# Existence of Oscillatory Solutions of Second Order Delay Differential Equations with Distributed Deviating Arguments

Authors: Youjun Liu; Jianwen Zhang; Jurang Yan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Under weaker hypothesis, we use the Schauder-Tychonoff theorem to obtain new sufficient condition for the global existence
of oscillatory solutions for forced second order nonlinear delay differential
equations with distributed deviating arguments.


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## Body

## 1. Introduction

In this paper, we study the existence of oscillatory solutions for the nonlinear second order delay differential equations with the perturbed term(1)[r(t)Φ(x′(t))]′+∫ab‍p(t,τ)f(x(t-τ))dτ=q(t),t≥t0.Under the following conditions:(1)
r∈C1([t0,∞),R+),p∈C([t0,∞)×[a,b],R),q∈C([t0,∞),R),fi∈C([t0,∞),R);(2)
Φ∈C1(R,R),Φ(u) is increasing function for all u∈R, Φ-1(u) satisfies the local Lipischitz condition.During the past three decades, the investigation of oscillatory theory for delay differential equations and delay dynamic equations has attracted attention of numerous researchers due to their significance in theory and applications. We mention here the monographs of Myshkis [1], Norkin [2], Shevelo [3], and Agarwal et al. [4]. The oscillation properties of second order delay differential equations were considered also in Koplatadze et al. [5], Shmul’yan [6], and Skubachevskii [7]. Distances between adjacent zeros of oscillating solutions are estimated in [8, 9] for delay and for neutral second order equation in [10]. Distances between zero of solution and zero of its derivative were estimated in [11]. Based on oscillation properties, asymptotic properties of second order delay differential equations were studied in [12]. For related work, we refer the reader to the references [13–24]. However, to the best of our knowledge, the existence of oscillatory solutions for differential equation with distributed deviating arguments has been scarcely investigated. Thus, the research presents its significance.As usual, a solution of (1) is a function x(t) defined on [t0-b,∞) such that x(t) and r(t)Φ(x′(t)) are continuously differentiable on [t0-b,∞) and [t0,∞) and (1) holds. Our attention will be restricted to the solution x(t) of (1) which satisfy sup|x(t)|>0, for t≥t0-b. Such a solution is said to be oscillatory if it has a sequence of zeros tending to infinity. Otherwise, it is said to be nonoscillatory.The purpose of this paper is to prove a general result for (1) on the existence of oscillatory solutions. Throughout this paper, we will use the following notations. For a constant γ>0,θγ=max|x|≤γ|f(x)|,t≥t0,Lγ denote the local lipischitz constants of functions Φ-1(u).


## 2. The Main Results

Lemma 1 (see [13]).
LetX be a locally convex space, K⊂X nonempty and convex, S⊂K, and S compact. Given a continuous map F:K→S, there exists x~∈S such that F(x~)=x~.Theorem 2.
Assume that there existη,γ>0 such that r(t)>η,
(2)1r(t)∫t∞‍q(s)dsisintegrableon[t0,∞),θγr(t)∫t∞‍∫ab‍p(s,τ)dτdsisintegrableon[t0,∞);
moreover, there exist two sequences {tn}, {sn} with tn→∞,sn→∞ such that
(3)∫tn∞‍Φ-1(1r(s)∫s∞‍(q(u)+θγ∫ab‍p(u,τ)dτ)du)ds<0,∫sn∞‍Φ-1(1r(s)∫s∞‍(q(u)-θγ∫ab‍p(u,τ)dτ)du)ds>0.
Then (1), has an oscillatory solution x(t) defined on [t0,∞) with |x|≤γ, and limt→∞x(t)=0.Proof.
The proof is based on an application of the well-known Schauder-Tychonoff fixed point theorem. From (2), for any γ>0, we have to choose a large Tγ≥T such that for all t≥Tγ,
(4)∫t∞‍Φ-1(1r(s)∫s∞‍(q(u)+θγ∫ab‍p(u,τ)dτ)du)ds≤γ,∫t∞‍Φ-1(1r(s)∫s∞‍(q(u)-θγ∫ab‍p(u,τ)dτ)du)ds≥-γ.
LetC[t0-b,∞) denote the locally convex space of all continuous functions with topology of uniform convergence on compact subsets of [t0-b,∞). Let S={x∈C[t0-b,∞),|x(t)|≤γ}. Clearly, S is a close convex subset of C[t0-b,∞).
Introduce an operatorF by
(5)(Fx)(t)={∫t∞Φ-1({∫ab‍p(u,τ)}∫ab‍1r(s)00000000×∫s∞‍(∫ab‍{∫ab‍p(u,τ)(x(t-τ))dτ∫abt‍}q(u)000000000000000-∫ab‍p(u,τ)00000000000000000×f(x(t-τ))dτ∫abt‍)du)ds,0000000000000000000000000000000000t>Tγ,(Fx)(Tγ),t0-b≤t≤Tγ.
It is easy to see that, for anyx∈S, (Fx)(t) is well defined on [t0-b,∞) continuously.
From (4), we obtain
(6)(Fx)(t)≤∫t∞‍Φ-1(1r(s)∫s∞‍({+θγ∫ab‍p(u,τ)dτ}q(u)+θγ∫ab‍p(u,τ)dτ)du)ds≤γ,t≥t0-b,(Fx)(t)≥∫t∞‍Φ-1(1r(s)∫s∞(∫ab‍q(u)-θγ‍×∫ab‍p(u,τ)dτ)du)ds≥-γ,t≥t0-b.
Hence, |(Fx)(t)|≤γ. Thus, we have FS⊂S and Fx is uniformly bounded on S.
Let{xn}n=1∞∈S be any sequence and x0∈S with limn→∞xn=x0. Let T1 be large constant with T1>T, for any ϵ>0 so that
(7)∫T1∞‍1r(s)∫s∞‍(∫ab‍p(s,τ)dτ)duds<ϵ3θγLγ.
From the compactness of the domain offi, there exists a large N(ϵ)>0 and a constant δ(ϵ)>0; let t∈[t0-b,T1] and n≥N when |xn-x0|<δ(ϵ),
(8)maxt0-b≤t≤T1{|f(xn(t-τ))-f(x0(t-τ))|}≤ϵ3LγM,
where M=∫t0-bT1‍(s-t0+b/r(s))∫ab‍p(s,τ)dτds. By virtue of (1)–(8), we have that for any t≥t0-b and |xn-x0|<δ,
(9)|(Fxn)(t)-(Fx0)(t)|=|∫t∞‍Φ-1(∫ab‍1r(s)×∫s∞‍(q(u)-∫ab‍p(u,τ)×f(xn(t-τ))dτ∫ab‍)du)ds-∫t∞‍Φ-1(∫ab‍1r(s)×∫s∞‍(q(u)-∫ab‍p(u,τ)×f(x0(t-τ))dτ∫ab‍)du)ds|≤∫t∞‍|Φ-1(∫ab‍1r(s)×∫s∞‍(q(u)-∫ab‍p(u,τ)×f(xn(t-τ))dτ∫ab‍)du)ds-Φ-1(∫ab‍1r(s)×∫s∞‍(q(u)-∫ab‍p(u,τ)×f(x0(t-τ))dτ∫ab‍)du)ds|ds≤Lγ∫t∞‍(∫ab‍1r(s)×∫s∞(|∫ab‍p(u,τ)|×|f(xn(u-τ))-f(x0(u-τ))|dτ|∫ab‍p(u,τ)|)du)ds≤Lγ∫t0-bT1‍(∫ab‍1r(s)×∫s∞‍(|∫ab‍p(u,τ)|×|f(xn(u-τ))-f(x0(u-τ))|dτ|∫ab‍p(u,τ)|)du)ds+Lγ∫T1∞‍(∫ab‍1r(s)×∫s∞‍(|∫ab‍p(u,τ)|×|f(xn(u-τ))-f(x0(u-τ))|dτ|∫ab‍p(u,τ)|)du)ds≤Lγ∫t0-bT1‍(∫ab‍1r(s)(s-t0+b)×(|∫ab‍p(s,τ)|×|f(xn(s-τ))-f(x0(s-τ))|dτ|∫ab‍p(u,τ)|))ds+2θγLγ∫T1∞‍1r(s)∫s∞‍(∫ab‍p(u,τ)dτ)du)ds<ϵ3+2ϵ3=ϵ.
The continuity of F on S is proved.
Moreover, for allt2,t1>t0-b,
(10)(Fx)(t2)-(Fx)(t1)=∫t1t2‍Φ-1({(u)-∫ab‍p(u,τ)}1r(s)×∫s∞‍(q(u)-∫ab‍p(u,τ)×f(x(t-τ))dτ∫ab‍)du1r(s))ds≤∫t1t2‍Φ-1(1r(s)∫s∞‍(q(u)+θγ∫ab‍p(u,τ)dτ)du)ds≤α(t2-t1),
where α=supt≥t0Φ-1(1/r(s)∫t∞‍(q(u)+θγ∫ab‍p(u,τ)dτ)du);
(11)(Fx)(t2)-(Fx)(t1)=∫t1t2‍Φ-1(∫ab‍1r(s)×∫s∞(q(u)+∫ab‍p(u,τ)‍×f(x(t-τ))dτ∫ab‍)du)ds≥∫t1t2‍Φ-1(1r(s)∫s∞‍(q(u)-θγ∫ab‍p(u,τ)dτ)du)ds≥β(t2-t1),
where β=inft≥t0Φ-1(1/r(s)∫t∞‍(q(u)-θγ∫ab‍p(u,τ)dτ)du). Thus,
(12)|(Fx)(t2)-(Fx)(t1)|≤M|t2-t1|,
where M=max{|α|,|β|}. This implies that Fx is equicontinuous. Hence, by the Ascoli-Arzela Theorem, the operator is completely continuous on S. By Lemma 1, there exists x~∈S satisfying
(13)x~(t)=(Fx~)(t)={∫t∞‍Φ-1(1r(s)00000000×∫s∞‍({∫t∞‍Φ-1∫t∞‍Φ-1∫t∞‍Φ-1(1r(s)}{∫ab‍p(u,τ)}q(u)00000000000-∫ab‍p(u,τ)00000000000000×f(x~(t-τ))dτ{∫t∞‍Φ-1∫t∞‍Φ-1(1r(s)})du{∫t∞‍Φ-1∫t∞‍Φ-1(1r(s)})ds,0000000000000000000000t≥T,(Fx~)(T),t0-b≤t<T.
On the other hand, from (3), we find
(14)x~(tn)≤∫tn∞‍Φ-1×(1r(s)∫s∞‍(q(u)+θγ∫ab‍p(u,τ)dτ)du)ds<0,x~(sn)≥∫sn∞‍Φ-1×(1r(s)∫s∞‍(q(u)-θγ∫ab‍p(u,τ)dτ)du)ds>0,
which implies that x~(t) is a bounded oscillatory solution of (1) and limt→∞x~(t)=0. The proof is complete.Corollary 3.
Assume that (2) of Theorem 2 holds, and specially Φ(u)=uα,α≥1 is the ratio of two positive odd integers, there exist two increasing divergent sequences {tn} and {sn},tn,sn such that
(15)∫tn∞(1r(s)∫s∞(q(u)+θγ∫abp(u,τ)dτ‍)du‍)1/αds<0,‍∫sn∞(1r(s)∫s∞(q(u)-θγ∫abp(u,τ)dτ‍)du‍)1/α‍ds>0.
Then, (1) has an oscillatory solution x(t) defined on [t0,∞) with |x|≤γ, and limt→∞x(t)=0.


## 3. Remark

Whenr(t)≡1,Φ(u)=u,f(v)=v, and q(t)=0. Let s=t-τ, (1) becomes
(16)x′′(t)-∫t-bt-a‍k(t,s)x(s)ds=0.According to the results of this paper,∫t∞‍∫ab‍k(u,u-τ)dτdu is integrable on [t0,∞), and there exist two sequences {tn}, {sn} with tn→∞,sn→∞ such that
(17)∫tn∞‍(∫s∞‍∫ab‍k(u,u-τ)dτdu)ds<0,∫sn∞‍(∫s∞‍∫ab‍k(u,u-τ)dτdu)ds>0.
Equation (16) has oscillatory solution.In paper [14], it was demonstrated that the inequality b(b-a)max0≤s≤t<∞k(t,s)≤2/e implied the existence of positive solutions. Neither of the two conditions can be deduced from each other. Moreover, when k(t,s)=s/1+t2(s=t-τ) is bounded, but ∫t∞‍∫ab‍k(u,u-τ)dτdu cannot exist. Likewise, when k(t,s)=5/1+t2 and b(b-a)≥2/e claim, ∫t∞‍∫ab‍k(u,u-τ)dτdu can exist, but b(b-a)max0≤s≤t<∞k(t,s)≤2/e cannot be hold.


## 4. Examples

Example 1.
Consider second order delay differential equations(18)(e-tx′(t))′+2e2π+eπ∫π2πe-2tx(t-τ)dτ‍=e-2t(sint-3cost)+e-3t(cost+sint).
Here,r(t)=e-t,Φ(u)=u,f(v)=v,p(t,τ)=2/((e2π+eπ)e-2t),a=π,b=2π, and q(t)=e-2t(sint-3cost)+e-3t(cost+sint).
It is easy to see thatr(t)≥η>0.
We have(19)Q(t)∶=∫t∞‍Φ-1×(1r(s)∫s∞‍(q(u)+2γe2π+eπ∫π2π‍e-2udτ)du)ds=∫t∞‍es∫s∞‍(2πγe2π+eπe-2u(sinu-3cosu)+e-3u(cosu+sinu)+2πγe2π+eπe-2u)duds=e-tsint+425e-2t(cost+34sint)-πγe2π+eπe-t,P(t)∶=∫t∞‍Φ-1×(1r(s)∫s∞‍(q(u)-2γe2π+eπ∫π2π‍e-2udτ)du)ds=∫t∞‍es∫s∞‍(2πγe2π+eπe-2u(sinu-3cosu)+e-3u(cosu+sinu)-2πγe2π+eπe-2u)duds=e-tsint+425e-2t(cost+34sint)+πγe2π+eπe-t.
Let tn=(2n-1)π,sn=(2n+1)π,n=1,2,…,
(20)Q(tn)=-e-tn(425e-tn+πγe2π+eπ)<0,P(sn)=e-sn(425e-sn+πγe2π+eπ)>0.
It is easy to see from (19) that there exists a N=1 such that for all n≥N, Q(tn)<0 and P(tn)>0. Thus, by Theorem 2, (18) has an oscillatory solution x(t) on [t0,∞) and limt→∞x(t)=0. It is not difficult to check that (18) has the oscillatory solution x(t)=e-tsint.Example 2.
Consider second order delay differential equations(21)(e-t(x′(t))1/3)′+67γ3∫12τ2e-2tx3(t-τ)dτ‍=e-2t(cost-2sint).
Here, r(t)=e-t,Φ(u)=u1/3,p(t,τ)=(6/7γ3)τ2e-2tf(v)=v3,a=1,b=2, and q(t)=e-2t(cost-2sint). It is easy to see that r(t)≥η>0,
(22)Q(t)∶=∫t∞‍Φ-1(1r(s)∫s∞‍(q(u)+67∫12‍τ2e-2udτ)du)ds=∫t∞‍(es∫s∞‍e-2u(cosu-2sinu)+2e-2udu)3ds=16e-3tsin2t(-sint-cost)+e-3t(-3sint-cost)-313e-3tsint(-3sint-2cost)+1939e-3t,P(t)∶=∫t∞‍Φ-1(1r(s)∫s∞‍(q(u)+67∫12‍τ2e-2udτ)du)ds=∫t∞‍(es∫s∞‍e-2u(cosu-2sinu)-2e-2udu)3ds=16e-3tsin2t(-sint-cost)+e-3t(-3sint-cost)+313e-3tsint(-3sint-2cost)+739e-3t.
Lettn=2nπ,sn=(2n+1)π,n=1,2,…,
(23)Q(tn)=-2039e-3tn,P(sn)=4639e-3sn.
It is easy to see from (22) that there exists a N=1 such that for all n≥N,Q(tn)<0 and P(sn)>0. Thus, by Theorem 2, (21) has an oscillatory solution x(t) on [t0,∞) and limt→∞x(t)=0.



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