Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

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R E S E A R C H

Open Access

Stability and square integrability of

derivatives of solutions of nonlinear fourth

order differential equations with delay

Erdal Korkmaz

*

*_{Correspondence:}

korkmazerdal36@hotmail.com Mathematics, Faculty of Arts and Sciences, Mu¸s Alparslan University, 49100, Mu¸s, Turkey

Abstract

In this paper, we give suﬃcient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order

non-autonomous diﬀerential equations with delay by using Lyapunov’s second method. The results obtained essentially improve, include and complement the results in the literature.

MSC: 34D20; 34C11

Keywords: stability; boundedness; Lyapunov functional; delay diﬀerential equations; fourth order; square integrability

1 Introduction

In mathematical literature, ordinary differential equations have been studied for more than  years since the seventeenth century after the concepts of differentiation and integration were formulated by Newton and Leibniz. By means of ordinary differential equations, researchers can explain many natural phenomena like gravity, projectiles, wave, vibration, nuclear physics, and so on. In addition, in Newtonian mechanics, the system’s state variable changes over time, and the law that governs the change of the system’s state is normally described by an ordinary differential equation. The question concerning the stability of ordinary differential equations has been originally raised by the general prob-lem of the stability of motion [].

However, thereafter along with the development of technology, it have been seen that the ordinary diﬀerential equations cannot respond to the needs arising in sciences and engineering. For example, in many applications, it can be seen that physical or biological background of a modeling system shows that the change rate of the system’s current status often depends not only on the current state but also on the history of the system. This usually leads to the so-called retarded functional diﬀerential equations [].

In particular, for more results on the stability, boundedness, convergence, etc. of ordi-nary or functional diﬀerential equations of fourth order, see the book of Reissig et al. [] as a good survey for the works done by  and the papers of Burton [], Cartwright [], Ezeilo [–], Harrow [, ], Tunç [–], Remili et al. [–], Wu [] and others and the references therein. This information indicates the importance of investigating the

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qualitative properties of solutions of retarded functional diﬀerential equations of fourth order.

In this paper, we study the uniform asymptotic stability of the solutions forp(t,x,x,x,

x)≡ and also square integrability and boundedness of solutions to the fourth order nonlinear diﬀerential equation with delay

x()+a(t)gx(t)x(t)+b(t)qx(t)x(t)+c(t)fx(t)x(t)

+d(t)hx(t–r)=pt,x,x,x,x. () For convenience, we get

θ(t) =g

x(t)x(t), θ(t) =q

x(t)x(t), θ(t) =f

x(t)x(t). We write () in the system form

x=y,

y=z,

z=w,

w= –a(t)g(x)w–b(t)q(x) +a(t)θ

z–b(t)θ+c(t)f(x)

y–d(t)h(x) +d(t)

t

t–r

h(x)y dη+p(t,x,y,z,w),

()

whereris a positive constant to be determined later, the functionsa,b,c,dare contin-uously diﬀerentiable functions and the functions f, h,g, q,pare continuous functions depending only on the arguments shown. Also derivativesg(x),q(x),f(x) andh(x) exist and are continuous. The continuity of the functionsa,b,c,d,p,g,g,q,q,f andh guar-antees the existence of the solutions of equation (). If the right-hand side of system () satisﬁes a Lipschitz condition inx(t),y(t),z(t),w(t) andx(t–r), and there exist solutions of system (), then it is the unique solution of system ().

Assume that there are positive constantsa,b,c,d,f,g,q,a,b,c,d,f,g,q,

m,M,δandηsuch that the following assumptions hold:

(A)  <a≤a(t)≤a; <b≤b(t)≤b; <c≤c(t)≤c; <d≤d(t)≤dfor

t≥.

(A)  <f≤f(x)≤f; <g≤g(x)≤g; <q≤q(x)≤qforx∈Rand

 <m<min{f,g, },M>max{f,g, }.

(A) h(_xx) ≥δ> forx= ,h() = .

(A) _∞(|a(t)|+|b(t)|+|c(t)|+|d(t)|)dt<η.

(A) |p(t,x,y,z,w)| ≤ |e(t)|.

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2 Preliminaries

We also consider the functional diﬀerential equation

.

x=f(t,xt), xt(θ) =x(t+θ), –r≤θ≤,t≥, ()

where f :I×CH→Rnis a continuous mapping,f(t, ) = ,CH:={φ∈(C[–r, ],Rn) :

φ ≤H}, and forH<H, there existsL(H) >  with|f(t,φ)|<L(H) whenφ<H.

Lemma ([]) Let V(t,φ) :I×CH→Rbe a continuous functional satisfying a local

Lipschitz condition,V(t, ) = ,and wedges Wisuch that

(i) W(φ)≤V(t,φ)≤W(φ).

(ii) V_() (t,φ)≤–W(φ).

Then the zero solution of equation()is uniformly asymptotically stable.

3 The main results

Lemma  ([]) Let h() = , xh(x) >  (x= ) and δ(t) –h(x)≥ (δ(t) > ), then

δ(t)H(x)≥h(x),where H(x) =_xh(s)ds.

Theorem  In addition to the basic assumptions imposed on the functions a,b,c,d,p,

f,h,g and q,suppose that there are positive constants h,δ,δ,ηandη such that the

following conditions are satisﬁed:

(i) h–a_dm_δ≤h(x)≤h_ forx∈R.

(ii) δ=dh_c_a_mM+c_aM_+_mδ<bq.

(iii) _–+_∞∞(|g(s)|+|q(s)|+|f(s)|)ds<η.

(iv) _∞|e(t)|dt<η.

Then any solution x(t) of equation () and its derivatives x(t), x(t) and x(t) are bounded and satisfy

∞



x(s) +x(s) +x(s)ds<∞,

provided that

r< 

dh

min

εcm

α+ β+ ,

bq–δ–εM(a+c)

,ε

α .

Proof To prove the theorem, we deﬁne a Lyapunov functional

W=W(t,x,y,z,w) =e–η t

γ(s)ds_V_, _()

where

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and

V= βd(t)H(x) +c(t)f(x)y+αb(t)q(x)z+a(t)g(x)z+ βa(t)g(x)yz

+βb(t)q(x) –αhd(t)

y–βz+αw+ d(t)h(x)y+ αd(t)h(x)z

+ αc(t)f(x)yz+ βyw+ zw+σ



–r

t

t+s

y(γ)dγds

withH(x) =_xh(s)ds,α=_a

m +ε,β= dh

cm +ε, andηare positive constants to be

deter-mined later in the proof. We can rearrange V as

V=a(t)g(x)

w

a(t)g(x)+z+βy



+c(t)f(x)

d(t)h(x)

c(t)f(x) +y+αz



+d

₍_t₎_h₍_x₎

c(t)f(x) + d(t)H(x) +σ



–r

t

t+s

y(γ)dγds+V+V+V,

where

V= d(t) x



h(s)

dh

cm

–  d(t)

c(t)f(x)h

₍_s₎_ds_,

V=

αb(t)q(x) –β–αc(t)f(x)z,

V=

βb(t)q(x) –αhd(t) –βa(t)g(x)

y+

α– 

a(t)g(x)

w.

Let

ε<min



am

,dh

cm

, bq–δ

M(a+c)

, ()

then



am

<α< 

am

, dh

cm

<β< dh

cm

. ()

By using conditions (A)-(A), (i)-(ii) and inequalities (), (), we have

V≥d(t)

d

cm x



h(s)

h

 –h

₍_s₎_ds_≥_,

V=

αb(t)q(x) –βa(t) –αc(t)f(x)+βαa(t) – z

≥α

bq–

dha

cm

–cM

am

– (a+cM)

z+β



m– 

z

≥αbq–δ–M(a+c)

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and

V≥β

bq–

α

βhd–βaM

y+

α– 

am

w

≥β

bq–

c

a

–a

dhM

cm

– (cm+aM)

y+w

≥βbq–δ–M(c+a)

y+w≥.

Thus, it is clear from the above inequalities that there exists a positive constantDsuch

that

V≥D

y+z+w+H(x). ()

From Lemma , (A) and (i), it follows that there is a positive constantDsuch that

V≥D

x+y+z+w. ()

In this way,Vis positive deﬁnite. From (A)-(A), it is clear that there is a positive constant

Usuch that

V≤U

x+y+z+w. ()

From (iii), we have

t



θ(s)+θ(s)+θ(s)ds

=

α(t)

α(t)

g(u)+q(u)+f(u)du

≤

+∞

–∞

g(u)+q(u)+f(u)du<η<∞, ()

whereα(t) =min{x(),x(t)}andα(t) =max{x(),x(t)}. From inequalities (), () and (),

it follows that

W≥D

x+y+z+w, ()

whereD=D_e–

η+η

η _{. Also, it is easy to see that there is a positive constant}_U

such that

W≤U

x+y+z+w ()

for allx,y,z,wand allt≥.

Now, we show thatW. is a negative deﬁnite function. The derivative of the functionV

along any solution (x(t),y(t),z(t),w(t)) of system (), with respect tot, is after simplifying

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where

V= –

dh

cm

c(t)f(x) –d(t)h(x)

y– αd(t)h–h(x)

yz,

V= –

b(t)q(x) –αc(t)f(x) –βa(t)g(x)z,

V= –

αa(t)g(x) – w,

V= αd(t)w t

t–r

hx(η)x(η)dη+ βd(t)y(t)

t

t–r

hx(η)x(η)dη

+ d(t)z(t)

t

t–r

hx(η)x(η)dη+σry(t) –σ

t

t–r

y(η)dη,

V= –a(t)θ

z+ αzw–b(t)θ

αz+ αzw+βy+ yz

+c(t)θ

y+ αyz,

V=d(t)

βH(x) –αhy+ h(x)y+ αh(x)z

+c(t)f(x)y+ αf(x)yz+b(t)αq(x)z+βq(x)y

+a(t)g(x)z+ βg(x)yz.

By regarding conditions (A), (A), (i), (ii) and inequality (), (), we have the following:

V≤–

d(t)h–d(t)h(x)

yz

≤–d(t)h–h(x)

yz

≤d(t)h–h(x)

y+α z  – α z 

≤α

 d(t)

h–h(x)

z. In that case,

V+V≤–

b(t)q(x) –αc(t)f(x) –βa(t)g(x) –α



d(t)

h–h(x)

z

≤–

bq–



am

+ε

cM–

dh

cm

+ε

aM–

α

 (amδ)

z

≤–

bq–

M am

c–

dhaM

cm

– δ

am

–εM(a+c)

z

≤–bq–δ–εM(a+c)

z≤, and

V≤–[αam– ]w= –εw≤.

By takingh=max{|h–a_dm_δ|,h_}, we get

V≤dhr

αw+βy+z+σry+dh(α+β+ ) –σ

t

t–r

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If we chooseσ=dh(α+β+ ), we have

V≤dhr

αw+ (α+ β+ )y+z.

Thus, there exists a positive constantDsuch that

–εc(t)f(x)y+V+V+V+V≤–D

y+z+w. From (), and the Cauchy-Schwarz inequality, we obtain

V≤a(t)|θ|

z+αz+w+b(t)|θ|

αz+αz+w+βy+y+z

+c(t)|θ|

y+αy+z

≤λ

|θ|+|θ|+|θ|

y+z+w+H(x)

≤λ

D

|θ|+|θ|+|θ|

V,

whereλ=max{a( +α),b( + α+β),c( +α)}. Using condition (iii) and Lemma , we

can write

h(x)≤hH(x),

hereby,

|V| ≤d(t)βH(x) +αhy+h(x) +y+α

h(x) +z

+c(t)y+αy+z+b(t)αz+βy

+a(t)z+ βy+z

≤λa(t)+b(t)+c(t)+d(t)y+z+w+H(x)

≤λ

D

_a₍_t₎₊_b₍_t₎₊_c₍_t₎₊_d₍_t₎_V_,

such thatλ=max{β+ (α+ )h,αh+ ,α+ }. By takingη = 

Dmax{λ,λ}, we obtain

.

V()≤–D

y+z+w+ (βy+z+αw)p(t,x,y,z,w) +

ηa

₍_t₎₊_b₍_t₎₊_c₍_t₎₊_d₍_t₎₊_|_θ

|+|θ|+|θ|

V. ()

From (A), (A),(iii), (), (), () and the Cauchy-Schwarz inequality, we get

.

W()= _.

V()–



ηγ(t)V

e–η

t

γ(s)ds

≤–D

y+z+w+ (βy+z+αw)p(t,x,y,z,w)e–η t

γ(s)ds _() ≤β|y|+|z|+α|w|p(t,x,y,z,w)

≤D

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≤D

 +y+z+we(t)

≤D

 + 

D

We(t)

≤De(t)+

D

D

We(t), ()

whereD=max{α,β, }. Integrating () from  totand using condition (iv) and the

Gron-wall inequality, we have

W ≤W,x(),y(),z(),w()+ Dη

+D

D t



Ws,x(s),y(s),z(s),w(s)e(s)ds

≤W,x(),y(),z(),w()+ Dη

e D_ D

t

|e(s)|ds

≤W,x(),y(),z(),w()+ Dη

e D

D_η

=K<∞. ()

Because of inequalities () and (), we write

x+y+z+w≤  D

W≤K, ()

whereK=_DK_. Clearly, () implies that x(t)≤K, y(t)≤

K, z(t)≤

K, w(t)≤

K for allt≥.

Hence

x(t)≤K, x(t)≤

K, x(t)≤

K, _x₍_t₎_≤_K

 for allt≥.

()

Now, we prove the square integrability of solutions and their derivatives. We deﬁneFt=

F(t,x(t),y(t),z(t),w(t)) as

Ft=W+ρ

t



y(s) +z(s) +w(s)ds,

whereρ> . It is easy to see thatFtis positive deﬁnite sinceW=W(t,x,y,z,w) is already

positive deﬁnite. Using the estimate

e–η+ηη ≤_e–η t

γ(s)ds≤_

by (), we have the following:

.

Ft()≤–D

y(t) +z(t) +w(t)e–η+ηη

+Dy(t)+z(t)+w(t)e(t)

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By choosingρ=De–

η_+η_

η _{, we obtain}

.

Ft()≤D

 +y(t) +z(t) +w(t)e(t)

≤D

 + 

D

We(t)

≤De(t)+

D

D

Fte(t). ()

Integrating inequality () from  totand using again the Gronwall inequality and con-dition (iv), we get

Ft≤F+ Dη+

D

D t



Fs|e(s)|ds

≤(F+ Dη)e D_ D

t

|e(s)|ds

≤(F+ Dη)e D_

Dη₌_K__<∞_. _()

Therefore,

∞



y(s)ds<K,

∞



z(s)ds<K,

∞



w(s)ds<K,

which implies that

∞



x(s)ds<K,

∞



x(s)ds<K,

∞



x(s)ds<K, ()

which completes the proof of the theorem.

Remark  Ifp(t,x,y,z,w)≡, similarly to the above proof, inequality () becomes

.

W() = _.

V()–



ηγ(t)V

e–η

t

γ(s)ds

≤–D

y+z+we–η t

γ(s)ds

≤–μy+z+w,

whereμ=De–

η+η

η _{. It can also be observed that the only solution of system () for which}

.

W()(t,x,y,z,w) =  is the solutionx=y=z=w= . The above discussion guarantees that

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Example  We consider the following fourth order nonlinear diﬀerential equation with delay:

x()+e–tsint+ x+ e

x_{+ }_e–x

ex₊_e–x

x

+

sint+ t+ 

t_{+ }

sinx+ ex+ e–x ex₊_e–x

x

+e–tsint+ xcosx+x

_{+ }

x_{+ }

x+

sint+t_{+ }

t_{+ }

x(t–_)

x₍_t_–  ) + 

= sint

t_{+  + (}_x_x₎_{+ (}_xx₎ ()

by taking g(x) = x+_exe₊x_e+–xe–x, q(x) =

sinx+ex_+_e–x

ex₊_e–x , f(x) = x cosx+x_+

x_+ , h(x) = _xx_+, a(t) =

e–t_sin__t_{+ ,}_b₍_t_{) =} sint+t_+

t_+ ,c(t) =e–tsint+ ,d(t) =

sin_t₊_t_+

t_+ ,r=_ andp(t,x,xx,

x) =_t_++(_x_xsin₎t₊₍_xx₎.

We obtain easily the following:g= .,g= .,f= .,f= .,q= .,q= .,

a= ,a= ,b= ,b= ,c= ,c= ,d= .,d= .,m= .,M= .,h= ,

α=_,β=_,δ=_ andδ= .. Also we have _∞

–∞

g(x)dx= 

_∞

–∞

_ex₊_e–x+x

e–x_–_ex

(ex₊_e–x₎ dx ≤  –∞

_ex₊_e–x–x

e–x–ex

(ex₊_e–x₎ dx +  ∞ 

_ex₊_e–x –x

e–x_–_ex

(ex₊_e–x₎ dx

= π,

∞

–∞

q(x)dx=

∞

–∞

(ex+e–x)cos₍_exx₊– (_e–exx₎–e–x)sinx dx ≤ ∞ –∞

_ex₊_e–x +x

ex_–_e–x

(ex₊_e–x₎ dx

=π,

∞

–∞

f(x)dx=

∞

–∞ cosx

x_{+ }– x  cosx

(x_{+ )}+ –x

sinx x_{+ }

dx ≤ ∞ –∞ _x_{+ }+

x

x_{+ } dx

= √π,

∞

 _p

t,x,x,x,xdt=

∞



_t_{+  + (}_x_xsin₎t_{+ (}_xx₎ dt ≤ ∞  _tsin_{+ }t

dt ≤ ∞  

t_{+ }dt

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∞



a(t)dt=

∞



–e–tsint+ e–tcostdt

≤

∞



e–tdt

= ,

∞



b(t)dt=

∞



_tcos_{+ }t– t

sint

(t_{+ )} dt

≤

∞





t_{+ }dt

=π  ,

∞



c(t)dt=

∞



–e–tsint+e–tcostdt

≤

∞



e–tdt

= ,

_∞



d(t)dt=

_∞



sin__tt_{+ }cost– t

sint

(t_{+ )} dt

≤ 



∞





t_{+ }dt

=π . Consequently,

+∞

–∞

g(s)+q(s)+f(s)ds<∞,

∞



a(t)+b(t)+c(t)+d(t)dt<∞.

Thus all the assumptions of Theorem  hold. This shows that every solution of equation () is bounded and square integrable.

4 Conclusion

A class of nonlinear retarded functional differential equations of fourth order is consid-ered. Sufficient conditions are established guaranteeing the uniform asymptotic stability of the solutions forp(t,x,x,x,x)≡ and also square integrability and boundedness of solutions of equation () with delay. In the proofs of the main results, we benefit from Lyapunov’s functional approach. The results obtained essentially improve, include and complement the results in the literature.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

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Acknowledgements

The author states his sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 27 December 2016 Accepted: 4 April 2017

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