# Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications

Authors: Shyam Sundar Santra; Omar Bazighifan; Hijaz Ahmad; Shao-Wen Yao
Journal: Complexity
 (2020)

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## Abstract

Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the formςyu′ya′+∑j=1mpjyucjϑjy=0fory≥y0, under the assumption ∫∞ςη−1/adη=∞. We consider two cases when a<cj and a>cj, where a and cj are the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.


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## Body

## 1. Introduction

The differential equation of second order appears in models concerning biological, physical, and chemical phenomena, optimization, the mathematics of networks, and dynamical systems, see [1].In this article, we consider the differential equation(1)ςyu′ya′+∑j=1mpjyucjϑjy=0,fory≥y0,where a and cj are the quotient of two positive odd integers and the functions pj,ς,andϑj are continuous that satisfy the conditions stated below:(A1)
ϑj∈C0,∞,ℝ, ϑjy<y, limy⟶∞ϑjy=∞(A2)
ς∈C10,∞,ℝ, pj∈C0,∞,ℝ; 0<ςy, 0≤pjy for all y≥0 and j=1,2,…,m; ∑pjy is not identically zero in any interval b,∞(A3)
ϒy=∫y1yς−1/aηdη with limy⟶∞Yy=∞(A4)
The existence of a differentiable functionϑ0 such that 0<ϑ0y=minϑjy,ϑ0′y≥ϑ0>0fory≥y0,j=1,2,…,mIn [2, 3], Baculǐkovǎ and Džurina have considered(2)ςyz′ya′+pyucϑy=0,zy=uy+qyuτy,y≥y0,and obtained oscillation criteria for the solutions of (2) using comparison techniques when a=c=1, 0≤qy<∞ and limy⟶∞Yy=∞. In the same technique, Baculikova and Džurina [4] have studied the oscillatory behavior of the solutions of (2) under the assumptions 0≤qy<∞ and limy⟶∞Υy=∞. In [?], Tripathy et al. have studied oscillatory and asymptotic behavior of (4) when limy⟶∞Υy=∞ and limy⟶∞Yy<∞ for different ranges of the neutral coefficient q. In [5], Bohner et al. have studied the oscillatory behavior of solutions of (2) under a=c, limy⟶∞Yy<∞ and 0≤qy<1. Grace et al. [6] have studied the oscillatory behavior of (4) when a=c and limy⟶∞Yy<∞, limy⟶∞Yy=∞ and 0≤qy<1. In [7], Li et al. have studied the oscillatory behavior of the solutions of (2), under the assumptions limy⟶∞Yy<∞ and qy≥0. Karpuz and Santra [8] have studied the oscillatory behavior of(3)ςyuyqyuτy′′+pyfuϑy=0,by considering the assumptions limy⟶∞Yy<∞ and limy⟶∞Yy<∞ for different ranges of q.For further work on the oscillation of this type of equations, we refer the readers to [6, 9–32]. Note that the majority of works consider only sufficient conditions, and merely a few consider both necessary and sufficient conditions. Hence, the objective of this work is to establish both necessary and sufficient conditions for the oscillation of solutions of (1) without using the comparison and the Riccati techniques. In this paper, we restrict our attention to study (1), which includes the class of functional differential equations of the delay type.Remark 1.
When the domain is not specified explicitly, all functional inequalities considered in this paper are assumed to hold eventually, i.e., they are satisfied for ally large enough.


## 2. Necessary and Sufficient Conditions

Lemma 1.
Let (A1)–(A3) hold and thatu is an eventually positive solution of (1). Then, there exist y1≥y0 and d>0 such that(4)0<uy≤dϒy,(5)ϒy∫y∞∑j=1mpjζucjϑjζdζ1/a≤uy,for y≥y1.Proof.
Letu be eventually positive solution of (1). Then, by (A1), there exists a y∗ such that uy>0 and uϑjy>0 for all y≥y∗ and j=1,2,…,m. From (1), it follows that(6)ςyu′ya′=−∑j=1mpjyucjϑjy≤0.
Therefore,ςyu′ya is nonincreasing for y≥y∗. Next, we show the ςyu′ya is positive. By contradiction, assume that ςyu′ya≤0 at a certain time y≥y∗. Using that ∑pj is not identically zero on any interval b,∞ and by (6), there exist y1≥y∗ such that(7)ςyu′ya≤ςy1u′y1a<0,for ally≥y1.
Recall thata is the quotient of two positive odd integers. Then,(8)u′y≤ςy1ςy1/au′y1,fory≥y1.
Integrating fromy1 to y, we have(9)uy≤uy1+ςy11/au′y1Yy.
By (A3), the right-hand side approaches−∞; then, limt⟶∞uy=−∞. This is a contradiction to the fact that uy>0. Therefore, ςyu′ya>0 for all y≥y∗. From ςyu′ya being nonincreasing, we have(10)u′y≤ςy1ςy1/au′y1,fory≥y1,integrating this inequality from y1 to y, and using that u is continuous,(11)uy≤uy1+ςy11/au′y1Yy.
Sincelimy⟶∞Yy=∞, there exists a positive constant d such that (4) holds.
Sinceςyu′ya is positive and nonincreasing, limy⟶∞ςyu′ya exists and is nonnegative. Integrating (1) from y to b, we have(12)ςbu′ba−ςyu′ya+∫yb∑j=1mpjηucjϑjηdη=0.
Letting limit asb⟶∞, we obtain(13)ςyu′ya≥∫y∞∑j=1mpjηucjϑjηdη.
Then,(14)u′y≥1ςy∫y∞∑j=1mpjηucjϑjηdη1/a.
Sinceuy1>0, integrating the above inequality yields(15)uy≥∫y1y1ςη∫η∞∑j=1mpjζucjϑjζdζ1/adη.
Since the integrand is positive, we can increase the lower limit of integration fromη to y and then use the definition of ϒy to obtain(16)uy≥ϒy∫y∞∑j=1mpjζucjϑjζdζ1/a,which yields (5).Theorem 1.
Assume that there exists a constantb1 and the quotient of two positive odd integers, such that 0<cj<b1<a. If (A1)–(A3) hold, then each solution of (1) is oscillatory if and only if(17)∫0∞∑j=1mpjζYcjϑjζdζ=∞.Proof.
On the contrary, we assume that a solutionu is eventually positive. So, Lemma 1 holds, and then there exists y1≥y0 such that(18)uy≥Yyw1/ay≥0,fory≥y1,where(19)wy=∫y∞∑j=1mpjζucjϑjζdζ.
Computing the derivative ofw, we have(20)w′y=−∑j=1mpjyucjϑjy.
Thus,w is nonnegative and nonincreasing. Since u>0, by (A2), it follows that ∑j=1mpjyucjϑjy cannot be identically zero in any interval b,∞; thus, w′ cannot be identically zero, and w cannot be constant on any interval b,∞. Therefore, wy>0 for y≥y1. Computing the derivative,(21)w1−b1/ay′=1−b1aw−b1/ayw′y.
Integrating (21) from y2 to y and using that w>0, we have(22)w1−b1/ay2≥1−b1a−∫y2yw−b1/aζw′ζdζ=1−b1a∫y2yw−b1/aζ∑j=1mpjζucjϑjζdζ.
Next, we find a lower bound for the right-hand side of (22), independent of the solution u. By (4) and (19), we have(23)ucjy=ucj−b1yub1y≥dYycj−b1ub1y≥dYycj−b1Yyw1/ayb1=dcj−b1Ycjywb1/ay,fory≥y2.
Sincew is nonincreasing, b1/a>0 and ϑjη<η, and it follows that(24)ucjϑjη≥dcj−b1ϒcjϑjηwb1/aϑjη≥dcj−b1Ycjϑjηwb1/aη.
Going back to (22), we have(25)w1−b1/ay2≥1−b1adcj−b1∫y2y∑j=1mpjηYcjϑjηdη.
Since1−b1/a>0, by (17), the right-hand side approaches +∞ as y⟶∞. This contradicts (25) and completes the proof of sufficiency for eventually positive solutions.
The eventually negative solution can be dealt with similarly by introducing the variablesv=−u.
Next, we show the necessity part by a contrapositive argument. If (17) does not hold, then, for each κ>0, there exists y1≥y0 such that(26)∫η∞∑j=1mpjζYcjϑjζdζ≤κ1−cj/a2,for all η≥y1. We define the set of continuous functions:(27)S=u∈C0,∞:κ21/aYy≤uy≤κ1/aYy,y≥y1.
We define an operatorΩ on S by(28)Ωuy=0,ify≤y1,∫y1y1ςηκ2+∫η∞∑j=1mpjζucjϑjζdζ1/adη,ify>y1.
Note that whenu is continuous, Ωu is also continuous on 0,∞. If u is a fixed point of Ω, i.e., Ωu=u, then u is a solution of (1).
First, we estimateΩuy from below. By (A3), we have(29)Ωuy≥∫y1y1ςηκ2+01/adη=κ21/aYy.
Now, we estimateΩuy from above. For u in S, we have ucjϑjζ≤κ1/aYϑjζcj. Then, by (26),(30)Ωuy≤∫y1y1ςηκ2+∫η∞∑j=1mpjζucjϑjζdζ1/adη≤κ1/aYy.
Therefore,Ω maps S to S.
Next, we find a fixed point forΩ in S. Let us define a sequence of functions in S by the recurrence relation:(31)v0y=0,fory≥y0,v1y=Ωv0y=0,ify<y1,κ1/aYy,ify≥y1,vn+1y=Ωvny,forn≥1,y≥y1.
Note that, for each fixedy, we have v1y≥v0y. Using mathematical induction, we can show that vn+1y≥vny. Therefore, the sequence vn converges pointwise to a function v. Using the Lebesgue Dominated Convergence Theorem, we can show that v is a fixed point of Ω in S. This shows under assumption (26) that there is a nonoscillatory solution that does not converge to zero. This completes the proof.Theorem 2.
Assume that there exists a constantb2 and the quotient of two positive odd integers such that 0<a<b2<cj. If (A1)–(A4) hold and ςy is nondecreasing, then each solution of (1) is oscillatory if and only if(32)∫y1∞1ςη∫η∞∑j=1mpjζdζ1/adη=∞.Proof.
On the contrary, we assume thatu is an eventually positive solution that does not converge to zero. Using the same argument as in Lemma 1, there exists y1≥y0 such that uϑjy>0 and ςyu′ya is positive and nonincreasing. Since ςy>0, so uy is increasing for y≥y1. Using uy≥uy1, we have(33)ucjy≥ucj−b2yub2y≥ucj−b2y1ub2y,and hence(34)ucjϑjy≥ucj−b2y1ub2ϑjy,fory≥y2.
Using (34) and ϑjy≥ϑ0y, from (13), we have(35)ςyu′ya≥ucj−b2y1ub2ϑ0y∫y∞∑j=1mpjηdη,for y≥y2. From ςyu′ya being nonincreasing and ϑ0y≤y, we have(36)ςϑ0yu′ϑ0ya≥ςyu′ya.
We use this in the left-hand side of (35). Then, dividing by ςϑ0yub2ϑ0y>0, raising both sides to the 1/a power, we have(37)u′ϑ0yub2/aϑ0y≥ucj−b2y1ςϑ0y∫y∞∑j=1mpjηdη1/a,for y≥y2. Multiplying the left-hand side by ϑ0′y/ϑ0≥1 and integrating from y2 to y,(38)1ϑ0∫y2yu′ϑ0ηϑ0′ηub2/aϑ0ηdη≥ucj−b2y1∫y2y1ςϑ0η∫η∞∑j=1mpjζdζ1/adη.
On the left-hand side, sincea<b2, integrating, we have(39)1α1−b2/az1−b2/aϑ0ηs=y2y≤1αb2/a−1z1−b2/aϑ0y2<∞.
On the right-hand side of (38), we use that ςϑ0η≤ςη, to conclude that (32) implies the right-hand side approaching +∞, as y⟶∞, which is a contradiction. Hence, the solution u cannot be eventually positive.
For eventually negative solutions, we use the same change of variables as in Theorem1 and proceed as above.
To prove the necessity part, we assume that (32) does not hold and obtain an eventually positive solution that does not converge to zero. If (32) does not hold, then for each κ>0 there exists y1≥y0 such that(40)∫y1∞1ςη∫η∞∑j=1mpjζdζ1/adη<κ1−cj/a2,∀y≥y1.
We define the set of continuous function(41)S=u∈C0,∞:κ2≤uy≤κ,fory≥y1.
Then, we define the operator:(42)Ωuy=0,ify≤y1,κ2+∫y1y1ςη∫η∞∑j=1mpjζucjϑjζdζ1/adη,ify>y1.
Note that ifu is continuous, Ωu is also continuous at y=y1. Also note that if Ωu=u, then u is solution of (1).
First, we estimateΩuy from below. Let u∈M, and we have Ωuy≥κ/2+0, on y1,∞.
Now, we estimateΩuy from above. Let u∈M. Then, u≤κ, and by (40), we have(43)Ωuy≤κ2+κcj/a∫y1y1ςη∫η∞∑j=1mpjζdζ1/adη≤κ2+κ2=κ.
Therefore,Ω maps S to S. To find a fixed point for Ω in S, we define a sequence of functions by the recurrence relation:(44)v0y=0,fory≥y0,v1y=Ωv0y=1,fory≥y1,vn+1y=Ωvny,forn≥1,y≥y1.
Note that, for each fixedy, we have v1y≥v0y. Using mathematical induction, we can prove that vn+1y≥vny. Therefore, vn converges pointwise to a function v in S. Then, v is a fixed point of Ω and a positive solution of (1). The proof is completed.Example 1.
Consider(45)e−yu′y11/3′+1y+1uy−21/3+1y+2uy−15/3=0.
Here,a=11/3, ςy=e−y, ϑ1y=y−2, ϑ2y=y−1, ϒy=∫y1ye11η/3dη=3/11e11y/3−e11y1/3, c1=1/3, and c2=5/3. For b=7/3, we have 0<maxc1,c2<b<a. To check (17), we have(46)∫0∞∑j=1mpjsϒcjϑjsds≥∫0∞p1sϒα1ϑ1sds=∫0∞1η+1311e11s−2/3−e11y1/31/3ds=∞.
So, every condition of Theorem1 holds true. Therefore, all solution of (45) is oscillatory.Example 2.
Consider(47)u′y1/3′+yuy−27/3+y+1uy−111/3=0.
Here,a=1/3, ςy=1, ϑ1y=y−2, ϑ2y=y−1, c1=7/3, and c2=11/3. For b=5/3, we have minc1,c2>b>a. To check (32), we have(48)∫y0∞1ςs∫s∞∑j=1mpjζdζ1/ads≥∫y0∞1ςs∫s∞p1ζdζ1/ads≥∫2∞∫s∞ζdζ3ds=∞.
So, every conditions of Theorem2 hold true. Thus, all solutions of (47) is oscillatory.


## 3. Conclusion

This work aims to study the oscillatory behavior of second-order neutral nonlinear differential equation. The obtained oscillation theorems complement the well-known oscillation results present in the literature.



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