Open Access

Existence of positive solutions of higher-order nonlinear neutral equations

Journal of Inequalities and Applications20132013:573

https://doi.org/10.1186/1029-242X-2013-573

Received: 16 August 2013

Accepted: 11 November 2013

Published: 5 December 2013

Abstract

In this work, we consider the existence of positive solutions of higher-order nonlinear neutral differential equations. In the special case, our results include some well-known results. In order to obtain new sufficient conditions for the existence of a positive solution, we use Schauder’s fixed point theorem.

Keywords

neutral equations fixed point higher-order positive solution

1 Introduction

The purpose of this article is to study higher-order neutral nonlinear differential equations of the form
[ r ( t ) [ x ( t ) P 1 ( t ) x ( t τ ) ] ( n 1 ) ] + ( 1 ) n Q 1 ( t ) f ( x ( t σ ) ) = 0 ,
(1)
[ r ( t ) [ x ( t ) P 1 ( t ) x ( t τ ) ] ( n 1 ) ] + ( 1 ) n c d Q 2 ( t , ξ ) f ( x ( t ξ ) ) d ξ = 0
(2)
and
[ r ( t ) [ x ( t ) a b P 2 ( t , ξ ) x ( t ξ ) d ξ ] ( n 1 ) ] + ( 1 ) n c d Q 2 ( t , ξ ) f ( x ( t ξ ) ) d ξ = 0 ,
(3)

where n 2 is an integer, τ > 0 , σ 0 , d > c 0 , b > a 0 , r, P 1 C ( [ t 0 , ) , ( 0 , ) ) , P 2 C ( [ t 0 , ) × [ a , b ] , ( 0 , ) ) , Q 1 C ( [ t 0 , ) , ( 0 , ) ) , Q 2 C ( [ t 0 , ) × [ c , d ] , ( 0 , ) ) , f C ( R , R ) , f is a nondecreasing function with x f ( x ) > 0 , x 0 .

The motivation for the present work was the recent work of Culáková et al. [1] in which the second-order neutral nonlinear differential equation of the form
[ r ( t ) [ x ( t ) P ( t ) x ( t τ ) ] ] + Q ( t ) f ( x ( t σ ) ) = 0
(4)

was considered. Note that when n = 2 in (1), we obtain (4). Thus, our results contain the results established in [1] for (1). The results for (2) and (3) are completely new.

Existence of nonoscillatory or positive solutions of higher-order neutral differential equations was investigated in [25], but in this work our results contain not only existence of solutions but also behavior of solutions. For books, we refer the reader to [611].

Let ρ 1 = max { τ , σ } . By a solution of (1) we understand a function x C ( [ t 1 ρ 1 , ) , R ) , for some t 1 t 0 , such that x ( t ) P 1 ( t ) x ( t τ ) is n 1 times continuously differentiable, r ( t ) ( x ( t ) P 1 ( t ) x ( t τ ) ) ( n 1 ) is continuously differentiable on [ t 1 , ) and (1) is satisfied for t t 1 . Similarly, let ρ 2 = max { τ , d } . By a solution of (2) we understand a function x C ( [ t 1 ρ 2 , ) , R ) , for some t 1 t 0 , such that x ( t ) P 1 ( t ) x ( t τ ) is n 1 times continuously differentiable, r ( t ) ( x ( t ) P 1 ( t ) x ( t τ ) ) ( n 1 ) is continuously differentiable on [ t 1 , ) and (2) is satisfied for t t 1 . Finally, let ρ 3 = max { b , d } . By a solution of (3) we understand a function x C ( [ t 1 ρ 3 , ) , R ) , for some t 1 t 0 , such that x ( t ) a b P 2 ( t , ξ ) x ( t ξ ) d ξ is n 1 times continuously differentiable, r ( t ) [ x ( t ) a b P 2 ( t , ξ ) x ( t ξ ) d ξ ] ( n 1 ) is continuously differentiable on [ t 1 , ) and (3) is satisfied for t t 1 .

The following fixed point theorem will be used in proofs.

Theorem 1 (Schauder’s fixed point theorem [9])

Let A be a closed, convex and nonempty subset of a Banach space Ω. Let S : A A be a continuous mapping such that SA is a relatively compact subset of Ω. Then S has at least one fixed point in A. That is, there exists x A such that S x = x .

2 Main results

Theorem 2 Let
t 0 Q 1 ( t ) d t = .
(5)
Assume that 0 < k 1 k 2 and there exists γ 0 such that
k 1 k 2 exp ( ( k 2 k 1 ) t 0 γ t 0 Q 1 ( t ) d t ) 1 ,
(6)
exp ( k 2 t τ t Q 1 ( s ) d s ) + exp ( k 2 t 0 γ t τ Q 1 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( exp ( k 1 t 0 γ u σ Q 1 ( z ) d z ) ) d u d s P 1 ( t ) exp ( k 1 t τ t Q 1 ( s ) d s ) + exp ( k 1 t 0 γ t τ Q 1 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( exp ( k 2 t 0 γ u σ Q 1 ( z ) d z ) ) d u d s , t t 1 t 0 + max { τ , σ } .
(7)

Then (1) has a positive solution which tends to zero.

Proof Let Ω be the set of all continuous and bounded functions on [ t 0 , ) with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
A = { x Ω : v 1 ( t ) x ( t ) v 2 ( t ) , t t 0 } ,
where v 1 ( t ) and v 2 ( t ) are nonnegative functions such that
v 1 ( t ) = exp ( k 2 t 0 γ t Q 1 ( s ) d s ) , v 2 ( t ) = exp ( k 1 t 0 γ t Q 1 ( s ) d s ) , t t 0 .
(8)
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator S : A Ω as
( S x ) ( t ) = { P 1 ( t ) x ( t τ ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( x ( u σ ) ) d u d s , t t 1 , ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) , t 0 t t 1 .

We show that S satisfies the assumptions of Schauder’s fixed point theorem.

First, S maps A into A. For t t 1 and x A , using (7) and (8), we have
( S x ) ( t ) P 1 ( t ) v 2 ( t τ ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( v 1 ( u σ ) ) d u d s = P 1 ( t ) exp ( k 1 t 0 γ t τ Q 1 ( s ) d s ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( exp ( k 2 t 0 γ u σ Q 1 ( z ) d z ) ) d u d s v 2 ( t )
and
( S x ) ( t ) P 1 ( t ) v 1 ( t τ ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( v 2 ( u σ ) ) d u d s = P 1 ( t ) exp ( k 2 t 0 γ t τ Q 1 ( s ) d s ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) f ( exp ( k 1 t 0 γ u σ Q 1 ( z ) d z ) ) d u d s v 1 ( t ) .
For t [ t 0 , t 1 ] and x A , we obtain
( S x ) ( t ) = ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 2 ( t )
and in order to show ( S x ) ( t ) v 1 ( t ) , consider
H ( t ) = v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) + v 1 ( t 1 ) .
By making use of (6), it follows that
H ( t ) = v 2 ( t ) v 1 ( t ) = k 1 Q 1 ( t ) v 2 ( t ) + k 2 Q 1 ( t ) v 1 ( t ) = Q 1 ( t ) v 2 ( t ) [ k 1 + k 2 v 1 ( t ) exp ( k 1 t 0 γ t Q 1 ( s ) d s ) ] = Q 1 ( t ) v 2 ( t ) [ k 1 + k 2 exp ( ( k 1 k 2 ) t 0 γ t Q 1 ( s ) d s ) ] Q 1 ( t ) v 2 ( t ) [ k 1 + k 2 exp ( ( k 1 k 2 ) t 0 γ t 0 Q 1 ( s ) d s ) ] 0 , t 0 t t 1 .
Since H ( t 1 ) = 0 and H ( t ) 0 for t [ t 0 , t 1 ] , we conclude that
H ( t ) = v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) + v 1 ( t 1 ) 0 , t 0 t t 1 .
Then t [ t 0 , t 1 ] and for any x A ,
( S x ) ( t ) = ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 1 ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) , t 0 t t 1 .

Hence, S maps A into A.

Second, we show that S is continuous. Let { x i } be a convergent sequence of functions in A such that x i ( t ) x ( t ) as i . Since A is closed, we have x A . It is obvious that for t [ t 0 , t 1 ] and x A , S is continuous. For t t 1 ,
| ( S x i ) ( t ) ( S x ) ( t ) | P 1 ( t ) | x i ( t τ ) x ( t τ ) | + | 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) [ f ( x i ( u σ ) ) f ( x ( u σ ) ) ] d u d s | P 1 ( t ) | x i ( t τ ) x ( t τ ) | + 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s Q 1 ( u ) | f ( x i ( u σ ) ) f ( x ( u σ ) ) | d u d s .
Since | f ( x i ( t σ ) ) f ( x ( t σ ) ) | 0 as i , by making use of the Lebesgue dominated convergence theorem, we see that
lim t ( S x i ) ( t ) ( S x ) ( t ) = 0

and therefore S is continuous.

Third, we show that SA is relatively compact. In order to prove that SA is relatively compact, it suffices to show that the family of functions { S x : x A } is uniformly bounded and equicontinuous on [ t 0 , ) . Since uniform boundedness of { S x : x A } is obvious, we need only to show equicontinuity. For x A and any ϵ > 0 , we take T t 1 large enough such that ( S x ) ( T ) ϵ 2 . For x A and T 2 > T 1 T , we have
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | | ( S x ) ( T 2 ) | + | ( S x ) ( T 1 ) | ϵ 2 + ϵ 2 = ϵ .
Note that
X n Y n = ( X Y ) ( X n 1 + X n 2 Y + + X Y n 2 + Y n 1 ) n ( X Y ) X n 1 , X > Y > 0 .
(9)
For x A and t 1 T 1 < T 2 T , by using (9) we obtain
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | | P 1 ( T 2 ) x ( T 2 τ ) P 1 ( T 1 ) x ( T 1 τ ) | + 1 ( n 2 ) ! T 1 T 2 ( s T 1 ) n 2 r ( s ) s Q 1 ( u ) f ( x ( u σ ) ) d u d s + 1 ( n 2 ) ! T 2 ( s T 1 ) n 2 ( s T 2 ) n 2 r ( s ) s Q 1 ( u ) f ( x ( u σ ) ) d u d s | P 1 ( T 2 ) x ( T 2 τ ) P 1 ( T 1 ) x ( T 1 τ ) | + max T 1 s T 2 { 1 ( n 2 ) ! s n 2 r ( s ) s Q 1 ( u ) f ( x ( u σ ) ) d u } ( T 2 T 1 ) + 1 ( n 3 ) ! T 2 ( s T 1 ) n 3 r ( s ) s Q 1 ( u ) f ( x ( u σ ) ) d u d s ( T 2 T 1 ) .
Thus there exits δ > 0 such that
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | < ϵ if  0 < T 2 T 1 < δ .
Finally, for x A and t 0 T 1 < T 2 t 1 , there exits δ > 0 such that
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | = | v 2 ( T 1 ) v 2 ( T 2 ) | < ϵ if  0 < T 2 T 1 < δ .

Therefore SA is relatively compact. In view of Schauder’s fixed point theorem, we can conclude that there exists x A such that S x = x . That is, x is a positive solution of (1) which tends to zero. The proof is complete. □

Theorem 3 Let
t 0 Q ˜ 2 ( t ) d t = ,
(10)
where Q ˜ 2 ( t ) = c d Q 2 ( t , ξ ) d ξ . Assume that 0 < k 1 k 2 and there exists γ 0 such that
k 1 k 2 exp ( ( k 2 k 1 ) t 0 γ t 0 Q ˜ 2 ( t ) d t ) 1 , exp ( k 2 t τ t Q ˜ 2 ( s ) d s ) + exp ( k 2 t 0 γ t τ Q ˜ 2 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 1 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s P 1 ( t ) exp ( k 1 t τ t Q ˜ 2 ( s ) d s ) + exp ( k 1 t 0 γ t τ Q ˜ 2 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 2 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s , t t 1 t 0 + max { τ , d } .
(11)

Then (2) has a positive solution which tends to zero.

Proof Let Ω be the set of all continuous and bounded functions on [ t 0 , ) with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
A = { x Ω : v 1 ( t ) x ( t ) v 2 ( t ) , t t 0 } ,
where v 1 ( t ) and v 2 ( t ) are nonnegative functions such that
v 1 ( t ) = exp ( k 2 t 0 γ t Q ˜ 2 ( s ) d s ) , v 2 ( t ) = exp ( k 1 t 0 γ t Q ˜ 2 ( s ) d s ) , t t 0 .
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator S : A Ω as follows:
( S x ) ( t ) = { P 1 ( t ) x ( t τ ) 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u d s , t t 1 , ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) , t 0 t t 1 .

Since the remaining part of the proof is similar to those in the proof of Theorem 2, it is omitted. Thus the theorem is proved. □

Theorem 4 Suppose that (10) and (11) hold. In addition, assume that
exp ( k 2 t a t Q ˜ 2 ( s ) d s ) + exp ( k 2 t 0 γ t a Q ˜ 2 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 1 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s P ˜ 2 ( t ) exp ( k 1 t b t Q ˜ 2 ( s ) d s ) + exp ( k 1 t 0 γ t b Q ˜ 2 ( s ) d s ) × 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 2 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s , t t 1 t 0 + max { b , d } ,
(12)

where P ˜ 2 ( t ) = a b P 2 ( t , ξ ) d ξ . Then (3) has a positive solution which tends to zero.

Proof Let Ω be the set of all continuous and bounded functions on [ t 0 , ) with the sup norm. Then Ω is a Banach space. Define a subset A of Ω by
A = { x Ω : v 1 ( t ) x ( t ) v 2 ( t ) , t t 0 } ,
where v 1 ( t ) and v 2 ( t ) are nonnegative functions such that
v 1 ( t ) = exp ( k 2 t 0 γ t Q ˜ 2 ( s ) d s ) , v 2 ( t ) = exp ( k 1 t 0 γ t Q ˜ 2 ( s ) d s ) , t t 0 .
(13)
It is clear that A is a bounded, closed and convex subset of Ω. We define the operator S : A Ω as
( S x ) ( t ) = { a b P 2 ( t , ξ ) x ( t ξ ) d ξ 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u d s , t t 1 , ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) , t 0 t t 1 .

We show that S satisfies the assumptions of Schauder’s fixed point theorem.

First of all, S maps A into A. For t t 1 and x A , using (12), (13), the decreasing nature of v 2 and v 1 , we have
( S x ) ( t ) a b P 2 ( t , ξ ) v 2 ( t ξ ) d ξ 1 ( n 2 ) ! × t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( v 1 ( u ξ ) ) d ξ d u d s P ˜ 2 ( t ) exp ( k 1 t 0 γ t b Q ˜ 2 ( s ) d s ) 1 ( n 2 ) ! × t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 2 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s v 2 ( t )
and
( S x ) ( t ) a b P 2 ( t , ξ ) v 1 ( t ξ ) d ξ 1 ( n 2 ) ! × t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( v 2 ( u ξ ) ) d ξ d u d s P ˜ 2 ( t ) exp ( k 2 t 0 γ t a Q ˜ 2 ( s ) d s ) 1 ( n 2 ) ! × t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( exp ( k 1 t 0 γ u ξ Q ˜ 2 ( z ) d z ) ) d ξ d u d s v 1 ( t ) .
For t [ t 0 , t 1 ] and x A , we obtain
( S x ) ( t ) = ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 2 ( t )
and to show ( S x ) ( t ) v 1 ( t ) , consider
H ( t ) = v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) + v 1 ( t 1 ) .
By making use of (11), it follows that
H ( t ) = v 2 ( t ) v 1 ( t ) = k 1 Q ˜ 2 ( t ) v 2 ( t ) + k 2 Q ˜ 2 ( t ) v 1 ( t ) = Q ˜ 2 ( t ) v 2 ( t ) [ k 1 + k 2 v 1 ( t ) exp ( k 1 t 0 γ t Q ˜ 2 ( s ) d s ) ] Q ˜ 2 ( t ) v 2 ( t ) [ k 1 + k 2 exp ( ( k 1 k 2 ) t 0 γ t 0 Q ˜ 2 ( s ) d s ) ] 0 , t 0 t t 1 .
Since H ( t 1 ) = 0 and H ( t ) 0 for t [ t 0 , t 1 ] , we conclude that
H ( t ) = v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) + v 1 ( t 1 ) 0 , t 0 t t 1 .
Then t [ t 0 , t 1 ] and for any x A ,
( S x ) ( t ) = ( S x ) ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 1 ( t 1 ) + v 2 ( t ) v 2 ( t 1 ) v 1 ( t ) , t 0 t t 1 .

Hence, S maps A into A.

Next, we show that S is continuous. Let { x i } be a convergent sequence of functions in A such that x i ( t ) x ( t ) as i . Since A is closed, we have x A . It is obvious that for t [ t 0 , t 1 ] and x A , S is continuous. For t t 1 ,
| ( S x i ) ( t ) ( S x ) ( t ) | a b P 2 ( t , ξ ) | x i ( t ξ ) x ( t ξ ) | d ξ + 1 ( n 2 ) ! t ( s t ) n 2 r ( s ) s c d Q 2 ( u , ξ ) | f ( x i ( u ξ ) ) f ( x ( u ξ ) ) | d ξ d u d s .
Since | f ( x i ( t ξ ) ) f ( x ( t ξ ) ) | 0 as i and ξ [ c , d ] , by making use of the Lebesgue dominated convergence theorem, we see that
lim t ( S x i ) ( t ) ( S x ) ( t ) = 0 .

Thus S is continuous.

Finally, we show that SA is relatively compact. In order to prove that SA is relatively compact, it suffices to show that the family of functions { S x : x A } is uniformly bounded and equicontinuous on [ t 0 , ) . Since uniform boundedness of { S x : x A } is obvious, we need only to show equicontinuity. For x A and any ϵ > 0 , we take T t 1 large enough such that ( S x ) ( T ) ϵ 2 . For x A and T 2 > T 1 T , we have
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | | ( S x ) ( T 2 ) | + | ( S x ) ( T 1 ) | ϵ 2 + ϵ 2 = ϵ .
For x A and t 1 T 1 < T 2 T , by using (9) we obtain
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | a b | P 2 ( T 2 , ξ ) x ( T 2 ξ ) P 2 ( T 1 , ξ ) x ( T 1 ξ ) | d ξ + 1 ( n 2 ) ! T 1 T 2 ( s T 1 ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u d s + 1 ( n 2 ) ! T 2 ( s T 1 ) n 2 ( s T 2 ) n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u d s a b | P 2 ( T 2 , ξ ) x ( T 2 ξ ) P 2 ( T 1 , ξ ) x ( T 1 ξ ) | d ξ + max T 1 s T 2 { 1 ( n 2 ) ! s n 2 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u } ( T 2 T 1 ) + 1 ( n 3 ) ! T 2 ( s T 1 ) n 3 r ( s ) s c d Q 2 ( u , ξ ) f ( x ( u ξ ) ) d ξ d u d s ( T 2 T 1 ) .
Thus there exits δ > 0 such that
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | < ϵ if  0 < T 2 T 1 < δ .
For x A and t 0 T 1 < T 2 t 1 , there exits δ > 0 such that
| ( S x ) ( T 2 ) ( S x ) ( T 1 ) | = | v 2 ( T 1 ) v 2 ( T 2 ) | < ϵ if  0 < T 2 T 1 < δ .

Therefore SA is relatively compact. In view of Schauder’s fixed point theorem, we can conclude that there exists x A such that S x = x . That is, x is a positive solution of (1) which tends to zero. The proof is complete. □

Example 1 Consider the neutral differential equation
[ e t / 2 [ x ( t ) P 1 ( t ) x ( t 3 2 ) ] ( 2 ) ] q x ( t 1 ) = 0 , t t 0 ,
(14)
where q ( 0 , ) and
exp ( k 2 q τ ) + exp ( q [ k 2 ( t + γ τ t 0 ) k 1 ( γ σ t 0 ) ] ) k 1 exp ( ( q k 1 1 2 ) t ) ( k 1 q + 1 2 ) 2 P 1 ( t ) exp ( k 1 q τ ) + exp ( q [ k 1 ( t + γ τ t 0 ) k 2 ( γ σ t 0 ) ] ) k 2 × exp ( ( q k 2 1 2 ) t ) ( k 2 q + 1 2 ) 2 .
Note that for k 1 = 2 3 , k 2 = 1 , q = 1 and t 0 = γ = 13 2 , we have
k 1 k 2 exp ( ( k 2 k 1 ) t 0 γ t 0 Q 1 ( t ) d t ) = 2 3 exp ( 1 3 0 13 2 1 d t ) = 5.8194 1
and
exp ( 3 2 ) + 54 49 exp ( t 5 6 ) P 1 ( t ) exp ( 1 ) + 4 9 exp ( 5 t 6 ) , t 8 .

If P 1 ( t ) fulfils the last inequality above, a straightforward verification yields that the conditions of Theorem 2 are satisfied and therefore (14) has a positive solution which tends to zero.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Arts and Sciences, Niğde University

References

  1. Culáková I, Hanuštiaková L’, Olach R: Existence for positive solutions of second-order neutral nonlinear differential equations. Appl. Math. Lett. 2009, 22: 1007–1010. 10.1016/j.aml.2009.01.009MathSciNetView ArticleGoogle Scholar
  2. Candan T: The existence of nonoscillatory solutions of higher order nonlinear neutral equations. Appl. Math. Lett. 2012, 25(3):412–416. 10.1016/j.aml.2011.09.025MathSciNetView ArticleGoogle Scholar
  3. Candan T, Dahiya RS: Existence of nonoscillatory solutions of higher order neutral differential equations with distributed deviating arguments. Math. Slovaca 2013, 63(1):183–190. 10.2478/s12175-012-0091-0MathSciNetView ArticleGoogle Scholar
  4. Li W-T, Fei X-L: Classifications and existence of positive solutions of higher-order nonlinear delay differential equations. Nonlinear Anal. 2000, 41: 433–445. 10.1016/S0362-546X(98)00286-7MathSciNetView ArticleGoogle Scholar
  5. Zhou Y, Zhang BG: Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients. Appl. Math. Lett. 2002, 15: 867–874. 10.1016/S0893-9659(02)00055-1MathSciNetView ArticleGoogle Scholar
  6. Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar
  7. Agarwal RP, Bohner M, Li W-T: Nonoscillation and Oscillation: Theory for Functional Differential Equations. Dekker, New York; 2004.View ArticleGoogle Scholar
  8. Erbe LH, Kong QK, Zhang BG: Oscillation Theory for Functional Differential Equations. Dekker, New York; 1995.Google Scholar
  9. Györi I, Ladas G: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford; 1991.Google Scholar
  10. Bainov DD, Mishev DP: Oscillation Theory for Neutral Differential Equations with Delay. Hilger, Bristol; 1991.Google Scholar
  11. Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York; 1987.Google Scholar

Copyright

© Candan; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.