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

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.

1. Introduction

In this paper, we study the boundedness of solutions to fourth-order nonlinear differential equation: 𝑥^′′′′+𝑎𝑥^′′′+𝑓𝑥,𝑥^′,𝑥^′′+𝑔𝑥,𝑥^′+ℎ(𝑥)=𝑝(𝑡), (1) 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 𝜑𝛼₁<0,𝜑𝛽₁>0,𝜑𝛼₋₁<0,𝜑𝛽₋₁>0, (2) 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 constants 𝑎, 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 solution 𝑥(𝑡) of (1) satisfies limsup_𝑡→∞𝑥^′′′(𝑡≤4())𝐻+𝑃𝑎∶=𝐷^′′′, (3) provided that limsup_𝑡→∞𝑥^′(𝑡)≤𝐻+𝑃𝑐∶=𝐷^′,limsup_𝑡→∞𝑥^′′(𝑡≤2())𝐻+𝑃𝑏∶=𝐷^′′. (4)(5)
Note that the constants 𝑎, 𝑏, and 𝑐 satisfy the conditions ensuring that the auxiliary equation 𝜆³+𝑎𝜆²+𝑏𝜆+𝑐=0 (6) has negative real roots.

Proof. Substituting 𝑧∶=𝑥^′′′, we get from (1) that 𝑧^′+𝑎𝑧=𝑝(𝑡)−𝑓𝑥,𝑥^′,𝑥^′′−𝑔𝑥,𝑥^′−ℎ(𝑥), (7) with the solutions of the form 𝑥^′(𝑡)=𝑧(𝑡)=𝐶𝑒^−𝑎𝑡+^𝑡_𝑇𝑥𝑒^{−𝑎(𝑡−𝜏)}𝑝(𝜏)−𝑓𝑥(𝜏),𝑥^′(𝜏),𝑥^′′(𝜏)−𝑔𝑥(𝜏),𝑥^′(𝜏)−ℎ(𝑥(𝜏))𝑑𝜏, (8) where 𝐶 is an arbitrary constant and 𝑇_𝑥 is a great enough number. Let us assume that the assumptions (4) and (5) hold. Thus, by the conditions of Lemma 1, for 𝑡≥𝑇_𝑥, we have not only ^𝑡_𝑇𝑥𝑒^{−𝑎(𝑡−𝜏)}𝑝(𝜏)−𝑓𝑥(𝜏),𝑥^′(𝜏),𝑥^′′(𝜏)−𝑔𝑥(𝜏),𝑥^′(𝜏)−ℎ(𝑥(𝜏))𝑑𝜏≤^𝑡_𝑇𝑥𝑝(𝜏+𝑓𝑥(𝜏)),𝑥^′(𝜏),𝑥^′′(𝜏)𝑥^′′(𝜏)×𝑥^′′(𝜏)𝑒^{−𝑎(𝑡−𝜏)}+𝑑𝜏_𝑡𝑇𝑥𝑔𝑥(𝜏),𝑥^′(𝜏)𝑥^′(𝜏)𝑥^′(𝜏+|ℎ(𝑥(𝜏𝑒)))|^{−𝑎(𝑡−𝜏)}≤𝑑𝜏_𝑡𝑇𝑥𝑃+𝑏𝑥^′′(𝜏)+𝑐𝑥^′(𝜏)+𝐻𝑒^{−𝑎(𝑡−𝜏)}≤4()𝑑𝜏𝐻+𝑃𝑎1−𝑒^{−𝑎(𝑡−𝑇𝑥)}, (9) but also limsup_𝑡→∞𝑥^′′′(𝑡≤4())𝐻+𝑃𝑎. (10)
This completes the proof of Lemma 1.

Lemma 2. In addition to the assumptions of Lemma 1, one assumes that the following conditions hold: (i)|ℎ^′(𝑥)|≤𝐻^′, (ii)|∫^∞₀𝑝(𝑡)𝑑𝑡|<∞,
where 𝐻^′ is a suitable constant. Then, every bounded solution 𝑥(𝑡) of (1) either satisfies the relation lim_𝑡→∞𝑥(𝑡)=𝑥,lim_𝑡→∞𝑥^′(𝑡)=lim_𝑡→∞𝑥^′′(𝑡)=lim_𝑡→∞𝑥^′′′(𝑡)=0,(ℎ(𝑥)=0) (11) or there exists a root 𝑥 of ℎ(𝑥) such that (𝑥(𝑡)−𝑥) oscillates.

Proof. Let 𝑥(𝑡) be a fixed bounded solution of (1). Substituting this solution into (1) and integrating the result from 𝑇_𝑥 to 𝑡 (𝑇_𝑥—a great enough number), we obtain the following: ^𝑡_𝑇𝑥ℎ(𝑥(𝜏))𝑑𝜏=−^𝑡_𝑇𝑥𝑓𝑥(𝜏),𝑥^′(𝜏),𝑥^′′(𝜏)−𝑑𝜏_𝑡𝑇𝑥𝑔𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏−𝑥^′′′(𝑡)+𝑥^′′′𝑇_𝑥+𝑎−𝑥^′′(𝑡)+𝑥^′′𝑇_𝑥+^𝑡_𝑇𝑥𝑝(𝜏)𝑑𝜏(∶=𝐼(𝑡)). (12)
By noting the assumption (ii) of Lemma 1 and the boundedness of solution 𝑥(𝑡), it follows that there exists a constant 𝑀_𝑥 for 𝑡≥𝑇_𝑥 such that |𝐼(𝑡)|≤𝑀_𝑥,thatis,^𝑡_𝑇𝑥ℎ(𝑥(𝜏))𝑑𝜏≤𝑀_𝑥. (13)
Now let us assume that 𝑥(𝑡) does not converge to any root 𝑥 of ℎ(𝑥), that is, limsup_𝑡→∞|𝑥(𝑡)−𝑥|>0 (14) and simultaneously, ℎ(𝑥(𝑡))≥0orℎ(𝑥(𝑡))≤0for𝑡≥𝑇_𝑥. (15)
Then, 𝐻(𝑡)∶=^𝑡_𝑇𝑥ℎ(𝑥(𝜏))𝑑𝜏,𝑡≥𝑇_𝑥 (16) 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 lim_𝑡→∞^𝑡_𝑇𝑥|ℎ(𝑥(𝜏))|𝑑𝜏=lim_𝑡→∞^𝑡_𝑇𝑥ℎ(𝑥(𝜏))𝑑𝜏≤𝑀_𝑥 (17) but also liminf_𝑡→∞|𝑥(𝑡)−𝑥|=0 (18) holds, because otherwise if (i.e., liminf_𝑡→∞|𝑥(𝑡)−𝑥|>0) (15) together with the fact that the roots of ℎ(𝑥) are isolated would yield liminf_𝑡→∞|ℎ(𝑥(𝑡))|=liminf_𝑡→∞|ℎ(𝑥(𝑡))−ℎ(𝑥)|>0, (19) which is a contradiction to (17).
Thus, the estimates (14) and (18) imply that limsup_𝑡→∞|ℎ(𝑥(𝑡))|=limsup_𝑡→∞|ℎ(𝑥(𝑡))−ℎ(𝑥)|>0, (20) and consequently, there exist such a sequence {𝑡_𝑖}≥𝑇_𝑥 and a constant 𝐻>0 such that (in what follows, 𝑑(𝑥,𝑦) denotes the distance between 𝑥 and 𝑦)(α)liminf_{𝑡→∞⇒𝑡𝑖→∞}𝑑(𝑡_𝑖,𝑡_𝑖−1)>0, (β)|ℎ(𝑥(𝑡_𝑖))|≥𝐻
hold. Hence, the estimate 𝑀_𝑥≥lim_𝑡→∞^𝑡_𝑡1|ℎ(𝑥(𝜏))|𝑑𝜏=^∞_𝑖=2^𝑡𝑖_𝑡𝑖−1ℎ(𝑥(𝜏))𝑑𝜏 (21) implies that limsup_{𝑡→∞⇒𝑡𝑖→∞}^𝑡𝑖_𝑡𝑖−1|ℎ(𝑥(𝑡))|𝑑𝑡=0 (22) or 𝐻^′limsup_𝑡→∞𝑥^′(𝑡)≥limsup_𝑡→∞𝑑ℎ(𝑥(𝑡))(𝑡)𝑥𝑑𝑥^′(𝑡)=limsup_𝑡→∞𝑑ℎ(𝑥(𝑡))𝑑𝑡=∞. (23)
However, according to the assertion of Lemma 1, this case is impossible, and that is why (𝑥(𝑡)−𝑥) necessarily oscillates. The remaining part of Lemma 2 is followed from the assertion 𝑥(𝑡)∈𝐶^(𝑛)[0,∞),lim_𝑡→∞𝑥^(𝑛)(𝑡)<∞,lim_𝑡→∞|𝑥(𝑡)|<∞⟹lim_𝑡→∞𝑥^(𝑘)(𝑡)=0, (24) where 𝑛≥2 is a natural number and 𝑘=1,…,(𝑛−1). This completes the proof.

Lemma 3. In addition to the assumptions of Lemma 2, one assumes that the following conditions hold: (i)|𝑝^′(𝑡)|≤𝑃^′, (ii)limsup_𝑡→∞|𝑝(𝑡)|>0, (iii)max{|𝑓_𝑥(𝑥,𝑥^′,𝑥^′′)|,|𝑓_𝑥′(𝑥,𝑥^′,𝑥^′′)|,|𝑓_𝑥′′(𝑥,𝑥^′,𝑥^′′)|}≤𝑏₀, (iv)max{|𝑔_𝑥(𝑥,𝑥^′)|,|𝑔_𝑥′(𝑥,𝑥^′)|}≤𝑐₀,
where 𝑏₀, 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 Lemma 3 does not hold, then according to Lemma 2, (11) holds and the fifth derivative of 𝑥(𝑡) satisfies 𝑥^(𝑣)(𝑡)=𝑝^′(𝑡)−𝑎𝑥^′′′′(𝑡)−𝑓_𝑥𝑥,𝑥^′,𝑥^′′𝑥^′(𝑡)−𝑓_𝑥′𝑥,𝑥^′,𝑥^′′𝑥^′′(𝑡)−𝑓_𝑥′′𝑥,𝑥^′,𝑥^′′𝑥^′′′(𝑡)−𝑔_𝑥𝑥,𝑥^′𝑥^′(𝑡)−𝑔_𝑥′𝑥,𝑥^′𝑥^′′(𝑡)−ℎ^′(𝑥(𝑡))𝑥^′(𝑡). (25)
Thus, by the assumptions of Lemmas 2 and 3, we have 𝑥^(𝑣)(𝑡)≤𝑃^′+𝑎𝑥^′′′′(𝑡)+𝑏₀𝑥^′′′(𝑡+𝑏₀)+𝑐₀𝑥^′′(𝑡)+𝐻^′+𝑏₀+𝑐₀𝑥^′(𝑡). (26)
Hence, by the boundedness of 𝑥^′(𝑡), 𝑥^′′(𝑡), 𝑥^′′′(𝑡), and 𝑥^′′′′(𝑡), it follows that there exists a constant 𝐾 such that limsup_𝑡→∞𝑥^(𝑣)(𝑡)≤𝐾, (27) which according to (24), gives the following estimates: lim_𝑡→∞𝑥(𝑡)=𝑥⟹lim_𝑡→∞ℎ(𝑥(𝑡))=ℎ(𝑥)=0,lim_𝑡→∞𝑥^(𝑗)(𝑡)=0,𝑗=1,2,3 (28) or limsup_𝑡→∞𝑝(𝑡)=limsup_𝑡→∞𝑥^′′′′+𝑎𝑥^′′′+𝑓𝑥,𝑥^′,𝑥^′′+𝑔𝑥,𝑥^′+ℎ(𝑥)=0, (29) which is a contradiction to limsup_𝑡→∞|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 constants 𝐻, 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^′, |∫^𝑡₀𝑝(𝜏)𝑑𝜏|≤𝑃₀, limsup_t→∞|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,
where 𝑥_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, that 𝑥(𝑡) is an unbounded solution of (1), that is,limsup_𝑡→∞𝑥(𝑡)=∞. Then, Lemma 1 implies the existence of a number 𝑇₀≥0 such that 𝑡≥𝑇₀, 𝑥^′(𝑡)≤𝐷^′+𝜀₁,𝑥^′′(𝑡)≤𝐷^′′+𝜀₂,𝑥^′′′(𝑡)≤𝐷^′′′+𝜀₃, (30) with 𝜀_i>0, (𝑖=1,2,3), small enough constants.
Let 𝑇₁≥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 𝑥^′′′(𝑡)−𝑥^′′′𝑇₁+𝑎𝑥^′′(𝑡)−𝑥^′′𝑇₁+^𝑡_𝑇1𝑓𝑥(𝜏),𝑥^′(𝜏),𝑥^′′(𝜏)𝑑𝜏+^𝑡_𝑇1𝑔𝑥(𝜏),𝑥^′(𝜏)+𝑑𝜏_𝑡𝑇1ℎ(𝑥(𝜏))𝑑𝜏=^𝑡_𝑇1𝑝(𝜏)𝑑𝜏. (31)
Therefore, on replacing ∫^𝑡_𝑇1𝑓(𝑥(𝜏),𝑥^′(𝜏),𝑥^′′(𝜏))𝑑𝜏 and ∫^𝑡_𝑇1𝑔(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 |𝑥(𝑡)|≤𝑥𝑇₁+2𝑐𝐷^′′′+𝑎𝐷^′′+𝑏𝐷^′+𝑃₀2+𝜀, (32) 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 𝑥^′′′′(𝑡)+2𝑥^′′′(𝑡)+5𝑥^′′(𝑡)41+𝑥²(𝑡)+𝑥^′2(𝑡+𝑥)^′(𝑡)4+𝑥²(𝑡)+1(𝑡)=1100sin𝑥100cos𝑡, (33) 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 𝜆³+2𝜆²+541𝜆+4=0 (34) 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, min𝑑𝑥_𝑘,𝑥_𝑘+1,𝑑𝑥_𝑘,𝑥_𝑘−1=𝜋,𝐻+𝑃𝑐4𝑎+2𝑎𝑏+𝑏𝑐+𝑃₀𝑐=4(0.01+0.01)2+165+5+0.01=0.856<1. (35)
Since 𝜋>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.