# On the Convergence of Solutions of Certain Third-Order Differential Equations

Authors: Ercan Tunç
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We establish sufficient conditions for the convergence of solutions of a certain third-order nonlinear differential equations. By constructing a Lyapunov function as the basic tool, some results which exist in the relevant literature are generalized.


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## Body

## 1. Introduction

As well known, in the investigation of qualitative behaviors of solutions, stability, convergence, boundedness, oscillation, and so forth of solutions are very important problems in theory and applications of differential equations. For example, in applied sciences, some practical problems concerning mechanics, the engineering technique fields, economy, control theory, physics, chemistry, biology, medicine, atomic energy, information theory, and so forth are associated with certain higher-order linear or nonlinear differential equations. Ever since Lyapunov [1] proposed his famous theory on the stability of motion, For some papers published on the qualitative behaviors of solutions of nonlinear second-and third-order differential equations, the readers can referee to the papers of Afuwape and Omeike [2, 3], Ezeilo [4, 5], Meng [6], Tejumola [7, 8], Tunç [9–11], Omeike [12], and the references listed in these papers as well as one can refer to the books of Reissig et al. [13, 14]. The motivation for the present work has been inspired basically by the paper of Afuwape and Omeike [2]  and the papers listed above. Our aim here is to extend the results established by Afuwape and Omeike [2] to nonlinear differential equation (1.4) for the convergence of all solutions of this equation.  In 2008, Afuwape and Omeike [2] considered third-order nonlinear differential equations of the form
(1.1)x⃛+aẍ+g(ẋ)+h(x)=p(t,x,ẋ,ẍ),
and by introducing a Lyapunov function they discussed the convergence of solutions for this equation. During establishment of the results, Afuwape and Omeike [2] defined the following relations with respect to the functions g and h:
(1.2)0<b≤g(y2)-g(y1)y2-y1≤b0<∞,
for any pair of constants y2,y1(y2≠y1) and
(1.3)0<δ≤h(x2)-h(x1)x2-x1≤kab,
for any pair of constants x2,x1(x2≠x1), where k<1 is a positive constant.In this paper, we consider nonlinear differential equation of the form(1.4)x⃛+f(ẍ)+g(ẋ)+h(x)=p(t,x,ẋ,ẍ),
where the functions f,g,h, and p are continuous in their respective arguments, with the functions f, g, and h are not necessarily differentiable. In addition to (1.2) and (1.3) we assume that
(1.5)0<a≤f(z2)-f(z1)z2-z1≤a0<∞,
for any pair of constants z2,z1(z2≠z1).By convergence of solutions we mean, any two solutionsx1(t),x2(t) of (1.4) are said to converge to each other if
(1.6)x2(t)-x1(t)→0,ẋ2(t)-ẋ1(t)→0,ẍ2(t)-ẍ1(t)→0
as t→∞.


## 2. Main Results

The following results are established.Theorem 2.1.
Suppose thatf(0)=g(0)=h(0), and that(i)
there are constantsa>0,a0>0  such that f(z) satisfies inequalities (1.5),(ii)
there are constantsb>0,b0>0  such that g(y) satisfies inequalities (1.2),(iii)
there are constantsδ>0,  k<1  such that for any ξ,η(η≠0), the incrementary ratio for h satisfies
(2.1)(h(ξ+η)-h(ξ))ηliesinI0
with I0=[δ,kab],(iv)
there is a continuous functionϕ(t) such that
(2.2)|p(t,x2,y2,z2)-p(t,x1,y1,z1)|≤ϕ(t){|x2-x1|+|y2-y1|+|z2-z1|}holds for arbitrary t,x1,y1,z1,x2,y2,z2, and satisfies
(2.3)∫0t‍ϕν(τ)dτ≤D1t
for some constant D1>0, where ν is a constant in the range 1≤ν≤2.
Then all solutions of (1.4) converge.A very important step in the proof of Theorem2.1 will be to give estimate for any two solutions of (1.4). This in itself, being of independent interest, is giving as follows.Theorem 2.2.
Letx1(t),x2(t) be any two solutions of (1.4). Suppose that all the conditions of Theorem 2.1 are satisfied, then for each fixed ν, in the range 1≤ν≤2, there exist constants D2,D3, and D4 such that for t2≥t1,
(2.4)S(t2)≤D2S(t1)exp{-D3(t2-t1)+D4‍∫t1t2ϕν(τ)dτ},
where
(2.5)S(t)={[x2(t)-x1(t)]2+[ẋ2(t)-ẋ1(t)]2+[ẍ2(t)-ẍ1(t)]2}.We have the following corollaries whenx1(t)=0 and t1=0.Corollary 2.3.
Suppose thatp=0 in (1.4) and suppose further that conditions (i), (ii), and (iii) of Theorem 2.1 hold, then the trivial solution of (1.4) is exponentially stable in the large.Also, if we putξ=0 in (2.1) with η(η≠0) arbitrary, we get the following.Corollary 2.4.
Ifp≠0 and hypotheses (i), (ii), and (iii) of Theorem 2.1 hold for arbitrary η(η≠0) , and ξ=0, then there exists a constant D5>0 such that every solution x(t) of (1.4) satisfies
(2.6)|x(t)|≤D5,|ẋ(t)|≤D5,|ẍ(t)|≤D5.


## 3. Preliminary Results

On settingẋ=y,ẏ=z, (1.4) can be replaced by an equivalent system
(3.1)ẋ=y,ẏ=z,ż=-f(z)-g(y)-h(x)+p(t,x,y,z).
Let (xi(t),yi(t),zi(t)),  i=1,2, be any two solutions of (3.1) such that
(3.2)a≤f(z2)-f(z1)z2-z1≤a0(z2≠z1),b≤g(y2)-g(y1)y2-y1≤b0(y2≠y1),δ≤h(x2)-h(x1)x2-x1≤kab(x2≠x1),
where a0,a,b0,b,δ, and k are finite constants, and k will be determined later.Our investigation rests mainly on the properties of the function,W=W(x2-x1,y2-y1,z2-z1) defined by
(3.3)2W=β(1-β)b2(x2-x1)2+βb(y2-y1)2+αba-1(y2-y1)2+αa-1(z2-z1)2+{(z2-z1)+a(y2-y1)+(1-β)b(x2-x1)}2,
where 0<β<1 and α>0 are constants.Following the argument used in [5], we can easily verify the following for W.Lemma 3.1.
(i)W(0,0,0)=0.
(ii) There exist finite positive constantsD6,D7 such that
(3.4)D6{(x2-x1)2+(y2-y1)2+(z2-z1)2}≤W≤D7{(x2-x1)2+(y2-y1)2+(z2-z1)2},
where
(3.5)D6=12min{β(1-β)b2,b(β+αa-1),αa-1},
and using the inequality |x||y|≤(1/2)(x2+y2),(3.6)D7=12max{b(1-β)(1+b+a),b(β+αa-1)+a(1+a+b(1-β)),1+αa-1+a+b(1-β)}.If we define the functionW(t) by
(3.7)W(x2(t)-x1(t),y2(t)-y1(t),z2(t)-z1(t))
and using the fact that the solutions (xi,yi,zi),i=1,2, satisfy (3.1), then S(t) as defined in (2.5) becomes
(3.8)S(t)={[x2(t)-x1(t)]2+[y2(t)-y1(t)]2+[z2(t)-z1(t)]2}.Lemma 3.2.
Assume that the conditions (i), (ii), and (iii) of Theorem2.1 are satisfied. Then, there exist positive finite constants D8 and D9 such that
(3.9)dWdt≤-2D8S+D9S1/2|θ|,
where θ=p(t,x2,y2,z2)-p(t,x1,y1,z1).Proof of Lemma3.2
Differentiating the functionW in (3.3) along the system (3.1) we obtain
(3.10)Ẇ=dWdt=-W1-W2-W3-W4-W5-W6-W7+W8,
in which
(3.11)W1={γ1b(1-β)H(x2,x1)(x2-x1)2+η1a[G(y2,y1)-b(1-β)](y2-y1)2+ξ1αa-1F(z2,z1)(z2-z1)2+(F(z2,z1)-a)(z2-z1)2},W2={γ2b(1-β)H(x2,x1)(x2-x1)2+ξ2αa-1F(z2,z1)(z2-z1)2+(1+αa-1)(x2-x1)(z2-z1)H(x2,x1)},W3={γ3b(1-β)H(x2,x1)(x2-x1)2+η2a[G(y2,y1)-b(1-β)](y2-y1)2+a(x2-x1)(y2-y1)H(x2,x1)},W4={γ4b(1-β)H(x2,x1)(x2-x1)2+ξ3αa-1F(z2,z1)(z2-z1)2+b(1-β)(x2-x1)(z2-z1)[F(z2,z1)-a]},W5={γ5b(1-β)H(x2,x1)(x2-x1)2+η3a[G(y2,y1)-b(1-β)](y2-y1)2+b(1-β)(x2-x1)(y2-y1)[G(y2,y1)-b]},W6={ξ4αa-1F(z2,z1)(z2-z1)2+η4a[G(y2,y1)-b(1-β)](y2-y1)2+(1+αa-1)(y2-y1)(z2-z1)[G(y2,y1)-b]},W7={ξ5αa-1F(z2,z1)(z2-z1)2+η5a[G(y2,y1)-b(1-β)](y2-y1)2+a(y2-y1)(z2-z1)[F(z2,z1)-a]},W8={b(1-β)(x2-x1)+a(y2-y1)+(1+αa-1)(z2-z1)}θ(t),
with
(3.12)F(z2,z1)=f(z2)-f(z1)z2-z1(z2≠z1),G(y2,y1)=g(y2)-g(y1)y2-y1(y2≠y1),H(x2,x1)=h(x2)-h(x1)x2-x1(x2≠x1),
and ξi,ηi, and γi,(i=1,2,3,4,5) are strictly positive constants such that
(3.13)∑i=15‍ξi=1,∑i=15‍ηi=1,∑i=15‍γi=1.
Also, let us denote F(z2,z1),G(y2,y1), and H(x2,x1) simply by F,G, and H, respectively. For strictly positive constants k1,k2,k3,k4,k5, and k6 conveniently chosen later, we get
(3.14)(1+αa-1)(x2-x1)(z2-z1)H={k1(1+αa-1)1/2H1/2(x2-x1)+12k1-1(1+αa-1)1/2H1/2(z2-z1)}2-k12(1+αa-1)H(x2-x1)2-14k1-2(1+αa-1)H(z2-z1)2,a(x2-x1)(y2-y1)H={k2a1/2H1/2(x2-x1)+12k2-1a1/2H1/2(y2-y1)}2-k22aH(x2-x1)2-14k2-2aH(y2-y1)2,b(1-β)(x2-x1)(z2-z1)[F-a]={12k3-1b1/2(1-β)1/2[F-a]1/2(x2-x1)+k3b1/2(1-β)1/2[F-a]1/2(z2-z1)}2-14k3-2b(1-β)[F-a](x2-x1)2-k32b(1-β)[F-a](z2-z1)2,b(1-β)(x2-x1)(y2-y1)[G-b]={k4b1/2(1-β)1/2[G-b]1/2(x2-x1)+12k4-1b1/2(1-β)1/2[G-b]1/2(y2-y1)}2-k42b(1-β)[G-b](x2-x1)2-14k4-2b(1-β)[G-b](y2-y1)2,(1+αa-1)(y2-y1)(z2-z1)[G-b]={k5(1+αa-1)1/2[G-b]1/2(y2-y1)+12k5-1(1+αa-1)1/2[G-b]1/2(z2-z1)}2-k52(1+αa-1)[G-b](y2-y1)2-14k5-2(1+αa-1)[G-b](z2-z1)2,a(y2-y1)(z2-z1)[F-a]={12k6-1a1/2[F-a]1/2(y2-y1)+k6a1/2[F-a]1/2(z2-z1)}2-14k6-2a[F-a](y2-y1)2-k62a[F-a](z2-z1)2
Thus,
(3.15)W2={k1(1+αa-1)1/2H1/2(x2-x1)+12k1-1(1+αa-1)1/2H1/2(z2-z1)}2+{γ2b(1-β)H-k12(1+αa-1)H}(x2-x1)2+{ξ2αa-1F-14k1-2(1+αa-1)H}(z2-z1)2,W3={k2a1/2H1/2(x2-x1)+12k2-1a1/2H1/2(y2-y1)}2+{γ3b(1-β)H-k22aH}(x2-x1)2+{η2a[G-b(1-β)]-14k2-2aH}(y2-y1)2,W4={12k3-1b1/2(1-β)1/2[F-a]1/2(x2-x1)+k3b1/2(1-β)1/2[F-a]1/2(z2-z1)}2+{γ4b(1-β)H-14k3-2b(1-β)[F-a]}(x2-x1)2+{ξ3αa-1F-k32b(1-β)[F-a]}(z2-z1)2,W5={k4b1/2(1-β)1/2[G-b]1/2(x2-x1)+12k4-1b1/2(1-β)1/2[G-b]1/2(y2-y1)}2+{γ5b(1-β)H-k42b(1-β)[G-b]}(x2-x1)2+{η3a[G-b(1-β)]-14k4-2b(1-β)[G-b]}(y2-y1)2,W6={k5(1+αa-1)1/2[G-b]1/2(y2-y1)+12k5-1(1+αa-1)1/2[G-b]1/2(z2-z1)}2+{η4a[G-b(1-β)]-k52(1+αa-1)[G-b]}(y2-y1)2+{ξ4αa-1F-14k5-2(1+αa-1)[G-b]}(z2-z1)2,W7={12k6-1a1/2[F-a]1/2(y2-y1)+k6a1/2[F-a]1/2(z2-z1)}2+{η5a[G-b(1-β)]-14k6-2a[F-a]}(y2-y1)2+{ξ5αa-1F-k62a[F-a]}(z2-z1)2.
Moreover, in view of (3.2),  we obtain for all xi,zi(i=1,2)  in  ℜ,(3.16)W2≥0,
if
(3.17)k12≤γ2(1-β)ab(α+a)withH≤4ξ2γ2α(1-β)a2b(α+a)2,
and for all xi,yi(i=1,2)  in ℜ,(3.18)W3≥0,
if
(3.19)k22≤γ3(1-β)bawithH≤4η2γ3β(1-β)b2a.
Combining all the inequalities in (3.16) and (3.18), we have for all xi,yi,zi(i=1,2) in ℜ,(3.20)W2≥0,W3≥0,
if
(3.21)H≤kabwithk=min{4ξ2γ2α(1-β)a(α+a)2,4η2γ3β(1-β)ba2}<1.
Also, for all xi,zi(i=1,2)in ℜ,(3.22)W4≥0,
if
(3.23)a0-a4γ4δ≤k32≤ξ3α(1-β)b(a0-a),
for all xi,yi(i=1,2) in ℜ,(3.24)W5≥0,
if
(3.25)(1-β)(b0-b)4βaη3≤k42≤δγ5(b0-b),
for all yi,zi(i=1,2) in ℜ,(3.26)W6≥0,
if
(3.27)(α+a)(b0-b)4ξ4αa≤k52≤η4βba2(α+a)(b0-b),
and for all yi,zi(i=1,2) in ℜ,(3.28)W7≥0,
if
(3.29)a0-a4η5βb≤k62≤ξ5αa(a0-a).
Further
(3.30)W1≥2D10{(x2-x1)2+(y2-y1)2+(z2-z1)2},
where 2D10=min{γ1bδ(1-β),η1abβ,ξ1α}, on the other hand
(3.31)W8≤D11{(x2-x1)2+(y2-y1)2+(z2-z1)2}1/2|θ(t)|,
where D11=2max{b(1-β),a,(1+αa-1)}.
Bringing together the estimates just obtained forW1,W2,W3,W4,W5,W6,W7, and W8 in (3.10) and using (3.8), we have
(3.32)dWdt≤-2D10S(t)+D11S1/2(t)|θ(t)|.
This completes the proof of Lemma 3.2.


## 4. Proof of Theorem2.2

This follows directly from [5], on using inequality (3.32). Let ν be any constant in the range 1≤ν≤2. Set 2μ=2-ν, so that 0≤μ≤1/2. We rewrite (3.32) in the form
(4.1)dWdt+D10S≤-D10S+D11S1/2|θ|=D11SμW*,
where
(4.2)W*=(|θ|-D12S1/2)S1/2-μ,
with D12=D10D11-1, considering the two cases(i)
|θ|<D12S1/2  and(ii)
|θ|≥D12S1/2separately. If|θ|<D12S1/2, then W*<0. On the other hand, if |θ|≥D12S1/2, then the definition of W* in (4.2) gives at least
(4.3)W*≤S(1/2-μ)|θ|
and also S1/2≤|θ|/D12. This implies that
(4.4)S1/2(1-2μ)≤[|θ|D12](1-2μ).
Therefore
(4.5)S1/2(1-2μ)|θ|≤[|θ|D12](1-2μ)×|θ|,
from which together with W*, we obtain
(4.6)W*≤D13|θ|2(1-μ),
where D13=D12(2μ-1). Again due to (4.1) and using the estimate on W* for W*, we have
(4.7)dWdt+D10S≤D11D13Sμ|θ|2(1-μ)≤D14Sμϕ2(1-μ)S(1-μ),
where D14=31-μD11D13, which follows from
(4.8)|θ|=|p(t,x2,y2,z2)-p(t,x1,y1,z1)|≤ϕ(t){|x2-x1|+|y2-y1|+|z2-z1|}.
In view of the fact that ν=2(1-μ), we obtain
(4.9)dWdt≤-D10S+D14ϕνS,
and on using inequality (3.4), we have
(4.10)dWdt+(D15-D16ϕν(t))W≤0
for some positive constants D15 and D16. On integrating (4.10) from t1 to t2(t2≥t1), we have
(4.11)W(t2)≤W(t1)exp{-D15(t2-t1)+D16∫t2t2‍ϕν(τ)dτ}.
Again, using Lemma 3.1, we obtain (2.4), with D2=D7D6-1,D3=D15, and D4=D16. This completes the proof of Theorem 2.2.


## 5. Proof of Theorem2.1

This follows from the estimate (2.4) and the condition (2.3) on ϕ(t). Choose D1=D3D4-1 in (2.3). From the estimate (2.4), if
(5.1)‍∫t1t2ϕν(τ)dτ≤D3D4-1(t2-t1),
then the exponential index remains negative for all t2-t1≥0. Then, as t=(t2-t1)→∞, we have S(t)→0, and this gives
(5.2)x2-x1→0,y2-y1→0,z2-z1→0,
as t→∞. This completes the proof of Theorem 2.1.



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