Citation: William R. B. Lionheart. Histogram tomography[J]. Mathematics in Engineering, 2020, 2(1): 55-74. doi: 10.3934/mine.2020004
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In this article, we study the oscillatory behavior of the fourth-order neutral nonlinear differential equation of the form
{(r(t)Φp1[w′′′(t)])′+q(t)Φp2(u(ϑ(t)))=0,r(t)>0, r′(t)≥0, t≥t0>0, | (1.1) |
where w(t):=u(t)+a(t)u(τ(t)) and the first term means the p-Laplace type operator (1<p<∞). The main results are obtained under the following conditions:
L1: Φpi[s]=|s|pi−2s, i=1,2,
L2: r∈C[t0,∞) and under the condition
∫∞t01r1/(p1−1)(s)ds=∞. | (1.2) |
L3: a,q∈C[t0,∞), q(t)>0, 0≤a(t)<a0<∞, τ,ϑ∈C[t0,∞), τ(t)≤t, limt→∞τ(t)=limt→∞ϑ(t)=∞
By a solution of (1.1) we mean a function u ∈C3[tu,∞), tu≥t0, which has the property r(t)(w′′′(t))p1−1∈C1[tu,∞), and satisfies (1.1) on [tu,∞). We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [tu,∞), and otherwise it is called to be nonoscillatory. (1.1) is said to be oscillatory if all its solutions are oscillatory.
We point out that delay differential equations have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems, such as electrical power systems, control systems, networks, materials, see [1]. The p-Laplace equations have some significant applications in elasticity theory and continuum mechanics.
During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations with p-Laplacian like operator in the canonical case and the noncanonical case, see [2,3,4,11] and the numerical solution of the neutral delay differential equations, see [5,6,7]. The oscillatory properties of differential equations are fairly well studied by authors in [16,17,18,19,20,21,22,23,24,25,26,27]. We collect some relevant facts and auxiliary results from the existing literature.
Liu et al. [4] studied the oscillation of even-order half-linear functional differential equations with damping of the form
{(r(t)Φ(y(n−1)(t)))′+a(t)Φ(y(n−1)(t))+q(t)Φ(y(g(t)))=0,Φ=|s|p−2s, t≥t0>0, |
where n is even. This time, the authors used comparison method with second order equations.
The authors in [9,10] have established sufficient conditions for the oscillation of the solutions of
{(r(t)|y(n−1)(t)|p−2y(n−1)(t))′+∑ji=1qi(t)g(y(ϑi(t)))=0,j≥1, t≥t0>0, |
where n is even and p>1 is a real number, in the case where ϑi(t)≥υ (with r∈C1((0,∞),R), qi∈C([0,∞),R), i=1,2,..,j).
We point out that Li et al. [3] using the Riccati transformation together with integral averaging technique, focuses on the oscillation of equation
{(r(t)|w′′′(t)|p−2w′′′(t))′+∑ji=1qi(t)|y(δi(t))|p−2y(δi(t))=0,1<p<∞, , t≥t0>0. |
Park et al. [8] have obtained sufficient conditions for oscillation of solutions of
{(r(t)|y(n−1)(t)|p−2y(n−1)(t))′+q(t)g(y(δ(t)))=0,1<p<∞, , t≥t0>0. |
As we already mentioned in the Introduction, our aim here is complement results in [8,9,10]. For this purpose we discussed briefly these results.
In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next sections, we will mention some auxiliary lemmas, also, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give some examples to illustrate the main results.
For convenience, we denote
A(t)=q(t)(1−a0)p2−1Mp1−p2(ϑ(t)), B(t)=(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t), ϕ1(t)=∫∞tA(s)ds,R1(t):=(p1−1)μt22r1/(p1−1)(t),ξ(t):=q(t)(1−a0)p2−1Mp2−p11ε1(ϑ(t)t)3(p2−1),η(t):=(1−a0)p2/p1Mp2/(p1−2)2∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ,ξ∗(t)=∫∞tξ(s)ds, η∗(t)=∫∞tη(s)ds, |
for some μ∈(0,1) and every M1,M2 are positive constants.
Definition 1. A sequence of functions {δn(t)}∞n=0 and {σn(t)}∞n=0 as
δ0(t)=ξ∗(t), and σ0(t)=η∗(t),δn(t)=δ0(t)+∫∞tR1(t)δp1/(p1−1)n−1(s)ds, n>1σn(t)=σ0(t)+∫∞tσp1/(p1−1)n−1(s)ds, n>1. | (2.1) |
We see by induction that δn(t)≤δn+1(t) and σn(t)≤σn+1(t) for t≥t0, n>1.
In order to discuss our main results, we need the following lemmas:
Lemma 2.1. [12] If the function w satisfies w(i)(ν)>0, i=0,1,...,n, and w(n+1)(ν)<0 eventually. Then, for every ε1∈(0,1), w(ν)/w′(ν)≥ε1ν/n eventually.
Lemma 2.2. [13] Let u(t) be a positive and n-times differentiable function on an interval [T,∞) with its nth derivative u(n)(t) non-positive on [T,∞) and not identically zero on any interval of the form [T′,∞), T′≥T and u(n−1)(t)u(n)(t)≤0, t≥tu then there exist constants θ, 0<θ<1 and ε>0 such that
u′(θt)≥εtn−2u(n−1)(t), |
for all sufficient large t.
Lemma 2.3 [14] Let u∈Cn([t0,∞),(0,∞)). Assume that u(n)(t) is of fixed sign and not identically zero on [t0,∞) and that there exists a t1≥t0 such that u(n−1)(t)u(n)(t)≤0 for all t≥t1. If limt→∞u(t)≠0, then for every μ∈(0,1) there exists tμ≥t1 such that
u(t)≥μ(n−1)!tn−1|u(n−1)(t)| for t≥tμ. |
Lemma 2.4. [15] Assume that (1.2) holds and u is an eventually positive solution of (1.1). Then, (r(t)(w′′′(t))p1−1)′<0 and there are the following two possible cases eventually:
(G1) w(k)(t)>0, k=1,2,3,(G2) w(k)(t)>0, k=1,3, and w′′(t)<0. |
Theorem 2.1. Assume that
liminft→∞1ϕ1(t)∫∞tB(s)ϕp1(p1−1)1(s)ds>p1−1pp1(p1−1)1. | (2.2) |
Then (1.1) is oscillatory.
proof. Assume that u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. Since r′(t)>0, we have
w(t)>0, w′(t)>0, w′′′(t)>0, w(4)(t)<0 and (r(t)(w′′′(t))p1−1)′≤0, | (2.3) |
for t≥t1. From definition of w, we get
u(t)≥w(t)−a0u(τ(t))≥w(t)−a0w(τ(t))≥(1−a0)w(t), |
which with (1.1) gives
(r(t)(w′′′(t))p1−1)′≤−q(t)(1−a0)p2−1wp2−1(ϑ(t)). | (2.4) |
Define
ϖ(t):=r(t)(w′′′(t))p1−1wp1−1(ζϑ(t)). | (2.5) |
for some a constant ζ∈(0,1). By differentiating and using (2.4), we obtain
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−1(ϑ(t)).wp1−1(ζϑ(t))−(p1−1)r(t)(w′′′(t))p1−1w′(ζϑ(t))ζϑ′(t)wp1(ζϑ(t)). |
From Lemma 2.2, there exist constant ε>0, we have
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)r(t)(w′′′(t))p1−1εϑ2(t)w′′′(ϑ(t))ζϑ′(t)wp1(ζϑ(t)). |
Which is
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)εr(t)ϑ2(t)ζϑ′(t)(w′′′(t))p1wp1(ζϑ(t)), |
by using (2.5) we have
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)ϖp1/(p1−1)(t). | (2.6) |
Since w′(t)>0, there exist a t2≥t1 and a constant M>0 such that
w(t)>M. |
Then, (2.6), turns to
ϖ′(t)≤−q(t)(1−a0)p2−1Mp2−p1(ϑ(t))−(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)ϖp1/(p1−1)(t), |
that is
ϖ′(t)+A(t)+B(t)ϖp1/(p1−1)(t)≤0. |
Integrating the above inequality from t to l, we get
ϖ(l)−ϖ(t)+∫ltA(s)ds+∫ltB(s)ϖp1/(p1−1)(s)ds≤0. |
Letting l→∞ and using ϖ>0 and ϖ′<0, we have
ϖ(t)≥ϕ1(t)+∫∞tB(s)ϖp1/(p1−1)(s)ds. |
This implies
ϖ(t)ϕ1(t)≥1+1ϕ1(t)∫∞tB(s)ϕp1/(p1−1)1(s)(ϖ(s)ϕ1(s))p1/(p1−1)ds. | (2.7) |
Let λ=inft≥Tϖ(t)/ϕ1(t) then obviously λ≥1. Thus, from (2.2) and (2.7) we see that
λ≥1+(p1−1)(λp1)p1/(p1−1) |
or
λp1≥1p1+(p1−1)p1(λp1)p1/(p1−1), |
which contradicts the admissible value of λ≥1 and (p1−1)>0.
Therefore, the proof is complete.
Theorem 2.2. Assume that
liminft→∞1ξ∗(t)∫∞tR1(s)ξp1/(p1−1)∗(s)ds>(p1−1)pp1/(p1−1)1 | (2.8) |
and
liminft→∞1η∗(t)∫∞t0η2∗(s)ds>14. | (2.9) |
Then (1.1) is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases (G1) and (G2).
For case (G1). Define
ω(t):=r(t)(w′′′(t))p1−1wp1−1(t). |
By differentiating ω and using (2.4), we obtain
ω′(t)≤−q(t)(1−a0)p2−1wp2−1(ϑ(t))wp1−1(t)−(p1−1)r(t)(w′′′(t))p1−1wp1(t)w′(t). | (2.10) |
From Lemma 2.1, we get
w′(t)w(t)≤3ε1t. |
Integrating again from t to ϑ(t), we find
w(ϑ(t))w(t)≥ε1ϑ3(t)t3. | (2.11) |
It follows from Lemma 2.3 that
w′(t)≥μ12t2w′′′(t), | (2.12) |
for all μ1∈(0,1) and every sufficiently large t. Since w′(t)>0, there exist a t2≥t1 and a constant M>0 such that
w(t)>M, | (2.13) |
for t≥t2. Thus, by (2.10), (2.11), (2.12) and (2.13), we get
ω′(t)+q(t)(1−a0)p2−1Mp2−p11ε1(ϑ(t)t)3(p2−1)+(p1−1)μt22r1/(p1−1)(t)ωp1/(p1−1)(t)≤0, |
that is
ω′(t)+ξ(t)+R1(t)ωp1/(p1−1)(t)≤0. | (2.14) |
Integrating (2.14) from t to l, we get
ω(l)−ω(t)+∫ltξ(s)ds+∫ltR1(s)ωp1/(p1−1)(s)ds≤0. |
Letting l→∞ and using ω>0 and ω′<0, we have
ω(t)≥ξ∗(t)+∫∞tR1(s)ωp1/(p1−1)(s)ds. | (2.15) |
This implies
ω(t)ξ∗(t)≥1+1ξ∗(t)∫∞tR1(s)ξp1/(p1−1)∗(s)(ω(s)ξ∗(s))p1/(p1−1)ds. | (2.16) |
Let λ=inft≥Tω(t)/ξ∗(t) then obviously λ≥1. Thus, from (2.8) and (2.16) we see that
λ≥1+(p1−1)(λp1)p1/(p1−1) |
or
λp1≥1p1+(p1−1)p1(λp1)p1/(p1−1), |
which contradicts the admissible value of λ≥1 and (p1−1)>0.
For case (G2). Integrating (2.4) from t to m, we obtain
r(m)(w′′′(m))p1−1−r(t)(w′′′(t))p1−1≤−∫mtq(s)(1−a0)p2−1wp2−1(ϑ(s))ds. | (2.17) |
From Lemma 2.1, we get that
w(t)≥ε1tw′(t) and hence w(ϑ(t))≥ε1ϑ(t)tw(t). | (2.18) |
For (2.17), letting m→∞and using (2.18), we see that
r(t)(w′′′(t))p1−1≥ε1(1−a0)p2−1wp2−1(t)∫∞tq(s)ϑp2−1(s)sp2−1ds. |
Integrating this inequality again from t to ∞, we get
w′′(t)≤−ε1(1−a0)p2/p1wp2/p1(t)∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ, | (2.19) |
for all ε1∈(0,1). Define
y(t)=w′(t)w(t). |
By differentiating y and using (2.13) and (2.19), we find
y′(t)=w′′(t)w(t)−(w′(t)w(t))2≤−y2(t)−(1−a0)p2/p1M(p2/p1)−1∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ, | (2.20) |
hence
y′(t)+η(t)+y2(t)≤0. | (2.21) |
The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Theorem 2.3. Let δn(t) and σn(t) be defined as in (2.1). If
limsupt→∞(μ1t36r1/(p1−1)(t))p1−1δn(t)>1 | (2.22) |
and
limsupt→∞λtσn(t)>1, | (2.23) |
for some n, then (1.1)is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases.
In the case (G1), proceeding as in the proof of Theorem 2.2, we get that (2.12) holds. It follows from Lemma 2.3 that
w(t)≥μ16t3w′′′(t). | (2.24) |
From definition of ω(t) and (2.24), we have
1ω(t)=1r(t)(w(t)w′′′(t))p1−1≥1r(t)(μ16t3)p1−1. |
Thus,
ω(t)(μ1t36r1/(p1−1)(t))p1−1≤1. |
Therefore,
limsupt→∞ω(t)(μ1t36r1/(p1−1)(t))p1−1≤1, |
which contradicts (2.22).
The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Corollary 2.1. Let δn(t) and σn(t) be defined as in (2.1). If
∫∞t0ξ(t)exp(∫tt0R1(s)δ1/(p1−1)n(s)ds)dt=∞ | (2.25) |
and
∫∞t0η(t)exp(∫tt0σ1/(p1−1)n(s)ds)dt=∞, | (2.26) |
for some n, then (1.1) is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases (G1) and (G2).
In the case (G1), proceeding as in the proof of Theorem 2, we get that (2.15) holds. It follows from (2.15) that ω(t)≥δ0(t). Moreover, by induction we can also see that ω(t)≥δn(t) for t≥t0, n>1. Since the sequence {δn(t)}∞n=0 monotone increasing and bounded above, it converges to δ(t). Thus, by using Lebesgue's monotone convergence theorem, we see that
δ(t)=limn→∞δn(t)=∫∞tR1(t)δp1/(p1−1)(s)ds+δ0(t) |
and
δ′(t)=−R1(t)δp1/(p1−1)(t)−ξ(t). | (2.27) |
Since δn(t)≤δ(t), it follows from (2.27) that
δ′(t)≤−R1(t)δ1/(p1−1)n(t)δ(t)−ξ(t). |
Hence, we get
δ(t)≤exp(−∫tTR1(s)δ1/(p1−1)n(s)ds)(δ(T)−∫tTξ(s)exp(∫sTR1(δ)δ1/(p1−1)n(δ)dδ)ds). |
This implies
∫tTξ(s)exp(∫sTR1(δ)δ1/(p1−1)n(δ)dδ)ds≤δ(T)<∞, |
which contradicts (2.25). The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Example 2.1. Consider the differential equation
(u(t)+12u(t2))(4)+q0t4u(t3)=0, | (2.28) |
where q0>0 is a constant. Let p1=p2=2, r(t)=1, a(t)=1/2, τ(t)=t/2, ϑ(t)=t/3 and q(t)=q0/t4. Hence, it is easy to see that
A(t)=q(t)(1−a0)(p2−1)Mp2−p1(ϑ(t))=q02t4, B(t)=(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)=εt227 |
and
ϕ1(t)=q06t3, |
also, for some ε>0, we find
liminft→∞1ϕ1(t)∫∞tB(s)ϕp1/(p1−1)1(s)ds>(p1−1)pp1/(p1−1)1.liminft→∞6εq0t3972∫∞tdss4>14q0>121.5ε. |
Hence, by Theorem 2.1, every solution of Eq (2.28) is oscillatory if q0>121.5ε.
Example 2.2. Consider a differential equation
(u(t)+a0u(τ0t))(n)+q0tnu(ϑ0t)=0, | (2.29) |
where q0>0 is a constant. Note that p=2, t0=1, r(t)=1, a(t)=a0, τ(t)=τ0t, ϑ(t)=ϑ0t and q(t)=q0/tn.
Easily, we see that condition (2.8) holds and condition (2.9) satisfied.
Hence, by Theorem 2.2, every solution of Eq (2.29) is oscillatory.
Remark 2.1. Finally, we point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type:
{(r(t)|y′′′(t)|p1−2y′′′(t))′+a(t)f(y′′′(t))+∑ji=1qi(t)|y(σi(t))|p2−2y(σi(t))=0,t≥t0, σi(t)≤t, j≥1,, 1<p2≤p1<∞. |
The paper is devoted to the study of oscillation of fourth-order differential equations with p-Laplacian like operators. New oscillation criteria are established by using a Riccati transformations, and they essentially improves the related contributions to the subject.
Further, in the future work we get some Hille and Nehari type and Philos type oscillation criteria of (1.1) under the condition ∫∞υ01r1/(p1−1)(s)ds<∞.
The authors express their debt of gratitude to the editors and the anonymous referee for accurate reading of the manuscript and beneficial comments.
The author declares that there is no competing interest.
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