t)+|hk(t)-hk(t0)|sup⁡t≥t0∑k=1mak(t)<ɛ2+ɛ2=ɛ.
Hence, the set TM contains functions which are uniformly bounded and equicontinuous on [t0,∞), so it is compact in the space C[t0,∞); thus, the set HM is also compact in C[t0,∞).
By the Schauder fixed point theorem, there exists a continuous functionu∈M such that u=Hu, then the function
(2.24)x(t)=exp⁡{-∫t0tu(s)ds}
is a bounded positive solution of (2.16). Moreover, since u is nonnegative, this solution is nonincreasing on [t0,∞).Theorem 2.11.
Suppose that the conditions of Theorem2.10 hold,
(2.25)∫t0∞∑k=1mak(s)ds=∞,
and x is a nonoscillatory solution of (2.16), then limt→∞x(t)=0.Proof.
Letx(t)>0 for t≥t0, then ẋ(t)≤0 for t≥t0. Hence, x(t) is nonincreasing and thus has a finite limit. If limt→∞x(t)=d>0, then x(t)>d for any t, and thus ẋ(t)≤-d∑k=1mak(t) which implies limt→∞x(t)=-∞. This contradicts to the assumption that x(t) is positive, and therefore limt→∞x(t)=0.Let us note that we cannot guarantee any (exponential or polynomial) rate of convergence to zero even for constant coefficientsak, as the following example demonstrates.Example 2.12.
Consider the equationẋ(t)+x(h(t))=0, where h(t)=ttlnt, t≥3, x(3)=1/ln3. Then, x(t)=1/(lnt) is the solution which tends to zero slower than t-r for any r>0.Consider now the advanced equation with positive and negative coefficients(2.26)ẋ(t)-∑k=1m[ak(t)x(hk(t))-bk(t)x(gk(t))]=0,t≥0.Theorem 2.13.
Suppose thatak(t) and  bk(t) are Lebesgue measurable locally essentially bounded functions,  ak(t)≥bk(t)≥0,  hk(t) and  gk(t) are Lebesgue measurable functions,  hk(t)≥gk(t)≥t, and inequality (2.2) has a nonnegative solution, then (2.26) has a nonoscillatory solution; moreover, it has a positive nondecreasing and a negative nonincreasing solutions.Proof.
Letu0 be a nonnegative solution of (2.2) and denote
(2.27)un+1(t)=∑k=1m(ak(t)exp⁡{∫thk(t)un(s)ds}-bk(t)exp⁡{∫tgk(t)un(s)ds}),t≥t0,n≥0.
We have u0≥0, and by (2.2),
(2.28)u0≥∑k=1mak(t)exp⁡{∫thk(t)u0(s)ds}≥∑k=1m(ak(t)exp⁡{∫thk(t)u0(s)ds}-bk(t)exp⁡{∫tgk(t)u0(s)ds})=u1(t).
Since ak(t)≥bk(t)≥0 and t≤gk(t)≤hk(t), then u1(t)≥0.
Next, let us assume that0≤un(t)≤un-1(t). The assumptions of the theorem imply un+1≥0. Let us demonstrate that un+1(t)≤un(t). This inequality has the form
(2.29)∑k=1m(ak(t)exp⁡{∫thk(t)un(s)ds}-bk(t)exp⁡{∫tgk(t)un(s)ds})≤∑k=1m(ak(t)exp⁡{∫thk(t)un-1(s)ds}-bk(t)exp⁡{∫tgk(t)un-1(s)ds}),
which is equivalent to
(2.30)∑k=1mexp⁡{∫thk(t)un(s)ds}(ak(t)-bk(t)exp⁡{-∫gk(t)hk(t)un(s)ds})≤∑k=1mexp⁡{∫thk(t)un-1(s)ds}(ak(t)-bk(t)exp⁡{-∫gk(t)hk(t)un-1(s)ds}).
This inequality is evident for any 0≤un(t)≤un-1(t), ak(t)≥0, and bk≥0; thus, we have un+1(t)≤un(t).
By the Lebesgue convergence theorem, there is a pointwise limitu(t)=limn→∞un(t) satisfying
(2.31)u(t)=∑k=1m(ak(t)exp⁡{∫thk(t)u(s)ds}-bk(t)exp⁡{∫tgk(t)u(s)ds}),t≥t0,u(t)≥0, t≥t0. Then, the function
(2.32)x(t)=x(t0)exp⁡{∫t0tu(s)ds},t≥t0
is a positive nondecreasing solution of (2.26) for any x(t0)>0 and is a negative nonincreasing solution of (2.26) for any x(t0)<0.Corollary 2.14.
Letak(t) and bk(t) be Lebesgue measurable locally essentially bounded functions satisfying ak(t)≥bk(t)≥0, and let hk(t) and gk(t) be Lebesgue measurable functions, where hk(t)≥gk(t)≥t. Assume in addition that inequality (2.7) holds. Then, (2.26) has a nonoscillatory solution.Consider now the equation with constant deviations of advanced arguments(2.33)ẋ(t)-∑k=1m[ak(t)x(t+τk)-bk(t)x(t+σk)]=0,
where ak,bk are continuous functions, τk≥0, σk≥0.DenoteAk=supt≥t0ak(t), ak=inft≥t0ak(t), Bk=supt≥t0bk(t), bk=inft≥t0bk(t).Theorem 2.15.
Suppose thatak≥0,  bk≥0,  Ak<∞, and Bk<∞.
If there exists a numberλ0<0 such that
(2.34)∑k=1m(akeλ0τk-Bk)≥λ0,(2.35)∑k=1m(Ak-bkeλ0σk)≤0,
then (2.33) has a nonoscillatory solution; moreover, it has a positive nonincreasing and a negative nondecreasing solutions.Proof.
In the spaceC[t0,∞), consider the set M={u∣λ0≤u≤0} and the operator
(2.36)(Hu)(t)=∑k=1m(ak(t)exp⁡{∫tt+τku(s)ds}-bk(t)exp⁡{∫tt+σku(s)ds}).
For u∈M, we have from (2.34) and (2.35)
(2.37)(Hu)(t)≤∑k=1m(Ak-bkeλ0σk)≤0,(Hu)(t)≥∑k=1m(akeλ0τk-Bk)≥λ0.
Hence, HM⊂M.
Consider the integral operator(2.38)(Tu)(t)∶=∫tt+δu(s)ds,δ>0.
We will show that TM is a compact set in the space C[t0,∞). For u∈M, we have
(2.39)‖(Tu)(t)‖C[t0,∞)≤sup⁡t≥t0∫tt+δ|u(s)|ds≤|λ0|δ.
Hence, the functions in the set TM are uniformly bounded in the space C[t0,∞).
The equality∫t0t0+δ-∫tt+δ=∫t0t+∫tt0+δ-∫tt0+δ-∫t0+δt+δ=∫t0t-∫t0+δt+δ implies
(2.40)|(Tu)(t)-(Tu)(t0)|=|∫tt+δu(s)-∫t0t0+δu(s)ds|≤∫t0t|u(s)|ds+∫t0+δt+δ|u(s)|ds≤2|λ0||t-t0|.
Hence, the set TM and so the set HM are compact in the space C[t0,∞).
By the Schauder fixed point theorem, there exists a continuous functionu satisfying λ0≤u≤0 such that u=Hu; thus, the function
(2.41)x(t)=x(t0)exp⁡{∫t0tu(s)ds},t≥t0
is a positive nonincreasing solution of (2.33) for any x(t0)>0 and is a negative nondecreasing solution of (2.26) for any x(t0)<0.Let us remark that (2.35) for any λ0<0 implies ∑k=1m(Ak-bk)<0.Corollary 2.16.
Let∑k=1m(Ak-bk)<0, ∑k=1mAk>0, and for
(2.42)λ0=ln⁡(∑k=1mAk/∑k=1mbk)max⁡kσk,
the inequality
(2.43)∑k=1m(akeλ0τk-Bk)≥λ0
holds, then (2.33) has a bounded positive solution.Proof.
The negative numberλ0 defined in (2.42) is a solution of both (2.34) and (2.35); by definition, it satisfies (2.35), and (2.43) implies (2.34).Example 2.17.
Consider the equation with constant advances and coefficients(2.44)ẋ(t)-ax(t+r)+bx(t+d)=0,
where 0<a<b, d>0, r≥0. Then, λ0=(1/d)ln(a/b) is the minimal value of λ for which inequality (2.35) holds; for (2.44), it has the form a-beλd≤0.
Inequality (2.34) for (2.44) can be rewritten as
(2.45)f(λ)=aeλr-b-λ≥0,
where the function f(x) decreases on (-∞,-ln(ar)/r] if τ>0 and for any negative x if r=0; besides, f(0)<0. Thus, if f(λ1)<0 for some λ1<0, then f(λ)<0 for any λ∈[λ1,0). Hence, the inequality
(2.46)f(λ0)=a(ab)r/d-b-1dln⁡(ab)≥0
is necessary and sufficient for the conditions of Theorem 2.15 to be satisfied for (2.44).
Figure1 demonstrates possible values of advances d and r, such that Corollary 2.16 implies the existence of a positive solution in the case 1=a<b=2. Then, (2.46) has the form 0.5r/d≥2-(ln2)/d, which is possible only for d>0.5ln2≈0.347 and for these values is equivalent to
(2.47)r≤-dln⁡(2-ln⁡2/d)ln⁡2.Figure 1
The domain of values(d,r) satisfying inequality (2.47). If the values of advances d and r are under the curve, then (2.44) has a positive solution.


## 3. Comments and Open Problems

In this paper, we have developed nonoscillation theory for advanced equations with variable coefficients and advances. Most previous nonoscillation results deal with either oscillation or constant deviations of arguments. Among all cited papers, only [8] has a nonoscillation condition (Theorem 2.11) for a partial case of (2.1) (with hk(t)=t+τk), which in this case coincides with Corollary 2.4. The comparison of results of the present paper with the previous results of the authors was discussed in the introduction.Finally, let us state some open problems and topics for research.(1)
Prove or disprove: if (2.1), with ak(t)≥0, has a nonoscillatory solution, then (2.26) with positive and negative coefficients also has a nonoscillatory solution.As the first step in this direction, prove or disprove that ifh(t)≥t and the equation (3.1)ẋ(t)-a+(t)x(h(t))=0has a nonoscillatory solution, then the equation(3.2)ẋ(t)-a(t)x(h(t))=0also has a nonoscillatory solution, wherea+(t)=max{a(t),0}.If these conjectures are valid, obtain comparison results for advanced equations.(2)
Deduce nonoscillation conditions for (2.1) with oscillatory coefficients. Oscillation results for an equation with a constant advance and an oscillatory coefficient were recently obtained in [22].(3)
Consider advanced equations with positive and negative coefficients when the numbers of positive and negative terms do not coincide.(4)
Study existence and/or uniqueness problem for the initial value problem or boundary value problems for advanced differential equations.



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