# Oscillation of Second-Order Sublinear Impulsive Differential Equations

Authors: A. Zafer
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type:x″(t)+p(t)|x(τ(t))|α-1x(τ(t))=0, t≠θk; Δx′(t)|t=θk+qk|x(τ(θk))|α-1x(τ(θk))=0; Δx(t)|t=θk=0, (0<α<1) by considering the cases τ(t)≤t and τ(t)=t, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.


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## Body

## 1. Introduction

We deal with second-order sublinear impulsive differential equations of the form(1.1)x′′(t)+p(t)|x(τ(t))|α-1x(τ(t))=0,t≠θk,Δx′(t)|t=θk+qk|x(τ(θk))|α-1x(τ(θk))=0,Δx(t)|t=θk=0,
where 0<α<1, t≥t0, and k≥k0 for some t0∈ℝ+ and k0∈ℕ, {θk} is a strictly increasing unbounded sequence of positive real numbers, (1.2)Δz(t)|t=θ:=z(θ+)-z(θ-),z(θ∓):=lim⁡t→θ∓z(t).LetPLC(J,R) denote the set of all real-valued functions u defined on J such that u is continuous for all t∈J except possibly at t=θk where u(θk±) exists and u(θk):=u(θk-).We assume in the sequel that(a)
p∈PLC([t0,∞),ℝ),(b)
{qk} is a sequence of real numbers,(c)
τ∈C([t0,∞),ℝ+), τ(t)≤t, limt→∞τ(t)=∞.By a solution of (1.1) on an interval J⊂[t0,∞), we mean a function x(t) which is defined on J such that x,x′,x′′∈PLC(J) and which satisfies (1.1). Because of the requirement Δx(t)|t=θk=0 every solution of (1.1) is necessarily continuous.As usual we assume that (1.1) has solutions which are nontrivial for all large t. Such a solution of (1.1) is called oscillatory if it has no last zero and nonoscillatory otherwise.In case there is no impulse, (1.1) reduces to Emden-Fowler equation with delay(1.3)x′′(t)+p(t)|x(τ(t))|α-1x(τ(t))=0,0<α<1,
and without delay(1.4)x′′+p(t)|x|α-1x=0,0<α<1.The problem of oscillation of solutions of (1.3) and (1.4) has been considered by many authors. Kusano and Onose [1] see also [2, 3] proved the following necessary and sufficient condition for oscillation of (1.3).Theorem 1.1.
Ifp(t)≥0, then a necessary and sufficient condition for every solution of (1.3) to be oscillatory is that
(1.5)∫∞[τ(t)]αp(t)dt=∞.The conditionp(t)≥0 is required only for the sufficiency part, and no similar criteria is available for p(t) changing sign, except in the case τ(t)=t. Without imposing a sign condition on p(t), Belohorec [4] obtained the following sufficient condition for oscillation of (1.4).Theorem 1.2.
If(1.6)∫∞tβp(t)dt=∞
for some β∈[0,α], then every solution of (1.4) is oscillatory.Compared to the large body of papers on oscillation of differential equations, there is only little known about the oscillation of impulsive differential equations; see [5–7] for equations with delay and [8–13] for equations without delay. For some applications of such equations, we may refer to [14–18]. The books [19, 20] are good sources for a general theory of impulsive differential equations.The object of this paper is to extend Theorems1.1 and 1.2 to impulsive differential equations of the form (1.1). The results show that the impulsive perturbations may greatly change the oscillatory behavior of the solutions. A nonoscillatory solution of (1.3) or (1.4) may become oscillatory under impulsive perturbations.The following two lemmas are crucial in the proof of our main theorems. The first lemma is contained in [21] and the second one is extracted from [22].Lemma 1.3.
If eachAi is continuous on [a,b], then
(1.7)∫ab∑s≤θi<bAi(s)ds=∑a≤θi<b∫aθiAi(s)ds.Lemma 1.4.
FixJ=[a,b], let u,λ∈C(J,ℝ+), h∈C(ℝ+,ℝ+), and c∈ℝ+, and let {λk} a sequence of positive real numbers. If u(J)⊂I⊂ℝ+ and
(1.8)u(t)≤c+∫atλ(s)h(u(s))ds+∑a<θk<tλkh(u(θk)),t∈J,
then
(1.9)u(t)≤G-1{G(c)+∫atλ(s)ds+∑a<θk<tλk},t∈[a,β),
where
(1.10)G(u)=∫u0udxh(x),u,u0∈I,β=sup⁡{ν∈J:G(c)+∫atλ(s)ds+∑a<θk<tλk∈G(I),a≤t≤ν}.


## 2. The Main Results

We first establish a necessary and sufficient condition for oscillation of solutions of (1.1) when τ(t)≤t.Theorem 2.1.
If(2.1)∫∞[τ(t)]α|p(t)|dt+∑∞[τ(θk)]α|qk|<∞,
then (1.1) has a solution x(t) satisfying
(2.2)lim⁡t→∞x(t)t=a≠0.Proof.
Chooset1≥max{1,t0}. In view of Lemma 1.3 by integrating (1.1) twice from t0 to t, we obtain
(2.3)x(t)=x(t1)-x′(t1)(t-t1)-∑t1≤θk<tqk|x(τ(θk))|α-1x((τ(θk)))(t-θk)-∫t1t(t-s)p(s)|x(τ(s))|α-1x((τ(s)))ds,t≥t1.
Set
(2.4)u(t)=c+∑t1≤θk<t|qk||x(τ(θk))|α+∫t1t|p(s)||x(τ(s))|αds,t≥t1,
where c=|x(t1)|+|x′(t1)|. Then
(2.5)|x(t)|≤tu(t),t≥t1.
Let t2≥t1 be such that τ(t)≥t1 for all t≥t2. Replacing t by τ(t) in (2.5) and using the increasing character of u(t), we see that
(2.6)|x(τ(t))|≤τ(t)u(t),t≥t2.
From (2.4), we also see that
(2.7)u′(t)=|p(t)||x(τ(t))|α,t≠θk,(2.8)Δu(t)|t=θk=|qk||x(τ(θk))|α
for t≥t2 and θk≥t2. Now, in view of (2.6) and (2.8), an integration of (2.7) from t2 to t leads to
(2.9)u(t)≤c+∫t2t|p(s)|[τ(s)]α[u(s)]αds+∑t2≤θk<t|qk|[τ(θk)]α[u(θk)]α.
Applying Lemma 1.4 with
(2.10)h(x)=xα,λ(s)=|p(s)|[τ(s)]α,λk=|qk|[τ(θk)]α,
we easily see that
(2.11)u(t)≤G-1{G(c)+∫t2t|p(s)|[τ(s)]αds+∑t2≤θk<t|qk|[τ(θk)]α}.
Since
(2.12)G(u)=u1-α1-α-u01-α1-α,G-1(u)=[(1-α)u+u01-α]1/(1-α),
the inequality (2.11) becomes
(2.13)u(t)≤[c1-α+(1-α)∫t1t|p(s)|[τ(s)]αds+(1-α)∑t1≤θk<t|qk|[τ(θk)]α]1/(1-α),
from which, on using (2.1), we have
(2.14)u(t)≤c1,t≥t2,
where
(2.15)c1=[c1-α+(1-α)∫t1∞|p(s)|[τ(s)]αds+(1-α)∑t1≤θk<∞|qk|[τ(θk)]α]1/(1-α).
In view of (2.5), (2.6), and (2.14) we see that
(2.16)|x(t)|≤c1t,|x(τ(t))|≤c1τ(t),t≥t2.
To complete the proof it suffices to show thatx′(t) approaches a nonzero limit as t tends to ∞. To see this we integrate (1.1) from t2 to t to get(2.17)x′(t)=x′(t1)-∫t2tp(s)|x(τ(s))|α-1x(τ(s))ds-∑t2≤θk<tqk|x(τ(θk))|α-1x(τ(θk)).
Employing (2.16) we have
(2.18)∫t2∞|p(s)x(τ(s))|αds≤c1α∫t2∞|p(s)|[τ(s)]αds<∞,∑t2≤θk<∞|qkx(τ(θk))|α≤c1α∑t2≤θk<∞|qk|[τ(θk)]α<∞.
Therefore, limt→∞x′(t)=L exists. Clearly, we can make L≠0 by requiring that
(2.19)x′(t2)>c1α[∫t2∞|p(s)|[τ(s)]αds+∑t2≤θk<∞|qk|[τ(θk)]α],
which is always possible by arranging t2.Theorem 2.2.
Suppose thatp and {qk} are nonnegative. Then every solution of (1.1) is oscillatory if and only if
(2.20)∫∞[τ(t)]αp(t)dt+∑∞[τ(θk)]αqk=∞.Proof.
Let (2.20) fail to hold. Then, by Theorem 2.1 we see that there is a solution x(t) which satisfies (2.2). Clearly, such a solution is nonoscillatory. This proves the necessity.
To show the sufficiency, suppose that (2.20) is valid but there is a nonoscillatory solution x(t) of (1.1). We may assume that x(t) is eventually positive; the case x(t) being eventually negative is similar. Clearly, there exists t1≥t0 such that x(τ(t))>0 for all t≥t1. From (1.1), we have that(2.21)x′′(t)≤0fort≥t1,t≠θk.
Thus, x′(t) is decreasing on every interval not containing t=θk. From the impulse conditions in (1.1), we also have Δx′(θk)≤0. Therefore, we deduce that x′(t) is nondecreasing on [t1,∞).
We may claim thatx′(t) is eventually positive. Because if x′(t)<0 eventually, then x(t) becomes negative for large values of t. This is a contradiction.
It is now easy to show that(2.22)x(t)≥(t-t1)x′(t),t≥t1.
Therefore,
(2.23)x(t)≥t2x′(t),t≥t2=2t1.
Let t3≥t2 be such that τ(t)≥t2 for t≥t3. Using (2.23) and the nonincreasing character of x′(t), we have
(2.24)x(τ(t))≥τ(t)2x′(t),t≥t3,
and so, by (1.1),
(2.25)x′′(t)+2-αp(t)[τ(t)]α[x′(t)]α≤0,t≠θk.
Dividing (2.25) by [x′(t)]α and integrating from t3 to t, we obtain
(2.26)∑t3≤θk<t{[x′(θk)]1-α-[x′(θk)-qk[x(τ(θk))]α]1-α}+[x′(t)]1-α-[x′(t3)]1-α+(1-α)2-α∫t3t[τ(t)]αp(s)ds≤0
which clearly implies that
(2.27)∑t3≤θk<tak+(1-α)2-α∫t3t[τ(t)]αp(s)ds≤[x′(t3)]1-α,
where
(2.28)ak=[x′(θk)]1-α[1-(1-qk[x(τ(θk))]αx′(θk))]1-α.
Since 1-(1-u)1-α≥(1-α)u for u∈(0,∞) and 0<α<1, by taking
(2.29)u=qk[x(τ(θk))]αx′(θk),
we see from (2.28) that
(2.30)ak≥(1-α)qk[x(τ(θk))]α[x′(θk)]α.
But, (2.24) gives
(2.31)x(τ(θk))≥τ(θk)2x′(τ(θk))≥τ(θk)2x′(θk),
and hence
(2.32)ak≥(1-α)2-α[τ(θk)]αqk.
Finally, (2.27) and (2.32) result in(2.33)∫t3∞[τ(t)]αp(t)dt+∑t3<θk<∞[τ(θk)]αqk<∞,
which contradicts (2.20). The proof is complete.Example 2.3.
Consider the impulsive delay differential equation(2.34)x′′(t)+(t-1)-2|x(t-1)|-1/2x(t-1)=0,t≠k,Δx′(t)|t=k+(k-1)-1|x(k-1)|-1/2x(k-1)=0,Δx(t)|t=k=0,
where t≥2 and i≥2.
We see thatτ(t)=t-1, α=1/2, p(t)=(t-1)-2,  and qk=(k-1)-1, θk=k. Since(2.35)∫∞(t-1)-3/2dt+∑∞(k-1)-1/2=∞,
applying Theorem 2.2 we conclude that every solution of (2.34) is oscillatory.
We note that if the equation is not subject to any impulse condition, then, since(2.36)∫∞(t-1)-5/2dt<∞,
the equation
(2.37)x′′(t)+(t-1)-2|x(t-1)|-1/2x(t-1)=0
has a nonoscillatory solution by Theorem 1.1.Let us now consider (1.1) when τ(t)=t. That is,(2.38)x′′+p(t)|x|α-1x=0,t≠θk,Δx′|t=θk+qk|x|α-1x=0,Δx|t=θk=0,
where 0<α<1 and pqk are given by (a) and (b).The following theorem is an extension of Theorem1.2. Note that no sign condition is imposed on p(t) and {qk}.Theorem 2.4.
If(2.39)∫∞tβp(t)dt+∑∞θkβqk=∞
for some β∈[0,α], then every solution of (2.38) is oscillatory.Proof.
Assume on the contrary that (2.38) has a nonoscillatory solution x(t) such that x(t)>0 for all t≥t0 for some t0≥0. The proof is similar when x(t) is eventually negative. We set
(2.40)w(t)=(t-1x(t))1-α,t≥t0.
It is not difficult to see that
(2.41)w′(t)=(α-1)tα-2[x(t)]1-α+(1-α)tα-1[x(t)]-αx′(t),t≠θk,
and hence
(2.42)Δw′|t=θk=(1-α)qkθkα-1.
From (2.41), we have
(2.43)tβ-1-α(t2w′(t))′=(1-α)tβx′′(t)x-α(t)-α(1-α)tβ-2x-α-1(t)[tx′(t)-x(t)]2,
and so
(2.44)tβ-1-α(t2w′(t))′≤(1-α)tβp(t),t≠θk.
In view of (2.42), by a straightforward integration of (2.44), we have
(2.45)∫t0tsβ-1-α(s2w′(s))′ds=sβ-1-αs2w′(s)|t0t-∑t0≤θk<tΔ(tβ-α+1w′(t))|t=θk-∫t0t(β-1-α)sβ-αw′(s)ds=tβ-α+1w′(t)-t0β-α+1w′(t0)-∑t0≤θk<t(1-α)qkθkβ-(β-α-1)[sβ-αw(s)]|t0t+(β-α)(β-α-1)∫t0tsβ-1-αw(s)ds,
which combined with (2.44) leads to
(2.46)tβ-α+1w′(t)≤t0β-α+1w′(t0)-(β-α+1)t0β-αw(t0)+(1-α)[∑t0≤θk<tθkβqk+∫t0tsβp(s)ds].
Finally, by using (2.39) in the last inequality, we see that there is a t1>t0 such that
(2.47)w′(t)≤-tα-β-1,t≥t1,
which, however, implies that w(t)→-∞ as t→∞, a contradiction with x(t)>0. The proof is complete.Example 2.5.
Consider the impulsive differential equation(2.48)x′′+t-7/3|x|-1/2x=0,t≠k,Δx′|t=k+k-1/6|x|-1/2x=0,Δx|t=k=0,
where t≥1 and i≥1.
We have thatp(t)=t7/3, α=1/2, and qk=k-1/6, θk=k. Taking β=1/3 we see from (2.38) that(2.49)∫∞t-2dt+∑∞k-1/3=∞.
Since the conditions of Theorem 2.4 are satisfied, every solution of (2.48) is oscillatory.
Note that if the impulses are absent, then, since(2.50)∫∞t-2dt<∞,
the equation
(2.51)x′′+t-7/3|x|-1/2x=0
is oscillatory by Theorem 1.2.



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