# Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

Authors: Kandasamy Alagesan; Subaramaniyam Jaikumar; Govindasamy Ayyappan
Journal: Abstract and Applied Analysis
 (2020)

---

## Abstract

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equationΔc3nΔc2nΔc1nΔun+pnfun−k=0 under the conditions ∑n=n0∞cin<∞,i=1,2,3. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.


---

## Body

## 1. Introduction

In this paper, we are concerned with the fourth-order delay difference equation of the form(1)D4un+pnfun−k=0,n≥n0≥0,where(2)D0u=u,Diu=cinΔDi−1u,i=1,2,3,D4u=ΔD3u.In the sequel, we will assume the following:(H1) {c1(n)}, {c2(n)} and {c3(n)} are the positive real sequence for all n ≥ n0 and satisfy(3)win0=∑n=n0∞1cin<∞,i=1,2,3.(H2) {p(n)} is a nonnegative real sequence and does not vanish eventually.(H3) f∈Cℝ,ℝ is nondecreasing, and fx/x≥M1>0 for all x ≠ 0.(H4) k is a nonnegative integer.Under a solution of (1), we mean a real-valued sequence {u(n)} defined for all n ≥ n0 − k and satisfies equation (1) for all n ≥ n0. We restrict our attention to only those solutions of (1) which exist for all n ≥ N ≥ n0 and satisfy the following condition:(4)supun:n≥N1>0for anyN1≥N.A solution {u(n)} of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate.Fourth-order difference equations naturally appear in discrete-type models concerning physical, biological, and chemical phenomena (see for example [1]). In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the last several years, there has been a constant interest in getting sufficient conditions for oscillatory behavior of different classes of fourth-order difference equations with or without deviating arguments, (see [2–19], and the references cited therein). In particular, the authors in [5, 8, 13, 15, 18, 19] established oscillation results for (1) under the following assumptions:(5)c1n=c2n=1andw3n0<∞,c1n=c3n=1andw2n0<∞,respectively.In [3, 4, 6, 7, 9–12, 14, 16, 17], the authors studied the oscillation properties of solutions of equation (1) under the following condition:(6)win0=∞,i=1,2,3.From the review of the literature, it seems that there is nothing known about the oscillation of equation (1) when condition (3) holds. Inspired by the ideas adopted in [20], our main aim is to fill this gap by presenting easily verifiable criteria for the oscillation of all solutions of (1). Examples illustrating the importance of the results obtained are presented.


## 2. Main Results

For the sake of convenience, we use the following notations throughout the paper. We denote(7)w12n=∑s=n∞w2sc1s,w23n=∑s=n∞w3sc2s,w123n=∑s=n∞w23sc1s,Bn,N=∑s=Nn−11c2s∑i=Ns−11c3i∑t=Ni−1pt,B¯n,N=∑s=Nn−1psw123s−kc3s−k,where n ≥ N ≥ n0. In what follows, we only need to consider eventually positive solutions of (1), since if {u(n)} satisfies (1), then so does {−u(n)}.Lemma 1.
Let {u(n)} be an eventually positive solution of (1). Then, there is an integer n1 ≥ n0 such that {u(n)} satisfies one of the following cases:(i)
u(n) > 0, D1u(n) > 0, D2u(n) > 0, D3u(n) > 0, and D4u(n) ≤ 0(ii)
u(n) > 0, D1u(n) > 0, D2u(n) > 0, D3u(n) < 0, and D4u(n) ≤ 0(iii)
u(n) > 0, D1u(n) > 0, D2u(n) < 0, D3u(n) > 0, and D4u(n) ≤ 0(iv)
u(n) > 0, D1u(n) > 0, D2u(n) < 0, D3u(n) < 0, and D4u(n) ≤ 0(v)
u(n) > 0, D1u(n) < 0, D2u(n) > 0, D3u(n) > 0, and D4u(n) ≤ 0(vi)
u(n) > 0, D1u(n) < 0, D2u(n) > 0, D3u(n) < 0, and D4u(n) ≤ 0(vii)
u(n) > 0, D1u(n) < 0, D2u(n) < 0, D3u(n) > 0, D4u(n) ≤ 0(viii)
u(n) > 0, D1u(n) < 0, D2u(n) < 0, D3u(n) < 0, and D4u(n) ≤ 0for alln ≥ n1.Proof.
The proof is quite obvious, and hence, we omit it.
First, we start with a lemma which ensures the nonexistence of solutions of types (i) − (iv).Lemma 2.
Let {u(n)} be a positive solution of (1). If(8)B∞,n0=∞,then cases (i) − (iv) of Lemma 1 are not possible.Proof.
From (H1) and (8), one can see that(9)∑n=n0∞1c3n∑s=n0n−1ps=∑n=n0∞pn=∞.
Now, let us assume that {u(n)} is an eventually positive solution of (1) satisfying one of the cases (i) − (iv) from Lemma 1 and there is an integer n1 ≥ n0 such that u(n − k) > 0 for n ≥ n1. Since {u(n)} is increasing, there is a constant M > 0 and an integer n2 ≥ n1 such that u(n − k) ≥ M for all n ≥ n2. Substituting this in (1), one gets(10)−D4un≥fMpn,n≥n2.
Summing (10) from n2 to n − 1, one can find(11)−D3un+D3un2≥fM∑s=n2n−1ps.
If we assume that {u(n)} belongs to either case (i) or case (iii), then from (9) and (11), we obtain(12)D3un2≥fM∑s=n2n−1ps⟶∞asn⟶∞,which contradicts the fact that D3u(n) is nonincreasing.
Next, assume that case (ii) holds. From (11), we have(13)−D3un≥fM∑s=n2n−1ps,or(14)−ΔD2un≥fMc3n∑s=n2n−1ps.
Summing (14) from n2 to n − 1, we have(15)D2un2−D2un≥fM∑s=n2n−11c3s∑t=n2s−1pt,which, in view of (9), yields(16)D2un2≥fM∑s=n2n−11c3s∑t=n2s−1pt⟶∞ast⟶∞,which obviously contradicts the fact that D2u(n) is decreasing.
Finally, we assume that case (iv) holds. Proceeding the same way as in the above case, we get (15), that is,.(17)−ΔD1un≥fMc2n∑s=n2n−11c3s∑t=n2s−1pt.
Summing this inequality fromn2 to n − 1, one obtains(18)D1un2−D1un≥fM∑s=n2n−11c2s∑t=n2s−11c3t∑j=n2t−1pj=fMBn,n2,which in view of (8) implies that D1u(n2) ≥ f(M)B(n, n2) ⟶ ∞ as n ⟶ ∞, which contradicts the fact that D1u(n) is decreasing. This completes the proof.
In our first main result with a simple condition, we show that any nonoscillatory solution of (1) converges to zero as n ⟶ ∞.Theorem 1.
If(19)∑n=n0∞Bn,n0c1n=∞,then any solution of (1) is oscillatory or converges to zero as n ⟶ ∞.Proof.
Let {u(n)} be a nonoscillatory solution of (1). With no loss of generality, we may take n1 ≥ n0 such that u(n) > 0 and u(n − k) > 0 for all n ≥ n1. From Lemma 1, one may have eight possible cases for n ≥ n1.
From (19) and (H1), we see that ∑n=n0∞Bn,n0 cannot be bounded, and by Lemma 2, case (i) − (iv) are impossible.
Now, let us assume that one of the cases (v) − (viii) holds. Since {u(n)} is decreasing, there is a finite nonnegative limit u(∞) = limn ⟶ ∞u(n) = M. Assume that M > 0. Then, there is a n2 ≥ n1 such that u(n − k) ≥ M for n ≥ n2 and inequality (10) holds. Then, one can arrive at contradiction to (12) in cases (v) and (vii) and a contradiction to (16) in case (vi). Thus, we conclude that M = 0.
If we assume that case (viii) holds, then we get (18), that is,(20)−D1un≥fMBn,n2or−Δun≥fMc1nBn,n2.
Summing the last inequality fromn2 to n, we obtain(21)un2≥fM∑s=n2nBs,n2c1s.
But the term on the right side of the above inequality tends to ∞ asn ⟶ ∞ due to (19), which contradicts the fact that {u(n)} is decreasing. This completes the proof.
In the following, we present oscillation criteria for (1).Theorem 2.
If(22)limn⟶∞supAn,N>1M1,for any N ≥ n0, where(23)An,N=minw1nBn,N,w3nB¯n,N,then (1) oscillates.Proof.
Let {u(n)} be a nonoscillatory solution of (1). With no loss of generality, we may take n1 ≥ n0 such that u(n) > 0 and u(n − k) > 0 for all n ≥ n1. By Lemma 1, eight possible cases may occur for all n ≥ n1.
First, we note that, in view of (H1), it is necessary for the validity of (22) that(24)B∞,n0=B¯∞,n0=∞.
In view of Lemma2, the above condition implies that cases (i) − (iv) of Lemma 1 are not possible. Next, we shall consider the remaining possible cases (v) − (viii) separately.
Assume that case (v) holds. From the monotonicity ofD2u(n), we have(25)−D1un≥D1u∞−D1un=∑s=n∞1c2sD2us≥w2nD2un,that is,(26)−Δun≥D2unw2nc1n.
Summing the above inequality fromn to ∞, one gets(27)un≥D2un∑s=n∞w2sc1s=D2unw12n.
Using (27) and the increasing property of D2u(n) in (1), there is a constant M > 0 and an integer n2 ≥ n1 such that(28)−D4un=pnfun−k≥M1pnD2un−kw12n−k≥M2pnw12n−k,n≥n2,where M2 = MM1. Summing the above inequality from n2 to n − 1, we have(29)D3un2≥D3un+M2∑s=n2n−1psw12s−k.
From (H1) and (24), one can easily see that(30)∞=B¯∞,n0=∑n=n0∞pnw123n−kw3n−k≤∑n=n0∞pnw12n−k.
Using (30) in (29), we arrive at a contradiction with the fact that {D3u(n)} is nonincreasing.
Assume that case (vi) holds. From the monotonicity ofD3u(n), we have(31)D2un−D2u∞=−∑s=n∞1c3sD3us≥−D3unw3n.
Therefore,(32)ΔD2unw3n=D3unw3n+D2unw3nw3n+1c3n≥0,which implies that D2un/w3n is nondecreasing. Using this property, one obtains(33)−D1un≥∑s=n∞1c2sD2us≥D2unw3n∑s=n∞w3sc2s=D2unw3nw23n.
Hence,(34)Δ−D1unw23n=−D2unw23n−D1unw3nw23n+1w23nc2n≥0,and so −D1un/w23n is nondecreasing. Finally, we get(35)un≥−∑s=n∞1c1sD1us≥−D1unw23n∑s=n∞w23sc1s=−D1unw23nw123n.
Using (33) in the above inequality, we get(36)un≥D2unw3nw123n.
Therefore,(37)−D4un≥M1pnun−k≥M1pnw123n−kw3n−kD2un−k.
Summing this inequality fromn1 to n − 1 and using the monotonicity of D2u(n), we obtain(38)−D3un≥∑s=n1n−1M1psw123s−kw3s−kD2us−k≥M1D2un−k∑s=n1n−1psw123s−kw3s−k≥M1D2unB¯n,n1.
From (31) and (38), one obtains(39)−D3un≥−M1D3unB¯n,n1w3n.
Dividing the above inequality by −D3u(n) and then taking the lim sup on both sides, one arrives at a contradiction with (22).
Assume that case (vii) holds. From the decreasing property ofD1u(n), one obtains(40)un≥un−u∞=−∑s=n∞1c1sD1us≥−w1nD1un.
Thus,(41)Δunw1n=D1unw1n+unw1nw1n+1c1n≥0,which means that un/w1n is nondecreasing. Summing (1) from n1 to n − 1 and using the above property of un/w1n, we obtain(42)D3un1≥D3un+M1∑s=n1n−1psus−k≥un1w1n1M1∑s=n1n−1psw1s.
Furthermore, on using (H1) and (30), it is easy to see that for any constant b > 0,(43)∞=∑n=n1∞psw12s≤b∑n=n1∞psw1s.
This, by virtue of (42), contradicts the fact that {D3u(n)} is nonincreasing.
Finally, assume that case (viii) holds. Summing (1) from n1 to n − 1, we have(44)−D3un≥M1∑s=n1n−1psus−k≥M1un−k∑s=n1n−1ps.
Dividing both sides of the above inequality byc3(n) and summing the resulting inequality again from n1 to n − 1, one obtains(45)−D2un≥M1un−k∑s=n1n−11c3s∑t=n1s−1pt.
Similarly, we obtain(46)−D1un≥M1un−k∑s=n1n−11c2s∑t=n1s−11c3t∑j=n1t−1pj=M1un−kBn,n1≥M1unBn,n1≥−M1D1unw1nBn,n1or1M1≥w1nBn,n1,which contradicts (22). This completes the proof.
Finally, we obtain an oscillation criterion using classical Riccati-type transformation technique.Theorem 3.
If for all sufficiently largeN ≥ n0,(47)limn⟶∞∑s=NnM1psw123s+1−w23s4c1sw123s+1=∞,(48)limn⟶∞sup∑s=NnM1w1s+1c2s∑t=Ns−11c3t∑j=Nt−1pj−14w1s+1c1s=∞,then equation (1) oscillates.Proof.
Assume the contrary that {u(n)} is a nonoscillatory solution of (1) for all n ≥ n0. With no loss of generality, we may take n1 ≥ n0 such that u(n) > 0 and u(n − k) > 0 for all n ≥ n1. By Lemma 1, eight possible cases may occur for all n ≥ n1. From (48), one may see that(49)∑n=n0∞w1nc2n∑s=n1n−11c3s∑t=n1s−1pt=∞,which in view of (H1) implies that B(∞, n0) = ∞. So by Lemma 2, cases (i) − (iv) from Lemma 1 are not possible. Therefore, it is enough to consider cases (v) − (viii).
First, assume that case (v) holds. From (47), we have(50)∞=∑n=n0∞pnw123n+1≤∑n=n0pnw123n.
Then, arguing as in the proof of Theorem2 case (v), we obtain a contradiction.
Next, assume case (vi) holds. Define the function(51)yn=D3unun<0.
In view of (31) and (36), one can obtain(52)un≥D3unw123n,and hence,(53)−1≤ynw123n<0.
Also, arguing as in the proof of Theorem2 case (vi), we obtain from (31) and (33) that(54)Δun≥−D3unw23nc1n.
By (1), (54), and the monotonicity of u(n), we conclude that(55)Δyn=D4unun+1−D3unΔununun+1≤−M1pnun−kun+1−D3un2w23nc1nunun+1≤−M1pn−y2nw23nc1n,orΔyn+M1pn+y2nw23nc1n≤0.
Multiplying the above inequality byw123 (n + 1) and summing the resulting inequality from n1 to n − 1, we obtain(56)∑s=n1n−1w123s+1Δys+∑s=n1n−1M1w123s+1ps+∑s=n1n−1w123s+1y2sw23sc1s≤0.
Now, applying the summation by parts formula and then rearranging, we obtain(57)ynw123n−yn1w123n1+∑s=n1n−1M1w123s+1ps+∑s=n1n−1ysw23sc1s+∑s=n1n−1y2sw23sw123s+1c1s≤0.
Therefore, in view of (53) that(58)∑s=n1n−1M1w123s+1ps−w23s4c1sw123s≤yn1w123n1+1,which contradicts (47).
Assume now that case (vii) holds. Note that(59)∑n=n0∞pnw123n=∞is necessary for (47). Then, for any constant M > 0, we have(60)∞=∑n=n0∞pnw123n≤M∑n=n0∞pnw12n.
Proceeding as in the proof of Theorem2 case (vii), we obtain a contradiction.
Finally, assume that case (viii) holds. Define(61)vn=D1unun<0.
From (45), we have(62)−D2un≥M1un+1∑s=n1n−11c3s∑t=n1s−1pt.
On the other hand, from the monotonicity ofD1u(n), we have(63)u∞−un=∑s=n∞D1usc1s≤D1unw1n,or(64)−1≤vnw1n<0.
Then, using (62), we have(65)Δvn=D2unc2nun+1−D1un2c1nunun+1≤−M1c2n∑s=n1n−11c3s∑t=n1s−1pt−v2nc1n.
Now, multiplying the last inequality byw1 (n + 1) and then summing from n1 to n − 1, one obtains(66)vnw1n−vn1w1n1+∑s=n1n−1vsc1s+∑s=n1n−1M1w1s+1c2s∑t=n1s−11c3t∑j=n1t−1pj+∑s=n1n−1w1s+1v2sc1s≤0.
Hence, in view of (64),(67)∑s=n1n−1M1w1s+1c2s∑t=n1s−11c3t∑j=n1t−1pj−14w1s+1c1s≤vn1w1n1+1,which contradicts (48). The proof is now complete.


## 3. Examples

In this section, we provide some examples to illustrate the applicability and strength of the results obtained in the previous section.Example 1.
Let us consider the following fourth-order delay difference equation:(68)Δn2Δn2Δn2Δun+q0n2un−k=0,n≥1,where q0 > 0 and k is a positive integer. It is easy to verify that condition (19) is satisfied and by Theorem 1, one can conclude that any nonoscillatory solution of (68) converges to zero as n ⟶ ∞.Example 2.
Consider the following fourth-order delay difference equation:(69)Δnn+1Δnn+1Δnn+1Δun+q0n2un−k=0,n≥1,where q0 > 0 and k is a positive integer. By simple calculation, we see that w1n=w2n=w3n=1/n and Bn,1≈q0n/6. Hence, by Theorem 2, equation (69) is oscillatory if q0 > 6. The same conclusion follows from Theorem 3 if q0>3/2.



---

*Source: 1653081-2020-03-21.xml*