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On the asympto c behavior of solu ons

to nonlinear differen al equa ons of the second order

Cemil Tunç and Timur Ayhan

Summary We study the asympto c behavior of solu ons to a non- linear differen al equa on of the second order whose coefficient of nonlinearity is a bounded func on for arbitrarily large values of x in R.

We obtain certain sufficient condi ons which guarantee boundedness of solu ons, their convergence to zero as x→ ∞ and their unbound- edness.

Keywords

asympto c behavior;

nonlinear differen al equa on;

second order

MSC 2010 34C10; 34D05 Received: 2012/02/22; Accepted: 2014/03/15

1. Introduc on

Nonlinear differen al equa ons of the second order can be derived from many fields, such as physics, mechanics, and engineering. An important ques on is whether these equa ons have bounded solu ons, solu ons convergent to zero as x → ∞ or unbounded solu ons.

In recent years, especially boundedness of solu ons to certain nonlinear differen al equa- ons of the second order has been widely discussed, notably by Ademola and Arawomo [1], Bucur [2], Constan n [3], Kiguradze [5], Kusano et al. [6], Lipovan [7], Mingareilli and Sadarangani [8], Mustafa [9], Saker [10], Tong [11], Trench [12], Tunç [13–17], Tunç and Tunç [18], Waltman [19], and Wong [20]. However, there exist only a few papers con-

Cemil Tunç Department of Mathema cs, Faculty of Sciences, Yüzüncü Yıl University (e-mail:cemtunc@yahoo.com)

Timur Ayhan Department of Primary School-Mathema cs, Faculty of Educa on, Siirt University (e-mail: murayhan@mynet.com)

DOI 10.14708/cm.v55i1.812 © 2015Polish Mathema cal Society

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cerned with the convergence or unboundedness of solu ons of the same type of equa ons (Ezeilo [4], Qarawani [21]).

Recently, Qarawani [21] considered the following non-linear differen al equa on of the second order

z

′′

+ p(x)z

+ q(x)z = h(x) ∣z∣

α

e

(α−12 ) ∫ p(x) dx

sgn z (1) where α ∈ (−1

,

1 ) \ {0}, q ∈ C

0

[x

0,

∞), x

0

> 0, h

,

p ∈ C

1

[x

0,

∞), p > 0, and h(x) is bounded for all sufficiently large x in R. He discussed boundedness of solu ons of Eq. (1), their convergence to zero as x → ∞ as well as their unboundedness.

We may assume that z > 0, because if z < 0, simply set z = −u, u > 0. Therefore, instead of Eq. (1), we can consider the differen al equa on

z

′′

+ p(x)z

+ q(x)z = h(x) ∣z∣

α

e

(α−12 ) ∫ p(x) dx

,

α ∈ (−1

,

1 ) \ {0}

.

(2) It is clear that if z (x) = y(x) exp(−

12

∫ p(x) dx) then Eq. ( 2) reduces to the equa on

y

′′

(x) + y(x) = h(x)y

α

(x)

,

α ∈ (−1

,

1 ) \ {0} (3) provided that

q (x) − 1

4 p

2

(x) − 1

2 p

(x) = 1

.

(4)

Qarawani [21] proved that if h (x) is a con nuously differen able func on that is bounded for all sufficiently large x ∈ R and the integral ∫

x0

∣h

′′

(x)∣ dx is convergent, then any solu on of Eq. (3) is bounded as x → ∞. He also showed that if h(x) sa sfies the above condi ons and ∫

x0

∣h(x)∣ dx < ∞, then for any solu on y(x) of Eq. ( 3), the asympto c formula y (x) = A sin (x + w

0

) + O (∫

x0

∣h(x)∣ dx) holds. Finally, Qarawani proved that if h(x) is a con nuously differen able func on that is bounded for all sufficiently large x ∈ R and

x0

∣h

(x)∣ dx = ∞, then any solu on of Eq. ( 3) is unbounded as x → ∞.

In this paper, instead of the condi on (4) we discuss the results of Qarawani [21] under the following condi on

q (x) − 1

4 p

2

(x) − 1

2 p

(x) = 1

x

2.

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Condi on (4) is a special case of our condi on (5). Our aim is to improve the results

established in [21], with condi on (5) instead of condi on (4). This paper is inspired by the

results of Qarawani [21] and other results men oned above. It develops and complements

the work of Qarawani. The obtained results are useful for the study of the qualita ve

behavior of solu ons to differen al equa ons of higher order. It should be noted that the

assump ons and results of this paper are different from those found in the literature.

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2. Boundedness of solu ons

2.1. Lemma. Subs tu on z (x) = y(x) exp(−

12

∫ p(x) dx) reduces Eq. ( 2) to

x

2

y

′′

(x) + y(x) = x

2

h (x)y

α

(x) (6) where α ∈ (−1

,

1 ) \ {0} and q(x) −

14

p

2

(x) −

12

p

(x) =

x12

.

Proof. Let z (x) = y(x) exp(−

12

∫ p(x) dx). Then we have z

= y

(x) exp (− 1

2 ∫ p(x) dx) − 1

2 p (x) exp(− 1

2 ∫ p(x) dx)y(x) and

z

′′

= y

′′

(x) exp(− 1

2 ∫ p(x) dx) − y

(x)p(x) exp(− 1

2 ∫ p(x) dx) + 1

4 p

2

(x) exp(− 1

2 ∫ p(x) dx)y(x) − 1

2 p

(x) exp(− 1

2 ∫ p(x) dx)y(x)

.

Hence, subs tu ng the expressions for z (x)

,

z

(x)

,

z

′′

(x) into Eq. ( 2) and dividing by

exp ( − 1

2 ∫ p(x) dx)

,

we get Eq. (6).

2.2. Example. Consider the second-order differen al equa on z

′′

(x) + 4xz

(x) + (4x

2

+ 1

x

2

+ 2)z (x) = e

(

x22+x

) z

12

(x)

.

(7) From a comparison of Eq. (7) and Eq. (2) it follows that

p (x) = 4x

,

q (x) = 4x

2

+ 1

x

2

+ 2

,

α = 1 2

.

Le ng

z (x) = y(x)e

(−12∫ p(x) dx)

= y(x)e

(−12∫ 4x dx)

= y(x)e

−x2,

we get

z

= y

(x)e

−x2

− 2xy(x)e

−x2

and

z

′′

= y

′′

(x)e

−x2

− 4xy

(x)e

−x2

− 2y(x)e

−x2

+ 4x

2

y (x)e

−x2.

Subs tu ng z (x), z

(x)

,

z

′′

(x) into Eq. ( 7), we obtain

y

′′

(x)e

−x2

− 4xe

−x2

y

′′

(x) − 2e

−x2

y (x) + 4x

2

e

−x2

y (x) + 4xe

−x2

y

′′

(x)

− 8x

2

e

−x2

y (x) + 4x

2

e

−x2

y (x) + y (x)e

−x2

x

2

+ 2e

−x2

y (x) = e

−(x2+x)

y

12

(x)

,

(4)

so

x

2

y

′′

(x) + y(x) = x

2

e

−x

y

12

(x)

.

Note that condi on q (x) =

x12

+

14

p

2

(x) +

12

p

(x) holds for the last equa on.

2.3. Theorem. Assume that x

2

h (x) is a con nuously differen able func on that is bounded for all sufficiently large x ∈ R and that ∫

x0

»»»» » x

2

h

(x)»»»»» dx and ∫

x0

y

′2

(x) dx are convergent. Then any solu on of Eq. (6) is bounded as x → ∞.

Proof. By le ng x = e

t

, Eq. (6) reduces to

y

′′

(t) − y

(t) + y(t) = e

2t

h (e

t

)y

α

(t)

.

(8) Mul plying both sides of Eq. (8) by y

and integra ng the result with respect to t from some posi ve t

0

to t, we get

t

t0

y

(s) y

′′

(s) ds − ∫

t

t0

y

′2

(s)ds + ∫

t

t0

y (s) y

(s) ds = ∫

t

t0

e

2s

h (e

s

)y

α

(s)y

(s) ds y

′2

(s) ∣

tt0

+y

2

(s) ∣

tt0

= 2 ∫

t

t0

y

′2

(s)ds + 2 ∫

t

t0

e

2s

h (e

s

)y

α

(s)y

(s) ds y

′2

(t) − y

′2

(t

0

) + y

2

(t) − y

2

(t

0

) = 2 ∫

t

t0

y

′2

(s)ds + 2 ∫

t

t0

e

2s

h (e

s

)y

α

(s)y

(s) ds

.

Integra ng the integral on the right-hand side by parts yields

y

′2

(t) + y

2

(t) = y

′2

(t

0

) + y

2

(t

0

) + 2 ∫

t

t0

y

′2

(s)ds − 2e

2t0

h (e

t0

)y

α+1

(t

0

) α + 1 + 2e

2t

h (e

t

)y

α+1

(t)

α + 1 − 4

α + 1 ∫

t

t0

e

2s

h (e

s

)y

α+1

(s)ds − 2 α + 1 ∫

t

t0

e

3s

h

(e

s

)y

α+1

(s)ds

.

Hence

y

2

(t) ⩽ y

′2

(t) + y

2

(t)

⩽ A

t0

+ 2 »» »»»e

2t

h (e

t

)»»»»» »»»» » y

α+1

(t)»»»»»

α + 1 + 2 ∫

t

t0

»»»» » y

′2

(s)»»»»»ds + 2 α + 1 ∫

t

t0

»»»» » e

3s

h

(e

s

)»»»»» »»»» » y

α+1

(s)»»»»»ds where A

t0

⩾ 0 is an expression dependent only on t

0

.

Let M = max

t0⩽s⩽t

∣y (s)∣. Without loss of generality we may assume that M ⩾ a

0

> 0

,

otherwise the theorem is proved. Since e

2t

h(e

t

) is bounded, we have

M

2

⩽ A

t0

+ 2BM

α+1

α + 1 + 2 ∫

t

t0

»»»» » y

′2

(s)»»»»»ds + 2M

α+1

α + 1 ∫

t

t0

»»»» » e

3s

h

(e

s

)»»»»»ds

,

α ∈ (−1

,

1 ) \ {0}

M

1−α

A

t0

M

α+1

+ 2B α + 1 + 2

M

α+1

t

t0

»»»» » y

′2

(s)»»»»»ds + 2 α + 1 ∫

t

t0

»»»» » e

3s

h

(e

s

)»»»»»ds M

1−α

A

t0

(a

0

)

α+1

+ 2B α + 1 + 2

(a

0

)

α+1

t0

»»»» » y

′2

(s)»»»»»ds + 2 α + 1 ∫

t0

»»»» » e

3s

h

(e

s

)»»»»»ds

.

(5)

Since the integrals ∫

t0

»»»» » e

3s

h

(e

s

)»»»»»ds and ∫

t0

»»»» » y

′2

(s)»»»»»ds are convergent,

∣y (t)∣ ⩽ M ⩽ (C + 2D + 2E + 2F)

1−α1 ,

α ∈ (−1

,

1 ) \ {0}

.

Therefore, y (t) is bounded for t → ∞. Hence y (x) is also bounded as x → ∞.

We give an example illustra ng the theorem.

2.4. Example. Consider the second-order differen al equa on

x

2

y

′′

(x) + y(x) =

y x

.

We will show that all its solu ons are bounded for x → ∞. It is clear that

α = 1

2

,

h (x) = 1

x

3,

»»»» » x

2

h (x)»»»»» = »»»» »»» 1

x »»»» »»» ⩽ 1 for ∣x∣ ⩾ 1

,

1

»»»» » x

2

h

(x)»»»»» dx = ∫

1

»»»» »»» −3

x

2

»»»» »»» dx = ∫

1

3

x

2

dx converges to 3

.

Applying to the above differen al equa on the same approach as in the proof of the the- orem, we get

M

12

A

t0

M

32

+ 4B

3 + 2 M

32

t

t0

»»»» » y

′2

(s)»»»»»ds + 4 3 ∫

t

t0

»»»» » e

3s

h

(e

s

)»»»»»ds

.

A

t0

(a

0

)

32

+ 4B

3 + 2C (a

0

)

32

+ 4

3 ∫

t0

»»»» » e

3s

h

(e

s

)»»»»»ds Since the integral ∫

t0

»»»» » e

3s

h

(e

s

)»»»»»ds converges,

∣y (t)∣ ⩽ M ⩽ (D + 4)

2

as t → ∞

.

Therefore, y (t) is bounded for t → ∞, hence y (x) is also bounded as x → ∞.

(6)

3. Unboundedness of solu ons

3.1. Theorem. Suppose that x

2

h (x) is a con nuously differen able func on that is bounded for x ∈ [x

0,

∞) and assume that ∫

x0

»»»» » x

2

h

(x)»»»»» dx = ∞, ∫

x0

y

′2

(x) dx is convergent and ∣y (x)∣ ⩾ a for some a > 0 and all x ∈ [x

0,

∞). Then any solu on of Eq. (2) is unbounded as x → ∞.

Proof. Since the func on x

2

h (x) is bounded, there exists a posi ve constant L such that

»»»» » x

2

h (x)»»»»» ⩽ Lforallsufficientlylargex. Supposethatthesolu ony(t)ofEq.( 8) is bounded for sufficiently large t. Hence there exists a posi ve constant M such that M = max

t0⩽s⩽t

∣y (s)∣

for all sufficiently large t.

Mul plying both sides of Eq. (8) by y

and integra ng the result with respect to t from some posi ve t

0

to t, we get

0 ⩽ y

′2

(t) + y

2

(t) = y

′2

(t

0

) + y

2

(t

0

) + 2 ∫

t

t0

y

′2

(s)ds − 2e

2t0

h (e

t0

)y

α+1

(t

0

) α + 1 + 2e

2t

h (e

t

)y

α+1

(t)

α + 1 − 4

α + 1 ∫

t

t0

e

2s

h (e

s

)y

α+1

(s)ds − 2 α + 1 ∫

t

t0

e

3s

h

(e

s

)y

α+1

(s)ds

.

It follows that

y

′2

(t

0

) + y

2

(t

0

) + 2 ∫

t

t0

y

′2

(s)ds − 2e

2t0

h (e

t0

)y

α+1

(t

0

)

α + 1 + 2e

2t

h (e

t

)y

α+1

(t) α + 1

− 4

α + 1 ∫

t

t0

e

2s

h (e

s

)y

α+1

(s)ds ⩾ 2 α + 1 ∫

t

t0

e

3s

h

(e

s

)y

α+1

(s)ds ⩾ ∫

t

t0

e

3s

h

(e

s

)y

α+1

(s)ds

,

so

»»»» » y

′2

(t

0

) + y

2

(t

0

)»»»»» + »»»» »»»» »»

2e

2t0

h (e

t0

)y

α+1

(t

0

) α + 1 »»»» »»»» »» + »»»» »»»» »»

2e

2t

h (e

t

)y

α+1

(t) α + 1 »»»» »»»» »» + 2 ∫

t

t0

»»»» » y

′2

(s)»»»»»ds

⩾ »»»»

»»»»∫

t t0

e

3s

h

(e

s

)y

α+1

(s)ds »»»»

»»»»

.

Applying the mean value theorem to the integral on the right-hand side, we obtain

A

t0

+ 2LM

α+1

α + 1 + 2F ⩾ »»»» »»»»∫

t t0

e

3s

h

(e

s

)y

α+1

(s)ds»»»» »»»» = »»»» » y

α+1

(s

)»»»»» »»»» »»»»∫

t t0

e

3s

h

(e

s

)ds»»»» »»»»

where s

∈ [t

0,

t ]. Since »»»»»∫

t0

e

3s

h

(e

s

)ds»»»»» = ∞, sending t to ∞ in the last inequality shows that M = ∞. The contradic on proves that y (t) is unbounded as t → ∞. We conclude that the solu on y (x) is unbounded as x → ∞.

3.2. Example. Consider the differen al equa on of the second order x

2

y

′′

(x) + y(x) =

y

.

(7)

We will show that all the solu ons are unbounded for x → ∞. Here α = 1

2

,

h (x) = 1

x

2,

»»»» » x

2

h (x)»»»»» = 1 for ∣x∣ ⩾ 1 and

1

»»»» » x

2

h

(x)»»»»» dx = 2∫

1

dx x = ∞

.

Applying the same approach as in the proof of Theorem 3.1, we get

A

t0

+ 2LM

α+1

α + 1 + 2F ⩾ »»»»

»»»»∫ t

0t

e

3s

h

(e

s

)y

α+1

(s) ds »»»»

»»»» = »»»» » y

α+1

(s

)»»»»» »»»»

»»»»∫

t t0

e

3s

h

(e

s

) ds »»»»

»»»»

.

Since ∫

t0

»»»» » e

3s

h

(e

s

)»»»»»ds = ∞, we have M = ∞. The contradic on proves that the solu on y (x) is unbounded as x → ∞.

Acknowledgement

The authors of this paper would like to express their sincere apprecia on to the anonymous referees for their valuable comments and sugges ons which have led to an improvement in the presenta on of the paper.

References

[1] A. Ademola and P. Arawomo, Stability, boundedness and asympto c behaviour of solu ons of cer- tain nonlinear differen al equa on, Kragujevac J. Math. 35 (2010), no. 3, 431–445.

[2] A. Bucur, About asympto c behaviour of solu ons of differen al equa ons as x→ ∞, Gen. Math.

14 (2006), no. 2, 55–58.

[3] A. Constan n, A note on a second order nonlinear differen al system, Glasg. Math. J. 42 (2000), no. 2, 195–199.

[4] J. O. C. Ezeilo, A note on the convergence of solu ons of certain second order differen al equa ons, Portugal. Math. 24 (1965), 49–58.

[5] I. Kiguradze, The asympto c behaviour of the solu ons of a non linear differen al equa on of Emden–

–Fowler type, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), no. 5, 965–986.

[6] T. Kusano, M. Naito, and H. Usami, Asympto c behavior of solu ons of a class of second order nonlinear differen al equa ons, Hiroshima Math. J. 16 (1986), no. 1, 149–159.

[7] O. Lipovan, On the asympto c behaviour of the solu ons to a class of second order nonlinear differ- en al equa ons, Glasgow Math. J. 45 (2003), 179–187.

[8] A.B. Mingareilli and K. Sadarangani, Asympto c solu ons of forced nonlinear second order differen- al equa ons and their extensions, Electron. J. Differen al Equa ons 40 (2007), pp. 40.

[9] O. Mustafa, On the existence of solu ons with prescribed asympto c behaviour for perturbed non- linear differen al equa ons of second order, Glasgow Math. J. 47 (2005), 177–18.

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[11] J. Tong, The asympto c behaviour of a class of nonlinear differen al equa ons of second order, Proc.

Amer. Math. Soc. 84 (1982), no. 2, 235–236.

[12] W. Trench, Asympto c integra on of linear differen al equa ons subject to integral smallness condi- ons involving ordinary convergence, SIAM J. Math. Anal. 7 (1976), 213–221.

[13] C. Tunç, Some new stability and boundedness results on the solu ons of the nonlinear vector differ- en al equa ons of second order, Iran. J. Sci. Technol. Trans. A Sci. 30 (2006), no. 2, 213–221.

[14] C. Tunç, A new boundedness theorem for a class of second order differen al equa ons, Arab. J. Sci.

Eng. Sect. A Sci. 33 (2008), no. 1, 83–92.

[15] C. Tunç, A note on boundedness of solu ons to a class of non-autonomous differen al equa ons of second order, Appl. Anal. Discrete Math. 4 (2010), 361–372.

[16] C. Tunç, Boundedness results for solu ons of certain nonlinear differen al equa ons of second order, J. Indones. Math. Soc. 16 (2010), no. no.2, 115–127.

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[18] C. Tunç and E. Tunç, On the asympto c behavior of solu ons of certain second-order differen al equa ons, J. Franklin Inst. 344 (2007), no. 5, 391–398.

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Soc. 15 (1964), 918–923.

[20] J. Wong, Oscilla on Theorems for second order nonlinear differen alequa ons, Bull. Inst. Math.

Acad. Simca. 3 (1975), 283–309.

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Let M be an n-dimensional manifold with given linear connection T and let Jff : 2M -+M be the linearized tangent bundle of second order.. A pure - linear connection in the bundle J

The aim of this paper is to extend the result of [9] to the case when the multi- function F is contained in the Fr´echet subdifferential of a φ-convex function of order two.. Since