# Second-Order Differential Equation: Oscillation Theorems and Applications

Authors: Shyam S. Santra; Omar Bazighifan; Hijaz Ahmad; Yu-Ming Chu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the formςyu′ya′+pyucϑy=0,fory≥y0, under the assumption ∫∞ςη−1/a=∞. Two cases are considered for a<c and a>c, where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.


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## Body

## 1. Introduction

Differential equations (DEs) have received a lot of attention, and it is an active research area among scientists and engineers [1–8]. The DEs have ability to formulate many complex phenomena in various fields such as biology, fluid mechanics, plasma physics, fluid mechanics, and optics; many exact and numerical schemes have been being derived such as [9–15]. Differential equation of second order appears in models as well as in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations and quantum mechanics, biological, physical and chemical phenomena, optimization, mathematics of networks, and dynamical systems (see [16–24]).In this article, we consider the differential equation(1)ςyu′ya′+pyucϑy=0,fory≥y0,where a and c are the quotient of two positive odd integers, and the functions p,ς,andϑ are continuous that satisfy the conditions stated below:(A1)ϑ∈C0,∞,ℝ, ϑy<y, limy⟶∞ϑy=∞.(A2)ς∈C10,∞,ℝ, p∈C0,∞,ℝ; 0<ςy, 0≤py for all y≥0; py is not identically zero in any interval b,∞.(A3)ϒy=∫y1yς−1/aηdη with limy⟶∞ϒy=∞.(A4) the existence of a differentiable functionϑ0 such that 0<ϑ0y≤ϑy, for ϑ0′y≥ϑ0>0,fory≥y0.In [25, 26], 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⟶∞ϒy=∞. In the same technique, Džurina and Dzurina [27] have studied the oscillatory behavior of the solutions of (2) under the assumptions 0≤qy<∞ and limy⟶∞ϒy=∞. In [28], Bohner et al. have studied the oscillatory behavior of solutions of (2) under a=c, limy⟶∞ϒy<∞, and 0≤qy<1. Grace et al. [29] have studied the oscillatory behavior of (2) when a=c and limy⟶∞ϒy<∞ and limy⟶∞ϒy=∞ and 0≤qy<1. In [30], Ali has studied the oscillatory behavior of the solutions of (2), under the assumptions limy⟶∞ϒy<∞ and qy≥0. Karpuz and Santra [31] have studied the oscillatory behavior of(3)ςyuy+qyuτy′′+pyfuϑy=0,by considering the assumptions limy⟶∞ϒy<∞ and limy⟶∞ϒy=∞ for different ranges of the neutral coefficient q.For further work on the oscillation of this type of equations, we refer the readers to the references. 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 the study (1), which includes the class of functional differential equations of neutral 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∞pζucϑζdζ1/a≤uy,for y≥y1.Proof.
Letu be an eventually positive solution of (1). Then, by (A1), there exists a y∗ such that uy>0 and uϑy>0 for all y≥y∗. From (1) it follows that(6)ςyu′ya′=−pyucϑy≤0.
Therefore,ςyu′ya is nonincreasing for y≥y∗. Next, we show that ςyu′ya is positive. By contradiction, assume that ςyu′ya≤0 at a certain time y≥y∗. Using that p is not identically zero on any interval b,∞ and by (6), there exists 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 fromy2 to y, we have(9)uy≤uy1+ςy11/au′y1ϒy.
By (A3), the right-hand side approaches−∞; then, limy⟶∞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 fromy1 to y and using that u is continuous,(11)uy≤uy1+ςy11/au′y1ϒy.
Sincelimy⟶∞ϒy=∞, 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+∫y∞pηucϑηdη=0.
Letting limit asb⟶∞, we get(13)ςyu′ya≥∫y∞pηucϑηdη.
Then,(14)u′y≥1ςy∫y∞pηucϑηdη1/a.
Sinceuy1>0, integrating the above inequality yields(15)uy≥∫y1y1ςη∫η∞pζucϑζdζ1/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∞pζucϑζdζ1/a,which yields (5).Theorem 1.
Assume that there exists a constantb1, the quotient of two positive odd integers, such that 0<c<b1<a.If (A1)–(A3) hold, then each solution of (1) is oscillatory if and only if(17)∫0∞pζϒcϑζdζ=∞.Proof.
On the contrary, we assume thatu is eventually positive solution. So, Lemma 1 holds, and then there exists y1≥y0 such that(18)uy≥ϒyw1/ay≥0,fory≥y1,where(19)wy=∫y∞pζucϑζdζ.
Computing the derivative ofw, we have(20)w′y=−pyucϑy.
Thus,w is nonnegative and nonincreasing. Since u>0, by (A2), it follows that pyucϑy 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, we have(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/aζpζucϑζdζ.
Next, we find a lower bound for the right-hand side of (25), independent of the solution u. By (4) and (19), we have(23)ucy=uc−b1yub1y≥dϒyc−b1ub1y≥dϒyc−b1ϒyw1/ayb1=dc−b1ϒcywb1/ay,fory≥y2.
Sincew is nonincreasing, b1/a>0, and ϑη<η, it follows that(24)ucϑη≥dc−b1ϒcϑηwb1/aϑη≥dc−b1ϒcϑηwb1/aη.
Going back to (22), we have(25)w1−b1/ay2≥1−b1adc−b1∫y2ypηϒcϑηdη.
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 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∞pζϒcϑζdζ≤κ1−c/a2,for all η≥y1. We define the set of continuous functions(27)S=u∈C0,∞:κ21/aϒy≤uy≤κ1/aϒy,y≥y1.
We define an operatorΩ on S by(28)Ωuy=0,ify≤y1,∫y1y1ςηκ2+∫η∞pζucϑζdζ1/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/aϒy.
Now, we estimateΩuy from above. For u in S, we have ucϑζ≤κ1/aϒϑζc. Then, by (26),(30)Ωuy≤∫y1y1ςηκ2+∫η∞pζucϑζdζ1/adη≤κ1/aϒy.
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/aϒy,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), there is a nonoscillatory solution that does not converge to zero. This completes the proof.Theorem 2.
Assume that there exists a constantb2, the quotient of two positive odd integers such that 0<a<b2<c. If (A1)–(A4) hold and ςy is nondecreasing, then each solution of (1) is oscillatory if and only if(32)∫y1∞1ςs∫s∞pζdζ1/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 uϑy>0 and ςyu′ya is positive and nonincreasing. Since ςy>0, uy is increasing for y≥y1. Using uy≥uy1, we have(33)ucy≥uc−b2yub2y≥uc−b2y1ub2y,and hence(34)ucϑy≥uc−b2y1ub2ϑy,fory≥y2.
Using (34) and ϑy≥ϑ0y, from (13), we have(35)ςyu′ya≥uc−b2y1ub2ϑ0y∫y∞pηdη,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 and raising both sides to the 1/a power, we have(37)u′ϑ0yub2/aϑ0y≥uc−b2y1ςϑ0y∫y∞pηdη1/a,for y≥y2. Multiplying the left-hand side by ϑ0′y/ϑ0≥1 and integrating from y2 to y, we have(38)1ϑ0∫y2yu′ϑ0ηϑ0′ηub2/aϑ0ηdη≥uc−b2y1∫y2y1ςϑ0η∫η∞pζdζ1/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∞pζdζ1/adηκ1−c/a2,∀y≥y1.
We define the set of continuous function(41)S=u∈C0,∞:κ2≤uy≤κfory≥y1.
Then, we define the operator(42)Ωuy=0,ify≤y1,κ2+∫y1y1ςη∫η∞pζucϑζdζ1/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, 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+κc/a∫y1y1ςη∫η∞pζdζ1/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 the differential equations(45)e−yu′y11/3′+1y+1uy−21/3=0.
Here,a=11/3, ςy=e−y, ϑ1y=y−2, ϒy=∫y1ye11η/3dη=3/11e11y/3−e11y1/3, and c1=1/3. For b=7/3, we have 0<c<b<a. To check (17), we have(46)∫0∞psϒcϑsds=∫0∞1η+1311e11s−2/3−e11y1/31/3ds=∞.
So, every conditions of Theorem1 hold true, and therefore, all solutions of (45) are oscillatory or converge to zero.Example 2.
Consider the differential equations(47)u′y1/3′+yuy−27/3=0.
Here,a=1/3, ςy=1, ϑ1y=y−2, and c1=7/3. For b=5/3, we have c>b>a. To check (32), we have(48)∫y0∞1ςs∫s∞pζdζ1/ads=∫2∞∫s∞ζdζ3ds=∞.
So, every conditions of Theorem2 hold true. Thus, all solutions of (47) are oscillatory or converge to zero.


## 3. Conclusion

The aim of this work is to establish necessary and sufficient conditions for the oscillation of solution to second-order half-linear differential equation. The obtained oscillation theorems complement the well-known oscillation results present in the literature. This work, as well as [31–41], leads us to pose an open problem: Can we find necessary and sufficient conditions for the oscillation of solutions to second-order differential equation(49)rtyt+ptyτt′γ′+∑i=1mqityαiτit=0,forp∈Cℝ+,ℝ?.



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