Open Access

Existence of positive solutions for a kind of periodic boundary value problem at resonance

Boundary Value Problems20132013:19

DOI: 10.1186/1687-2770-2013-19

Received: 20 December 2012

Accepted: 21 January 2013

Published: 11 February 2013

Abstract

In the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.

Keywords

periodic boundary value problem positive solution coincidence equation

1 Introduction

In the paper we are interested in the existence of positive solutions for the periodic boundary value problem (PBVP)
{ x ( t ) + h ( t ) x ( t ) + f ( t , x ( t ) , x ( t ) ) = 0 , t [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) ,
(1)
where f : [ 0 , T ] × [ 0 , ) × R R and h : [ 0 , T ] ( 0 , ) are continuous functions. Our study is motivated by current activity of many researchers in the area of theory and applications of PVBPs; see, for example, [14] and references therein. In particular, in a recent paper [1], Chu, Fan and Torres have studied the existence of positive periodic solutions for the singular damped differential equation
x ( t ) + h ( t ) x ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) , x ( t ) )
by combining the properties of the Green’s function of the PBVP
{ x ( t ) + h ( t ) x ( t ) + a ( t ) x ( t ) = 0 , t [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) ,
(2)
with a nonlinear alternative of Leray-Schauder type (see, for example, [5]). It should be noted that a 0 was the key assumption used in [1]. If a 0 , then PBVP (2) has nontrivial solutions, which means that the problem we are concerned with here, that is, PBVP (1), is at resonance. There are several methods to deal with the resonant PBVPs. For example, in [6], Torres studied the existence of a positive solution for the PBVP
{ x ( t ) = f ( t , x ( t ) ) , t ( 0 , 2 π ) , x ( 0 ) = x ( 2 π ) , x ( 0 ) = x ( 2 π ) ,
by considering the equivalent problem
{ x ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) ) + a ( t ) x ( t ) , t ( 0 , 2 π ) , x ( 0 ) = x ( 2 π ) , x ( 0 ) = x ( 2 π ) ,
via Krasnoselskii’s theorem on cone expansion and compression. Further results in this direction can be found in [7] and [8]. In [9] Rachůnková, Tvrdý and Vrkoč applied the method of upper and lower solutions and topological degree arguments to establish the existence of nonnegative and nonpositive solutions for the PBVP
{ x ( t ) = f ( t , x ( t ) ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) .
(3)

The same PBVP was studied by Wang, Zhang and Wang in [10]. Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for A-proper semilinear operators on cones developed by Cremins [11].

The goal of our paper is to provide sufficient conditions that ensure the existence of positive solutions of (1) with the function h positive on [ 0 , T ] . Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation L x = N x , where L is a Fredholm operator of index zero and N is a nonlinear operator, and to apply the Leggett-Williams norm-type theorem for coincidences obtained by O’Regan and Zima [12]. We would like to emphasize that the idea of results of [11] and [12], as well as these of [1315], goes back to the celebrated Mawhin’s coincidence degree theory [16]. For more details on this significant tool, its modifications and wide applications, we refer the reader to [1722] and references therein.

In this paper, for the first time, the existence theorem from [12] is used for studying the boundary value problem with the nonlinearity f depending also on the derivative. In general, the presence of x in f makes the problem much harder to handle. We point out that, to the best of our knowledge, there are only a few papers on PBVPs that discuss such a nonlinearity; we refer the reader to [15, 2325] for some results of that type. We also complement several results in the literature, for example, in [1, 26] and [27]. It is evident that the existence theorems for PBVP (1) can be established by the shift method used in [6], that is, one can employ the results of [1] to the periodic problem we study here. However, the conditions imposed on f in [1] are not comparable with ours.

2 Coincidence equation

For the convenience of the reader, we begin this section by providing some background on cone theory and Fredholm operators in Banach spaces.

Definition 1 A nonempty subset C, C { 0 } , of a real Banach space X is called a cone if C is closed, convex and
  1. (i)

    λ x C for all x C and λ 0 ,

     
  2. (ii)

    x, x C implies x = 0 .

     
Every cone induces a partial ordering in X as follows: for x , y X , we say that
x y if and only if y x C .

The following property holds for every cone in a Banach space.

Lemma 1 [28]For every u C { 0 } , there exists a positive number σ ( u ) such that
x + u σ ( u ) x

for all x C .

Consider a linear mapping L : dom L X Y and a nonlinear operator N : X Y , where X and Y are Banach spaces. If L is a Fredholm operator of index zero, that is, ImL is closed and dim Ker L = codim Im L < , then there exist continuous projections P : X X and Q : Y Y such that Im P = Ker L and Ker Q = Im L (see, for example, [14, 16]). Moreover, since dim Im Q = codim Im L , there exists an isomorphism J : Im Q Ker L . Denote by L P the restriction of L to Ker P dom L . Then L P is an isomorphism from Ker P dom L to ImL and its inverse
K P : Im L Ker P dom L

is defined.

As a result, the coincidence equation L x = N x is equivalent to x = Ψ x , where
Ψ = P + J Q N + K P ( I Q ) N .
Let ρ : X C be a retraction, that is, a continuous mapping such that ρ ( x ) = x for all x C . Put
Ψ ρ = Ψ ρ .

Let Ω 1 , Ω 2 be open bounded subsets of X with Ω ¯ 1 Ω 2 and C ( Ω ¯ 2 Ω 1 ) . Assume that

1 L is a Fredholm operator of index zero,

2 Q N : X Y is continuous and bounded and K P ( I Q ) N : X X is compact on every bounded subset of X,

3 L x λ N x for all x C Ω 2 dom L and λ ( 0 , 1 ) ,

4 ρ maps subsets of Ω ¯ 2 into bounded subsets of C,

5 d B ( [ I ( P + J Q N ) ρ ] | Ker L , Ker L Ω 2 , 0 ) 0 , where d B stands for the Brouwer degree,

6 there exists u 0 C { 0 } such that x σ ( u 0 ) Ψ x for x C ( u 0 ) Ω 1 , where
C ( u 0 ) = { x C : μ u 0 x  for some  μ > 0 }

and σ ( u 0 ) is such that x + u 0 σ ( u 0 ) x for every x C ,

7 ( P + J Q N ) ρ ( Ω 2 ) C and Ψ ρ ( Ω ¯ 2 Ω 1 ) C .

Theorem 1 [12]

Under the assumptions 1-7 the equation L x = N x has a solution in the set C ( Ω ¯ 2 Ω 1 ) .

In the next section, we use Theorem 1 to prove the existence of a positive solution for PBVP (1). For applications of Theorem 1 to nonlocal boundary value problems at resonance, we refer the reader to [22], [29] and [30].

3 Periodic boundary value problem

We now provide sufficient conditions for the existence of positive solutions for PBVP (1). For convenience and ease of exposition, we make use of the following notation:
e ( t ) = exp ( 0 t h ( τ ) d τ ) , φ ( t ) = 0 t e ( τ ) d τ , Φ ( t ) = 0 t φ ( τ ) d τ ,
(4)
and
ψ ( t ) = 1 e ( t ) ( 1 1 e ( T ) φ ( t ) φ ( T ) ) , t [ 0 , T ] .
(5)
We observe that 0 < ψ ( t ) < 1 e ( T ) ( 1 e ( T ) ) on [ 0 , T ] . Moreover, we put
k ( t , s ) = 1 T e ( s ) { φ ( s ) φ ( T ) [ φ ( T ) s T φ ( t ) + Φ ( T ) ] Φ ( s ) , 0 s t T , φ ( s ) φ ( T ) [ φ ( T ) ( s T ) T φ ( t ) + Φ ( T ) ] + T φ ( t ) Φ ( s ) , 0 t s T ,
(6)
and
K ( t , s ) = k ( t , s ) + M 0 T k ( t , τ ) d τ 0 T ψ ( τ ) d τ ψ ( s ) , t , s [ 0 , T ] ,
(7)

where M is a positive constant.

We assume that

(H1) f : [ 0 , T ] × [ 0 , ) × R R and h : [ 0 , T ] ( 0 , ) are continuous functions.

We also assume that there exist R > 0 , 0 < α β , 0 < M e ( T ) ( 1 e ( T ) ) 0 T ψ ( τ ) d τ α T , r ( 0 , R ) , m ( 0 , 1 ) , η [ 0 , T ] and a continuous function g : [ 0 , T ] [ 0 , ) such that

(H2) f ( t , x , y ) > α x + β | y | for ( t , x , y ) [ 0 , T ] × [ 0 , R ] × [ R , R ] ,

(H3) f ( t , R , 0 ) < 0 for t [ 0 , T ] ,

(H4) f ( 0 , x , R ) = f ( T , x , R ) and f ( 0 , x , R ) = f ( T , x , R ) for x [ 0 , R ] ,

(H5) f ( t , x , R ) h ( t ) R for t [ 0 , T ] and x [ 0 , R ) ,

(H6) f ( t , x , y ) g ( t ) ( x + | y | ) for ( t , x , y ) [ 0 , T ] × ( 0 , r ] × [ r , r ] ,

(H7) 1 α T K ( t , s ) 0 for t , s [ 0 , T ] and m 0 T K ( η , s ) g ( s ) d s 1 .

Theorem 2 Under the assumptions (H1)-(H7), PBVP (1) has a positive solution on [ 0 , T ] .

Proof Let denote the supremum norm in the space C [ 0 , T ] , that is, x = sup t [ 0 , T ] | x ( t ) | . Consider the Banach spaces X = C 1 [ 0 , T ] with the norm x = max { x , x } , and Y = C [ 0 , T ] with the norm .

We write problem (1) as a coincidence equation
L x = N x ,
where
L x ( t ) = x ( t ) h ( t ) x ( t ) , t [ 0 , T ] ,
and
N x ( t ) = f ( t , x ( t ) , x ( t ) ) , t [ 0 , T ] ,
with dom L = { x X : x C [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) } . Then
Ker L = { x X : x ( t ) = c , t [ 0 , T ] , c R }
and
Im L = { y Y : 0 T ψ ( s ) y ( s ) d s = 0 } ,

where ψ is given by (5).

Clearly, ImL is closed and Y = Y 1 + Im L with
Y 1 = { y 1 Y : y 1 = 1 0 T ψ ( s ) d s 0 T ψ ( s ) y ( s ) d s , y Y } .

Since Y 1 Im L = { 0 } , we have Y = Y 1 Im L . Moreover, dim Y 1 = 1 , which gives codim Im L = 1 . Consequently, L is Fredholm of index zero, and the assumption 1 is satisfied.

Define the projections P : X X by
P x ( t ) = 1 T 0 T x ( s ) d s , t [ 0 , T ] ,
and Q : Y Y by
Q y ( t ) = 1 0 T ψ ( s ) d s 0 T ψ ( s ) y ( s ) d s , t [ 0 , T ] .
It is a routine matter to show that for y Im L , the inverse K P of L P is given by
( K P y ) ( t ) = 0 T k ( t , s ) y ( s ) d s , t [ 0 , T ] ,
with the kernel k defined by (6). Clearly, the assumption 2 is satisfied. For y Im Q , define
J ( y ) = M y .
Then J is an isomorphism from ImQ to KerL. Next, consider a cone
C = { x X : x ( t ) 0  on  [ 0 , T ] } .
For u 0 ( t ) 1 , we have σ ( u 0 ) = 1 and
C ( u 0 ) = { x C : x ( t ) > 0  on  [ 0 , T ] } .
Let
Ω 1 = { x X : x < r , | x ( t ) | > m x  and  | x ( t ) | > m x  on  [ 0 , T ] } ,
and
Ω 2 = { x X : x < R } .

Obviously, Ω 1 and Ω 2 are open bounded subsets of X, and Ω ¯ 1 Ω 2 .

To verify 3, suppose that there exist x 0 C Ω 2 dom L and λ 0 ( 0 , 1 ) such that L x 0 = λ 0 N x 0 . Then x ( t ) 0 on [ 0 , T ] , x 0 = R ,
x 0 ( t ) h ( t ) x 0 ( t ) = λ 0 f ( t , x 0 ( t ) , x 0 ( t ) ) , t [ 0 , T ] ,
(8)
and
x 0 ( 0 ) = x 0 ( T ) , x 0 ( 0 ) = x 0 ( T ) .
(9)
There are two cases to consider.
  1. 1.

    If x 0 = x 0 , then there exists t 0 [ 0 , T ] such that x ( t 0 ) = R . For t 0 ( 0 , T ) , we get 0 x ( t 0 ) = λ 0 f ( t 0 , R , 0 ) , contrary to the assumption (H3). Similarly, if t 0 = 0 or t 0 = T , BCs (9) imply x ( 0 ) = x ( T ) = 0 . Hence, 0 x ( t 0 ) = λ 0 f ( t 0 , R , 0 ) which contradicts (H3) again.

     
  2. 2.
    If x 0 = x 0 > x 0 , then there exists t 0 [ 0 , T ] such that | x ( t 0 ) | = R . Observe that (H2) implies f ( t , x , ± R ) > 0 for t [ 0 , T ] and x [ 0 , R ] . Suppose that t 0 ( 0 , T ) . If x ( t 0 ) = R , we get from (8)
    h ( t 0 ) R = λ 0 f ( t 0 , x 0 ( t 0 ) , R ) ,
    (10)
     
a contradiction. For x ( t 0 ) = R , we have
h ( t 0 ) R = λ 0 f ( t 0 , x 0 ( t 0 ) , R ) < f ( t 0 , x 0 ( t 0 ) , R ) ,
(11)

contrary to (H5). By similar arguments, if t 0 = 0 or t 0 = T , BCs (9) and (H4) imply either (10) or (11). Thus, 3 is fulfilled.

Next, for x X , define (see [15])
ρ x ( t ) = { x ( t ) if  x ( t ) 0  on  [ 0 , T ] , 1 2 ( x ( t ) min { x ( t ) : t [ 0 , T ] } ) if  x ( t ˜ ) < 0  for some  t ˜ [ 0 , T ] .

Clearly, ρ is a retraction and maps subsets of Ω ¯ 2 into bounded subsets of C, so 4 holds.

To verify 5, it is enough to consider, for x Ker L Ω 2 and λ [ 0 , 1 ] , the mapping
H ( x , λ ) ( t ) = x ( t ) λ ( 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s ) .
Observe that if x Ker L Ω 2 , then x ( t ) = c on [ 0 , T ] and x < R . Suppose H ( x , λ ) = 0 for x Ω 2 . Then c = ± R . For c = R , we have ( ρ x ) ( t ) = x ( t ) and in view of (H3), we get
0 R ( 1 λ ) = λ M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , R , 0 ) d s < 0 ,
which is a contradiction. If c = R , then ( ρ x ) ( t ) = 0 , hence
R = λ M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , 0 , 0 ) d s ,
which contradicts (H2). Thus, H ( x , λ ) 0 for x Ω 2 and λ [ 0 , 1 ] . This implies
d B ( H ( x , 0 ) , Ker L Ω 2 , 0 ) = d B ( H ( x , 1 ) , Ker L Ω 2 , 0 ) ,
and
d B ( [ I ( P + J Q N ) ρ ] | Ker L , Ker L Ω 2 , 0 ) = d B ( H ( c , 1 ) , Ker L Ω 2 , 0 ) 0 .
We next show that 6 holds. Let x C ( u 0 ) Ω 1 . Then for t [ 0 , T ] , we have r x ( t ) m x > 0 , r | x ( t ) | x , and by (H6) and (H7), we obtain
Ψ x ( η ) = 1 T 0 T x ( s ) d s + 0 T K ( η , s ) f ( s , x ( s ) , x ( s ) ) d s 0 T K ( η , s ) g ( s ) [ x ( s ) + | x ( s ) | ] d s m 0 T K ( η , s ) g ( s ) [ x + x ] d s m x 0 T K ( η , s ) g ( s ) d s x .

This implies x Ψ x for x C ( u 0 ) Ω 1 , so 6 is satisfied.

Finally, we must check if 7 holds. If x Ω 2 , then in view of (H2), we get
( P + J Q N ) ( ρ x ) ( t ) = 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) [ α ( ρ x ) ( s ) + β | ( ρ x ) ( s ) | ] d s 0 T [ 1 T α M ψ ( s ) 0 T ψ ( τ ) d τ ] ( ρ x ) ( s ) d s 0 T [ 1 T α M e ( T ) ( 1 e ( T ) ) 0 T ψ ( τ ) d τ ] ( ρ x ) ( s ) d s 0 .
Moreover, for x Ω ¯ 2 Ω 1 , we have from (H2) and (H7)
Ψ ρ x ( t ) = 1 T 0 T ( ρ x ) ( s ) d s + 0 T K ( t , s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s 1 T 0 T ( ρ x ) ( s ) d s + 0 T K ( t , s ) [ α ( ρ x ) ( s ) + β | ( ρ x ) ( s ) | ] d s 0 T [ 1 T α K ( t , s ) ] ( ρ x ) ( s ) d s 0 .

Thus, 7 is fulfilled and the assertion follows. □

We now give two examples illustrating Theorem 2. Some calculations have been made with Mathematica. In the first example, the function h is constant, while in the second h ( t ) = 1 / ( 1 + t ) and f is independent of t.

Example 1

Consider the following PBVP:
{ x ( t ) + x ( t ) + ( t ( 1 t ) + 1 ) ( 2 9 x ( t ) + 3 4 | x ( t ) | + 1 ) = 0 , t [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) .
(12)
Then e ( t ) = e t , φ ( t ) = 1 e t , Φ ( t ) = t + e t 1 , ψ ( t ) = e e 1 , and
k ( t , s ) = { s + e s t + 1 e 1 t e 1 , 0 s t 1 , s + 1 + e s t e 1 t e 1 , 0 t s 1 .
Moreover, (7) with M = 3 2 reads
K ( t , s ) = { t s + e s t + 1 e 1 , 0 s t 1 , t s + 1 + e s t e 1 , 0 t s 1 ,

and the assumptions (H2)-(H7) are met with R = 20 , α = 2 9 , β = 3 4 , r = 36 53 , m [ 12 ( e 1 ) 17 + 7 e , 1 ) , η = 0 and g ( t ) = t ( 1 t ) + 1 . By Theorem 2, problem (12) has a positive solution.

Example 2

Consider the PBVP
{ x ( t ) + 1 1 + t x ( t ) + 1 10 1 9 x ( t ) + ( x ( t ) ) 4 / 5 = 0 , t [ 0 , 1 2 ] , x ( 0 ) = x ( 1 2 ) , x ( 0 ) = x ( 1 2 ) .
(13)
In this case, we have e ( t ) = 1 1 + t , φ ( t ) = ln ( 1 + t ) , Φ ( t ) = t + ln ( 1 + t ) + t ln ( 1 + t ) and
ψ ( t ) = ( 1 + t ) ( 3 ln ( 1 + t ) ln ( 3 2 ) ) .

The assumptions of Theorem 2 are fulfilled with M = 1 , R = 10 , α = 1 3 , β = 1 2 , r = 1 100 , m = 0.9 , η = 1 4 and g ( t ) = 3 .

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

Authors’ Affiliations

(1)
Institute of Mathematics, University of Rzeszów

References

  1. Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 10.1016/j.jmaa.2011.09.061MathSciNetView Article
  2. Cabada A, Cid JÁ: On comparison principles for the periodic Hill’s equation. J. Lond. Math. Soc. 2012, 86: 272-290. 10.1112/jlms/jds001MathSciNetView Article
  3. Graef JR, Kong L, Wang H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ. 2008, 245: 1185-1197. 10.1016/j.jde.2008.06.012MathSciNetView Article
  4. Ma R, Xu J, Han X: Global structure of positive solutions for superlinear second-order periodic boundary value problems. Appl. Math. Comput. 2012, 218: 5982-5988. 10.1016/j.amc.2011.11.079MathSciNetView Article
  5. Meehan M, O’Regan D: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear Anal. 1998, 31: 317-341. 10.1016/S0362-546X(96)00313-6MathSciNetView Article
  6. Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3View Article
  7. Yao Q: Positive solutions of nonlinear second-order periodic boundary value problems. Appl. Math. Lett. 2007, 20: 583-590. 10.1016/j.aml.2006.08.003MathSciNetView Article
  8. Ma R, Gao C, Chen R: Existence of positive solutions of nonlinear second-order periodic boundary value problems. Bound. Value Probl. 2010., 2010: Article ID 626054. doi:10.1155/2010/626054
  9. Rachůnková I, Tvrdý M, Vrkoč I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 2001, 176: 445-469. 10.1006/jdeq.2000.3995View Article
  10. Wang F, Zhang F, Wang F: The existence and multiplicity of positive solutions for second-order periodic boundary value problem. J. Funct. Spaces Appl. 2012., 2012: Article ID 725646. doi:10.1155/2012/725646
  11. Cremins CT: A fixed point index and existence theorems for semilinear equations in cones. Nonlinear Anal. 2001, 46: 789-806. 10.1016/S0362-546X(00)00144-9MathSciNetView Article
  12. O’Regan D, Zima M: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 2006, 87: 233-244. 10.1007/s00013-006-1661-6MathSciNetView Article
  13. Gaines RE, Santanilla J: A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 1982, 12: 669-678. 10.1216/RMJ-1982-12-4-669View Article
  14. Santanilla J: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 1985, 105: 357-371. 10.1016/0022-247X(85)90053-8MathSciNetView Article
  15. Santanilla J: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 1987, 126: 397-408. 10.1016/0022-247X(87)90049-7MathSciNetView Article
  16. Mawhin J: Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces. J. Differ. Equ. 1972, 12: 610-636. 10.1016/0022-0396(72)90028-9MathSciNetView Article
  17. Gaines RE, Mawhin J Lect. Notes Math. 568. In Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.
  18. Webb JRL: Solutions of semilinear equations in cones and wedges. I-IV. In World Congress of Nonlinear Analysts ’92 (Tampa, FL 1992). de Gruyter, Berlin; 1996:137-147.
  19. Feng W, Webb JRL: Solvability of three-point boundary value problems at resonance. Nonlinear Anal. 1997, 30: 3227-3238. 10.1016/S0362-546X(96)00118-6MathSciNetView Article
  20. Liu B: Solvability of multi-point boundary value problems at resonance. IV. Appl. Math. Comput. 2003, 143: 275-299. 10.1016/S0096-3003(02)00361-2MathSciNetView Article
  21. Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158-2171. 10.1016/j.na.2007.01.038MathSciNetView Article
  22. Franco D, Infante G, Zima M: Second order nonlocal boundary value problems at resonance. Math. Nachr. 2011, 284: 875-884. 10.1002/mana.200810841MathSciNetView Article
  23. Cabada A, Pouso R:Existence result for the problem ( ϕ ( u ) ) = f ( t , u , u ) with periodic and Neumann boundary conditions. Nonlinear Anal. 1997, 30: 1733-1742. 10.1016/S0362-546X(97)00249-6MathSciNetView Article
  24. Sȩdziwy S: Nonlinear periodic boundary value problem for a second order ordinary differential equation. Nonlinear Anal. 1998, 32: 881-890. 10.1016/S0362-546X(97)00533-6MathSciNetView Article
  25. Kiguradze I, Staněk S: On periodic boundary value problem for the equation u = f ( t , u , u ) with one-sided growth restrictions on f . Nonlinear Anal. 2002, 48: 1065-1075. 10.1016/S0362-546X(00)00235-2MathSciNetView Article
  26. Torres PJ: Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle. Mediterr. J. Math. 2004, 1: 479-486. 10.1007/s00009-004-0025-3MathSciNetView Article
  27. Cheng Z, Ren J: Harmonic and subharmonic solutions for superlinear damped Duffing equation. Nonlinear Anal., Real World Appl. 2013, 14: 1155-1170. 10.1016/j.nonrwa.2012.09.007MathSciNetView Article
  28. Petryshyn WV: On the solvability of x T x + λ F x in quasinormal cones with T and F k -set contractive. Nonlinear Anal. 1981, 5: 585-591. 10.1016/0362-546X(81)90105-XMathSciNetView Article
  29. Infante G, Zima M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal. 2008, 69: 2458-2465. 10.1016/j.na.2007.08.024MathSciNetView Article
  30. Zhang HE, Sun JP: Positive solutions of third-order nonlocal boundary value problems at resonance. Bound. Value Probl. 2012., 2012: Article ID 102

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© Zima and Drygaś; licensee Springer. 2013

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