# Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

Authors: Cemil Tunç; Muzaffer Ateş
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution
of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.


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## Body

## 1. Introduction

In this paper, we study the boundedness of solutions to fourth-order nonlinear differential equation:(1)x′′′′+ax′′′+f(x,x′,x′′)+g(x,x′)+h(x)=p(t),
where x∈ℜ, t∈[0,∞), a>0, f, g, h, p, and their first derivatives are continuous functions depending on the arguments shown. In addition, the functions h and p are oscillatory in the following sense: for each argument u, there exist numbers β1>α1>u>α-1>β-1  such that
(2)φ(α1)<0,φ(β1)>0,φ(α-1)<0,φ(β-1)>0,
where φ is either h(x) or p(t), u is either x or t, and all roots of the restoring term h(x) are isolated.It should be noted that there exist many papers dealing with boundedness of solutions to certain nonlinear differential equations of third and fourth order in the literature [1–15]. For nonlinear differential equations of fourth order, Afuwape and Adesina [1] used the frequency-domain approach to discuss the stability and periodicity of solutions, while Tunç and Tiryaki [11, 12] used intrinsic method to study the boundedness and stability of solutions. On the same time, Tunç [13–15] used Lyapunov’s second method to investigate the stability and boundedness properties of solutions of certain fourth order nonlinear differential equations. Further, other papers in this connection include those of Andres [2], Ogundare [6], and Omeike [7, 8], where the Cauchy formula was applied to evaluate the boundedness of solutions to certain third and fourth order nonlinear differential equations with oscillatory restoring and forcing terms.The aim of this work is to extend and improve the previous studies and make some contributions to the literature since there are only a few papers on the boundedness of solutions of fourth order differential equations with oscillatory restoring and forcing terms (see [6–8]). It should be noted that the equation considered here, (1), includes and extends that of Ogundare [6] and Omeike [7, 8].


## 2. Preliminary Results

We need the following lemmas in the proof of our main result.Lemma 1.
One assumes that there exist positive constantsa, b, c, H, and P, (a2>4b) such that the following  conditions hold for all x∈R and t≥0: (i)
|h(x)|≤H,(ii)
|p(t)|≤P,(iii)
0<f(x,x′,x′′)/x′′≤b<∞, (x′′≠0), f(x,x′,0)=0,(iv)
0<g(x,x′)/x′≤c<∞, (x′≠0), g(x,0)=0.
Then, each solutionx(t) of (1) satisfies
(3)limsupt→∞|x′′′(t)|≤4(H+P)a∶=D′′′,
provided that
(4)limsupt→∞|x′(t)|≤H+Pc∶=D′,(5)limsupt→∞|x′′(t)|≤2(H+P)b∶=D′′.
Note that the constantsa, b, and c satisfy the conditions ensuring that the auxiliary equation
(6)λ3+aλ2+bλ+c=0
has negative real roots.Proof.
Substitutingz∶=x′′′, we get from (1) that
(7)z′+az=p(t)-f(x,x′,x′′)-g(x,x′)-h(x),
with the solutions of the form
(8)x′(t)=z(t)x′(t)=Ce-at+∫Txte-a(t-τ)[p(τ)-f(x(τ),x′(τ),x′′(τ))11111111111111111111111-g(x(τ),x′(τ))11111111111111111111111-h(x(τ))[p(τ)-f(x(τ),x′(τ),x′′(τ))]dτ,
where C is an arbitrary constant and Tx is a great enough number. Let us assume that the assumptions (4) and (5) hold. Thus, by the conditions of Lemma 1, for t≥Tx, we have not only
(9)|∫Txte-a(t-τ)[p(τ)-f(x(τ),x′(τ),x′′(τ))1111111111-g(x(τ),x′(τ))-h(x(τ))]dτ∫Txte-a(t-τ)[p(τ)-f(x(τ),x′(τ),x′′(τ))|≤∫Txt(|p(τ)|+|f(x(τ),x′(τ),x′′(τ))x′′(τ)|111111111×|x′′(τ)|(|p(τ)|+|f(x(τ),x′(τ),x′′(τ))x′′(τ)|)e-a(t-τ)dτ+∫Txt(|g(x(τ),x′(τ))x′(τ)||x′(τ)|1111111111+|h(x(τ))|(|g(x(τ),x′(τ))x′(τ)||x′(τ)|)e-a(t-τ)dτ≤∫Txt(P+b|x′′(τ)|+c|x′(τ)|+H)e-a(t-τ)dτ≤4(H+P)a(1-e-a(t-Tx)),
but also
(10)limsupt→∞|x′′′(t)|≤4(H+P)a.
This completes the proof of Lemma1.Lemma 2.
In addition to the assumptions of Lemma1, one assumes that the following conditions hold: (i)
|h′(x)|≤H′,(ii)
|∫0∞p(t)dt|<∞,
whereH′ is a suitable constant. Then, every bounded solution x(t) of (1) either satisfies the relation
(11)limt→∞x(t)=x-,limt→∞x′(t)=limt→∞x′′(t)=limt→∞x′′′(t)=0,(h(x-)=0)
or there exists a root x- of h(x) such that (x(t)-x-) oscillates.Proof.
Letx(t) be a fixed bounded solution of (1). Substituting this solution into (1) and integrating the result from Tx to t (Tx—a great enough number), we obtain the following:
(12)∫Txth(x(τ))dτ=-∫Txtf(x(τ),x′(τ),x′′(τ))dτ-∫Txtg(x(τ),x′(τ))dτ-x′′′(t)+x′′′(Tx)+a[-x′′(t)+x′′(Tx)]+∫Txtp(τ)dτ(∶=I(t)).
By noting the assumption (ii) of Lemma1 and the boundedness of solution x(t), it follows that there exists a constant Mx for t≥Tx such that
(13)|I(t)|≤Mx,thatis,|∫Txth(x(τ))dτ|≤Mx.
Now let us assume thatx(t) does not converge to any root x- of h(x), that is,
(14)limsupt→∞|x(t)-x-|>0
and simultaneously,
(15)h(x(t))≥0orh(x(t))≤0fort≥Tx.
Then,(16)H(t)∶=∫Txth(x(τ))dτ,(t≥Tx)
evidently is a composed monotone function with a finite or infinite limit for t→∞.
Since (13) implies that the “divergent case” can be disregarded, then it follows from (15) that not only
(17)limt→∞∫Txt|h(x(τ))|dτ=limt→∞|∫Txth(x(τ))dτ|≤Mx
but also
(18)liminft→∞|x(t)-x-|=0
holds, because otherwise if (i.e., liminft→∞|x(t)-x-|>0) (15) together with the fact that the roots of h(x) are isolated would yield
(19)liminft→∞|h(x(t))|=liminft→∞|h(x(t))-h(x-)|>0,
which is a contradiction to (17).
Thus, the estimates (14) and (18) imply that
(20)limsupt→∞|h(x(t))|=limsupt→∞|h(x(t))-h(x-)|>0,
and consequently, there exist such a sequence {ti}≥Tx and a constant H~>0 such that (in what follows, d(x,y) denotes the distance between x and y)(α)
liminft→∞⇒ti→∞d(ti,ti-1)>0,(β)
|h(x(ti))|≥H~
hold. Hence, the estimate(21)Mx≥limt→∞∫t1t|h(x(τ))|dτ=∑i=2∞|∫ti-1tih(x(τ))|dτ
implies that
(22)limsupt→∞⇒ti→∞∫ti-1ti|h(x(t))|dt=0
or
(23)H′limsupt→∞|x′(t)|≥limsupt→∞|dh(x(t))dx(t)x′(t)|H′limsupt→∞|x′(t)|=limsupt→∞|dh(x(t))dt|=∞.
However, according to the assertion of Lemma1, this case is impossible, and that is why (x(t)-x-) necessarily oscillates. The remaining part of Lemma 2 is followed from the assertion
(24)x(t)∈C(n)[0,∞),limt→∞|x(n)(t)|<∞,limt→∞|x(t)|<∞⟹limt→∞x(k)(t)=0,
where n≥2 is a natural number and k=1,…,(n-1). This completes the proof.Lemma 3.
In addition to the assumptions of Lemma2, one assumes that the following conditions hold:  (i)
|p′(t)|≤P′,(ii)
limsupt→∞|p(t)|>0,(iii)
max{|fx(x,x′,x′′)|,|fx′(x,x′,x′′)|,|fx′′(x,x′,x′′)|}≤b0,(iv)
max{|gx(x,x′)|,|gx′(x,x′)|}≤c0,
whereb0, c0, and P′ are suitable constants. Then, for every bounded solution x(t) of (1), there exists a root x- of h(x) such that (x(t)-x-) oscillates.Proof.
If Lemma3 does not hold, then according to Lemma 2, (11) holds and the fifth derivative of x(t) satisfies
(25)x(v)(t)=p′(t)-ax′′′′(t)-fx(x,x′,x′′)x′(t)x(v)(t)=-fx′(x,x′,x′′)x′′(t)-fx′′(x,x′,x′′)x′′′(t)x(v)(t)=-gx(x,x′)x′(t)-gx′(x,x′)x′′(t)x(v)(t)=-h′(x(t))x′(t).
Thus, by the assumptions of Lemmas2 and 3, we have
(26)|x(v)(t)|≤P′+a|x′′′′(t)|+b0|x′′′(t)||x(v)(t)|≤+(b0+c0)|x′′(t)|+(H′+b0+c0)|x′(t)|.
Hence, by the boundedness ofx′(t), x′′(t), x′′′(t), and x′′′′(t), it follows that there exists a constant K such that
(27)limsupt→∞|x(v)(t)|≤K,
which according to (24), gives the following estimates:
(28)limt→∞x(t)=x-⟹limt→∞h(x(t))=h(x-)=0,limt→∞x(j)(t)=0,j=1,2,3
or
(29)limsupt→∞|p(t)|=limsupt→∞|x′′′′+ax′′′+f(x,x′,x′′)111111111111+g(x,x′)+h(x)|=0,
which is a contradiction to limsupt→∞|p(t)|>0. This completes the proof of Lemma 3.We now give the main result of this paper.


## 3. Main Result

Theorem 4.
One supposes that there exist positive constantsH, H′, P, P′, P0, and R such that for |x|>R and t≥0 the following conditions hold:(i)
|h(x)|≤H, |h′(x)|≤H′,(ii)
0<f(x,x′,x′′)/x′′≤b<∞, (x′′≠0), f(x,x′,0)=0,(iii)
0<g(x,x′)/x′≤c<∞, (x′≠0), g(x,0)=0(iv)
|p(t)|≤P, |p′(t)|≤P′, |∫0tp(τ)dτ|≤P0, limsupt→∞|p(t)|>0,(v)
min[d(x-k,x-k+1),d(x-k,x-k-1)]>((H+P)/c)(4/a+2a/b+b/c)+P0/c,
wherex-k are roots of h(x) with h′(x-k)>0 and x-k-1, x-k+1 denote the couple of adjacent roots of x-k(k=0,±2,±4,…). Then, all solutions x(t) of (1) are bounded, and for each of them, there exists a root x- of h(x) such that (x(t)-x-) oscillates.Proof.
Let us assume, on the contrary, thatx(t) is an unbounded solution of (1), that is,limsupt→∞x(t)=∞. Then, Lemma 1 implies the existence of a number T0≥0 such that t≥T0,
(30)|x′(t)|≤D′+ε1,|x′′(t)|≤D′′+ε2,|x′′′(t)|≤D′′′+ε3,
with εi>0, (i=1,2,3), small enough constants.
LetT1≥T0 be the last point with x(T1)=x-k (k-even) and T2>T1 the first point with x(T2)=x-k+1. If we integrate (1) from T1 to t, T1≤t≤T2, we come to
(31)x′′′(t)-x′′′(T1)+a[x′′(t)-x′′(T1)]+∫T1tf(x(τ),x′(τ),x′′(τ))dτ+∫T1tg(x(τ),x′(τ))dτ+∫T1th(x(τ))dτ=∫T1tp(τ)dτ.
Therefore, on replacing∫T1tf(x(τ),x′(τ),x′′(τ))dτ and ∫T1tg(x(τ),x′(τ))dτ with b[x′(t)-x′(T1)] and c[x(t)-x(T1)], respectively, for T1≤t≤T2, we have h(x(t))sgnx(t)≥0. Multiplying (31) by sgnx(t), we obtain
(32)|x(t)|≤|x(T1)|+2c[D′′′+aD′′+bD′+P02]+ε,
where ε>0 is an arbitrary small constant, a contradiction to x(T2)=x-k+1. The remaining part of the theorem follows from Lemma 3; therefore, we omit the details of the proof. The proof is complete.Example 5.
Consider the differential equation(33)x′′′′(t)+2x′′′(t)+5x′′(t)4[1+x2(t)+x′2(t)]+x′(t)4+x2(t)+1100sinx(t)=1100cost,
where a=2, b=5/4, c=1/4, h(x(t))=(1/100)sinx(t), p(t)=(1/100)cost with sinx(t) and cost, for t≥0, being oscillatory, and the equation
(34)λ3+2λ2+54λ+14=0
has negative real roots. A simple calculation (with the earlier notation) gives that H=0.01, H′=0.01, P=0.01, P′=0.01, and P0=0.01. From the condition (v) of the theorem, since h(x(t))=(1/100)sinx(t), then the roots of h(x(t)) are x-k-1=(k-1)π, x-k=kπ, and x-k+1=(k+1)π, (k=0,±2,±4,…), where x-k-1 and x-k+1 are the couple of adjacent roots of x-k.
Thus,(35)min[d(x-k,x-k+1),d(x-k,x-k-1)]=π,H+Pc(4a+2ab+bc)+P0c=4[(0.01+0.01)(2+165+5)+0.01]=0.856<1.
Sinceπ>1, then all the conditions of the theorem are satisfied; thus, all solutions x(t) of the above differential equation and their derivatives up to order three are bounded, and for each of them, there exists a root x- of h(x(t)) such that (x(t)-x-) oscillates.



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