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Solvability for nonlocal boundary value problems on a half line with dim(kerL)=2

Abstract

The existence of at least one solution to the second-order nonlocal boundary value problems on a half line is investigated by using Mawhin’s continuation theorem.

MSC: 34B10, 34B40, 34B15.

1 Introduction

Boundary value problems on an infinite interval arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and in various applications such as an unsteady flow of gas through a semi-infinite porous media, theory of drain flows and plasma physics. For an extensive collection of results as regards boundary value problems on unbounded domains, we refer the reader to a monograph by Agarwal and O’Regan [1]. For more recent results on unbounded domains, we refer the reader to [2]–[15] and the references therein.

A boundary value problem is called to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Resonance problems can be expressed as an abstract equation Lx=Nx, where L is a noninvertible operator. When L is linear, Mawhin’s continuation theorem [16] is an efficient tool in finding solutions for these problems. Recently, there have been many works concerning the existence of solutions for multi-point boundary value problems at resonance. For example, see [7], [17]–[21] in the case that dim(kerL)=1, and see [4], [22]–[25] in the case that dim(kerL)=2.

In this paper, we consider the existence of solutions to the following second-order nonlinear differential equation with nonlocal boundary conditions that contain integral and multi-point boundary conditions:

{ ( c u ) ( t ) = f ( t , u ( t ) , u ( t ) ) , a.e.  t ( 0 , ) , ( c u ) ( 0 ) = 0 g ( s ) ( c u ) ( s ) d s , lim t ( c u ) ( t ) = i = 1 m α i ( c u ) ( ξ i ) ,
(1)

where 0 ξ 1 << ξ m <, α i (,), cC[0,), g L 1 (0,), and f:[0,)×(,)×(,)(,) is a Carathéodory function, i.e., f=f(t,u,v) is Lebesgue measurable in t for all (u,v)(,)×(,) and continuous in (u,v) for almost all t[0,). Throughout this paper, we assume that the following conditions hold:

(H1) i = 1 m α i =1, 0 g(s)ds=1, c(t)>0 for t(0,), and 1 c L loc 1 [0,);

(H2) let w(t):= 0 t 1 c ( s ) ds, and there exist non-negative measurable functions α, β, and γ such that (1+w)α,β/c,γ L 1 (0,), and |f(t,u,v)|α(t)|u|+β(t)|v|+γ(t), a.e. (t,u,v)[0,)×(,)×(,);

(H3) let k(t) be the function such that (1+w()) e k ( ) L 1 (0,). Then

= | Q 2 ( w ( ) e k ( ) ) Q 1 ( w ( ) e k ( ) ) Q 2 ( e k ( ) ) Q 1 ( e k ( ) ) | =: | a 11 a 12 a 21 a 22 | 0,

where the linear operators Q 1 , Q 2 : L 1 (0,)(,) will be defined later in Section 3.

If 1 c L loc 1 [0,), then w is continuous in [0,), and in (H3), there exists a function k satisfying (1+w()) e k ( ) L 1 (0,) (see, e.g., Remark 3.1(1)). The boundary conditions in problem (1) are crucial since the differential operator Lu= ( c u ) under the boundary conditions in (1) satisfies dim(kerL)=2. The purpose of this paper is to establish the sufficient conditions for the existence of solutions to problem (1) on a half line at resonance with dim(kerL)=2 by using Mawhin’s continuation theorem [16].

The remainder of this paper is organized as follows: some preliminaries are provided in Section 2, the main result is presented in Section 3, and finally an example to illustrate the main result is given in Section 4.

2 Preliminaries

In this section, we recall some notations and two theorems which will be used later. Let X and Y be two Banach spaces with the norms X and Y , respectively. Let L:dom(L)XY be a Fredholm operator with index zero, and let P:XX, Q:YY be projectors such that ImP=kerL and kerQ=ImL. Then X=kerLkerP and Y=ImLImQ. It follows that L | dom L ker P :domLkerPImL is invertible. We denote the inverse of it by  K P . If Ω is an open bounded subset of X with domL Ω ¯ , then the map N:XY will be called L-compact on Ω ¯ if QN( Ω ¯ ) is bounded in Y and K P (IQ)N: Ω ¯ X is compact.

Theorem 2.1

([16])

LetL:dom(L)XYbe a Fredholm operator with index zero andN:XYbe L-compact on Ω ¯ . Assume that the following conditions are satisfied:

  1. (1)

    LuλNu for every (u,λ)[(domLkerL)Ω]×(0,1);

  2. (2)

    NuImL for every ukerLΩ;

  3. (3)

    deg(QN | ker L ,ΩkerL,0)0, where Q:YY is a projector such that ImL=kerQ.

Then the equationLu=Nuhas at least one solution indomL Ω ¯ .

Since the Arzelá-Ascoli theorem fails in the noncompact interval case, we use the following result in order to show that K P (IQ)N: Ω ¯ X is compact.

Theorem 2.2

([1])

Let Z be the space of all bounded continuous vector-valued functions on[0,)andSZ. Then S is relatively compact in Z if the following conditions hold:

  1. (i)

    S is bounded in Z;

  2. (ii)

    S is equicontinuous on any compact interval of [0,);

  3. (iii)

    S is equiconvergent at ∞, that is, given ϵ>0, there exists a T=T(ϵ)>0 such that ϕ ( t ) ϕ ( ) ( , ) n <ϵ for all t>T and all ϕS.

3 Main results

Let X be the set of the functions uC[0,) C 1 (0,) such that u ( t ) 1 + w ( t ) and (c u )(t) are uniformly bounded in [0,). Here, w is the function in the assumption (H2). Then X is a Banach space equipped with a norm u X = u 1 + u 2 , where

u 1 = sup t [ 0 , ) | u ( t ) | 1 + w ( t ) and u 2 = sup t [ 0 , ) | ( c u ) (t)|.

Let Y denote the Banach space L 1 (0,) equipped with the usual norm, h Y = 0 |h(s)|ds.

Remark 3.1

  1. (1)

    For any non-negative continuous function w(t), we can choose a function k(t) which satisfies (1+w()) e k ( ) Y. For example, put

    k(t)= 0 t ( 1 + w ( s ) ) ds.

Then (1+w()) e k ( ) Y.

  1. (2)

    If 1 c Y, then w L (0,), and the norm X is equivalent to the norm u= u + u 2 . Here,

    u = sup t [ 0 , ) |u(t)|.

Define L:domLY by Lu:= ( c u ) , where

dom L : = { u X : ( c u ) Y , ( c u ) ( 0 ) = 0 g ( s ) ( c u ) ( s ) d s , lim t ( c u ) ( t ) = i = 1 m α i ( c u ) ( ξ i ) } .

Clearly, kerL:={a+bw:a,b(,)}. Now we define the linear operators Q 1 , Q 2 :Y(,) under the hypothesis (H3) as follows:

Q 1 (y):= i = 1 m α i ξ i y(s)dsand Q 2 (y):= 0 g(s) 0 s y(τ)dτds.

Lemma 3.2

Assume that (H1) holds. Then

ImL= { y Y : Q 1 ( y ) = Q 2 ( y ) = 0 } .

Proof

Let yImL. Then there exists xdomL such that ( c x ) =y. For t[0,),

( c x ) (t)= ( c x ) (0)+ 0 t y(s)ds

and

( c x ) ( ξ i )= ( c x ) (0)+ 0 ξ i y(s)ds,

which imply that

lim t ( c x ) (t)= ( c x ) (0)+ 0 y(s)ds= ( c x ) ( ξ i )+ ξ i y(s)ds,

and thus Q 1 (y)=0 by the fact that i = 1 m α i =1. In a similar manner, Q 2 (y)=0. On the other hand, let yY satisfying Q 1 (y)= Q 2 (y)=0. Take

x(t)= 0 t 1 c ( s ) 0 s y(τ)dτds.

Then xdomL, and ( c x ) =yImL. Thus the proof is complete. □

By Lemma 3.2, codim(ImL)=2. Since dim(kerL)=2, L is a Fredholm operator with index 0. Let T 1 , T 2 :YY be linear operators which are defined as follows:

T 1 y : = 1 ( a 11 Q 1 ( y ) + a 12 Q 2 ( y ) ) e k ( ) , T 2 y : = 1 ( a 21 Q 1 ( y ) + a 22 Q 2 ( y ) ) e k ( ) .

Then, by simple calculations, T 1 ( T 1 y)= T 1 y, T 1 (( T 2 y)w)=0, T 2 ( T 1 y)=0, and T 2 (( T 2 y)w)= T 2 y. Define a bounded linear operator Q:YY by

(Qy)(t):=( T 1 y)(t)+( T 2 y)(t)w(t),t(0,).
(2)

Then Q 2 y=Qy, i.e., Q:YY is a linear projector. Since 0, kerQ=ImL, and Y=ImLImQ.

Define a continuous projector P:XX by

Px:=x(0)+ ( c x ) (0)w().

Clearly, ImP=kerL, and consequently X=kerLkerP. Define an operator K P :ImLdomLkerP by

( K P y)(t):= 0 t 1 c ( s ) 0 s y(τ)dτds,t[0,).

Then K P is the inverse operator L | dom L ker P , and it satisfies

K P y X 2 y Y .
(3)

Let a nonlinear operator N:XY be defined by (Nx)(t):=f(t,x(t), x (t)), t[0,). Then problem (1) is equivalent to Lx=Nx, xdomL.

From now on, we consider the case 1 c Y. The case 1 c Y can be dealt in a similar manner.

Lemma 3.3

Let 1 c Y, and assume that (H1)-(H3) hold. Assume that Ω is a bounded open subset of X such thatdomL Ω ¯ . Then N is L-compact on Ω ¯ .

Proof

Since Ω is a bounded open subset of X, there exists a constant r>0 such that x X r for any x Ω ¯ . For any x Ω ¯ and for almost all t[0,), by (H2),

|(Nx)(t)|=|f ( t , x ( t ) , x ( t ) ) | ( ( 1 + w ( t ) ) α ( t ) + β c ( t ) ) r+γ(t)=: N r (t).

Then N r Y and N x Y N r Y for all x Ω ¯ . Thus N( Ω ¯ ) is bounded in Y.

For any x Ω ¯ ,

| Q 1 (Nx)| ( i = 1 m | α i | ) N x Y ( i = 1 m | α i | ) N r Y ,| Q 2 (Nx)| N r Y .
(4)

It follows from (2) and (4) that, for almost all t[0,),

| ( Q N x ) ( t ) | | ( T 1 N x ) ( t ) | + | ( T 2 N x ) ( t ) | w ( t ) 1 | | ( ( | a 11 | i = 1 m | α i | + | a 12 | ) + ( | a 21 | i = 1 m | α i | + | a 22 | ) w ( t ) ) e k ( t ) N r Y = : Q r ( t ) .

Then Q r Y and Q N x Y Q r Y for all x Ω ¯ . Thus, QN( Ω ¯ ) is bounded in Y.

Next we will prove that K P (IQ)N( Ω ¯ ) is a relatively compact set in X. For x Ω ¯ , by (3),

K P ( I Q ) N x X 2 ( N x Y + Q N x Y ) 2 ( N r Y + Q r Y ) .

Thus K P (IQ)N( Ω ¯ ) is bounded in X.

Let T>0 be given. For any x Ω ¯ and let t 1 , t 2 [0,T] with t 1 < t 2 ,

| ( K P ( I Q ) N x ) ( t 1 ) 1 + w ( t 1 ) ( K P ( I Q ) N x ) ( t 2 ) 1 + w ( t 2 ) | | w ( t 2 ) w ( t 1 ) ( 1 + w ( t 1 ) ) ( 1 + w ( t 2 ) ) | | 0 t 1 1 c ( s ) 0 s ( I Q ) N x ( τ ) d τ d s | + | t 1 t 2 1 c ( s ) 0 s ( I Q ) N x ( τ ) d τ d s 1 + w ( t 2 ) | 2 | w ( t 2 ) w ( t 1 ) | ( N r Y + Q r Y )

and

| ( c ( K P ( I Q ) N x ) ) ( t 1 ) ( c ( K P ( I Q ) N x ) ) ( t 2 )| t 1 t 2 ( N r ( τ ) + Q r ( τ ) ) dτ,

which imply that

{ K P ( I Q ) N x 1 + w : x Ω ¯ } and { c ( K P ( I Q ) N x ) : x Ω ¯ }

are equicontinuous on [0,T].

For any x Ω ¯ , by L’Hôspital’s rule,

lim t ( K P ( I Q ) N x ) ( t ) 1 + w ( t ) = lim t 0 t ( ( I Q ) N x ) ( s ) d s = 0 ( ( I Q ) N x ) ( s ) d s lim t t ( ( I Q ) N x ) ( s ) d s

and

| ( c ( K P ( I Q ) N x ) ) (t) 0 ( ( I Q ) N x ) (τ)dτ| t | ( ( I Q ) N x ) (τ)|dτ.

Since |(IQ)Nx| N r + Q r for all x Ω ¯ ,

( K P ( I Q ) N x ) ( t ) 1 + w ( t ) 0 ( ( I Q ) N x ) (s)ds

and

( c ( K P ( I Q ) N x ) ) (t) 0 ( ( I Q ) N x ) (τ)dτ

uniformly on Ω ¯ as t. In view of Theorem 2.2, K P (IQ)N( Ω ¯ ) is a relatively compact set in X, and thus N is L-compact on Ω ¯ . □

The following theorem is the main result in this paper.

Theorem 3.4

Let 1 c Y, and assume that (H1)-(H3) hold. Assume also that the following hold:

(H4) there exist positive constants A and B such that if|x(t)|>Afor everyt[0,B]or|(c x )(t)|>Afor everyt[0,), thenQNx0;

(H5) there exists a positive constant C such that if|a|>Cor|b|>C, thenQN(a+bw)0, and only one of the following inequalities is satisfied:

( i ) a Q 1 ( N ( a + b w ) ) + b Q 2 ( N ( a + b w ) ) 0 , ( ii ) a Q 1 ( N ( a + b w ) ) + b Q 2 ( N ( a + b w ) ) 0 .

If α and β satisfy

( 1 + w ) α Y + β c Y < 1 4 + w ( B ) ,
(5)

then problem (1) has at least one solution in X.

Proof

We divide the proof into four steps.

Step 1. Let

Ω 1 = { x dom L ker L : L x = λ N x  for some  λ ( 0 , 1 ) } .

Then Ω 1 is bounded. In fact, x Ω 1 means λ(0,1) and NxImL. Thus, by Lemma 3.2, Q 1 (Nx)= Q 2 (Nx)=0. By (H4), there exist t 0 [0,B] and t 1 [0,) such that |x( t 0 )|A, |(c x )( t 1 )|A. Then

| ( c x ) (0)|=| ( c x ) ( t 1 ) 0 t 1 ( c x ) (s)ds|A+ N x Y .

On the other hand,

| x ( 0 ) | = | x ( t 0 ) 0 t 0 1 c ( s ) [ ( c x ) ( t 1 ) + t 1 s ( c x ) ( τ ) d τ ] d s | | x ( t 0 ) | + ( | ( c x ) ( t 1 ) | + N x Y ) w ( B ) A ( 1 + w ( B ) ) + w ( B ) N x Y .

Thus,

P x X |x(0)|+2| ( c x ) (0)|A ( 3 + w ( B ) ) + ( 2 + w ( B ) ) N x Y .

Since kerL=ImP, LPx=0 for all xX, it follows from (3) and (H2) that

x X = P x + ( I P ) x X = P x + K P L ( I P ) x X P x X + 2 L x Y P x X + 2 N x Y A ( 3 + w ( B ) ) + ( 4 + w ( B ) ) N x Y A ( 3 + w ( B ) ) + ( 4 + w ( B ) ) ( ( ( 1 + w ) α Y + β c Y ) x X + γ Y ) ,

and, by (5),

x X A ( 3 + w ( B ) ) + ( 4 + w ( B ) ) γ Y 1 ( 4 + w ( B ) ) ( ( 1 + w ) α Y + β c Y ) ,

which implies that Ω 1 is bounded.

Step 2. Set

Ω 2 ={xkerL:NxImL}.

Then Ω 2 is bounded. In fact, x Ω 2 implies x=a+bw and Q 1 (Nx)= Q 2 (Nx)=0. By (H5), we obtain |a|C and |b|C. Thus Ω 2 is bounded.

Step 3. Define an isomorphism J:kerLImQ by

J ( a + b w ( ) ) = 1 ( a 11 a + a 12 b + ( a 21 a + a 22 b ) w ( ) ) e k ( ) .

Assume first that (i) in (H5) holds. Let

Ω 3 = { x ker L : λ J x + ( 1 λ ) Q N x = 0  for some  λ [ 0 , 1 ] } .

Then Ω 3 is bounded. Indeed, x Ω 3 means that there exist constants a,b(,) and λ[0,1] such that x=a+bw and λJx+(1λ)QNx=0. If λ=0, then QNx=0. It follows from 0 that Q 1 (Nx)= Q 2 (Nx)=0. By (H5), we obtain |a|C and |b|C. If λ=1, clearly a=b=0. For λ(0,1), by the facts that λJx=(1λ)QNx and 0, it follows that λa=(1λ) Q 1 (Nx) and λb=(1λ) Q 2 (Nx). If |a|>C or |b|>C, by (i) in (H5),

λ ( a 2 + b 2 ) =(1λ) ( a Q 1 N ( a + b w ) + b Q 2 N ( a + b w ) ) 0,

which is a contradiction. Thus Ω 3 is bounded in X. On the other hand, if (ii) in (H5) holds, taking

Ω 3 = { x ker L : λ J x + ( 1 λ ) Q N x = 0  for some  λ [ 0 , 1 ] } ,

one can show that Ω 3 is bounded in a similar manner.

Step 4. Take an open bounded set Ω i = 1 3 Ω ¯ i {0}. By Step 1 and Step 2, in view of Theorem 2.1, we only need to prove that deg(QN | ker L ,ΩkerL,0)0 in order to show that Lx=Nx has at least one solution in domL Ω ¯ . Let

H(x,λ)=±λJx+(1λ)QNx.

By Step 3, H(x,λ)0 for all (x,λ)(kerLΩ)×[0,1]. Thus, by the homotopy invariance property of degree,

deg ( Q N | ker L , Ω ker L , 0 ) = deg ( H ( , 0 ) , Ω ker L , 0 ) = deg ( H ( , 1 ) , Ω ker L , 0 ) = deg ( ± J , Ω ker L , 0 ) 0 .

By Theorem 2.1, Lx=Nx has at least one solution in domL Ω ¯ , i.e., problem (1) has at least one solution in X. □

In the case that 1 c Y, w L (0,), and using the norm on X in Remark 3.1(2), we have a similar result to Theorem 3.4. We omit the proof.

Theorem 3.5

Let 1 c Y, and assume that (H1)-(H5) hold. If α and β satisfy

α Y + β c Y < 1 2 + 3 w ,

then problem (1) has at least one solution in X.

4 Example

Consider the following second-order nonlinear differential equation:

{ ( c u ) ( t ) = f ( t , u ( t ) , u ( t ) ) , a.e.  t ( 0 , ) , ( c u ) ( 0 ) = 0 g ( s ) ( c u ) ( s ) d s , lim t ( c u ) ( t ) = ( c u ) ( 1 ) ,
(6)

where f(t,u,v)= α 1 (t)u+ β 1 (t)v+ γ 1 (t) for (t,u,v)[0,)×(,)×(,), γ 1 Y satisfies 0 1 (1 t 2 ) γ 1 (t)dt= 1 γ 1 (t)dt=0,

α 1 ( t ) = { K e t , t [ 0 , 1 ] , 0 , t ( 1 , ) , β 1 ( t ) = { 0 , t [ 0 , 1 ] , K ( e 4 ) e t , t ( 1 , ) , c ( t ) = { t ρ , t [ 0 , 1 ] , 1 , t ( 1 , ) , g ( t ) = { 2 t , t [ 0 , 1 ] , 0 , t ( 1 , ) , w ( t ) = { 1 1 ρ t 1 ρ , t [ 0 , 1 ] , t + ρ 1 ρ , t ( 1 , ) .

We also assume that the constants K and ρ satisfy

0<K< K ρ := 1 ρ ( 5 4 ρ ) ( 3 e 1 + 1 1 ρ 0 1 t 1 ρ e t d t )

and ρ[0,1). Then 1/cY, and (H1)-(H2) hold for m= α 1 = ξ 1 =1, α= α 1 , β=| β 1 |, and γ=| γ 1 |. For yY,

Q 1 (y)= 1 y(s)dsand Q 2 (y)= 0 1 ( 1 s 2 ) y(s)ds.

Taking k(t)=t, then

a 11 = 1 1 ρ 0 1 ( 1 t 2 ) t 1 ρ e t d t , a 12 = ρ 2 1 ρ e 1 , a 21 = 1 4 e 1 , a 22 = e 1

and

= a 11 a 22 a 12 a 21 = 1 ( 1 ρ ) e 0 1 ( 1 t 2 ) t 1 ρ e t d t + 2 ρ 1 ρ e 1 ( 1 4 e 1 ) = e 4 e 2 + 1 ( 1 ρ ) e ( e 4 e + 0 1 ( 1 t 2 ) t 1 ρ e t d t ) .

Since, for any ρ[0,1),

e 4 e + 0 1 ( 1 t 2 ) t 1 ρ e t dt e 4 e + 0 1 ( 1 t 2 ) e t dt=0,

and <0. Thus (H3) holds for any ρ[0,1).

Take B=1 and let A>0 be arbitrary. Then (H4) holds. In fact, if |x(t)|>A for t[0,1], then

Q 2 (Nx)=K 0 1 ( 1 t 2 ) e t x(t)dt0.

If |(c x )(t)|>A for t[0,), then | x (t)|>A for t[1,), and

Q 1 (Nx)=K(e4) 1 e t x (t)dt0.

Since

Q 1 ( N ( a + b w ) ) =K ( 1 4 e 1 ) b

and

Q 2 ( N ( a + b w ) ) =K ( ( 4 e 1 1 ) a + b 1 ρ 0 1 ( 1 t 2 ) e t t 1 ρ d t )

for (a,b)(0,0),

| Q 1 ( N ( a + b w ) ) |+| Q 2 ( N ( a + b w ) ) |0,

i.e., QN(a+bw)0 for (a,b)(0,0), and

a Q 1 ( N ( a + b w ) ) +b Q 2 ( N ( a + b w ) ) = b 2 K 1 ρ 0 1 ( 1 t 2 ) e t t 1 ρ dt0

for all (a,b)(,)×(,). Therefore (H5) holds for any C>0. Since 0<K< K ρ , (5) is satisfied, and thus there exists at least one solution to problem (6) in view of Theorem 3.4.

References

  1. 1.

    Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.

    Book  Google Scholar 

  2. 2.

    Agarwal RP, O’Regan D: Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory. Stud. Appl. Math. 2003, 111(3):339-358. 10.1111/1467-9590.t01-1-00237

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cupini G, Marcelli C, Papalini F: On the solvability of a boundary value problem on the real line. Bound. Value Probl. 2011., 2011: 10.1186/1687-2770-2011-26

    Google Scholar 

  4. 4.

    Jiang W, Wang B, Wang Z:Solvability of a second-order multi-point boundary-value problems at resonance on a half-line with dimkerL=2. Electron. J. Differ. Equ. 2011., 2011:

    Google Scholar 

  5. 5.

    Kim C-G: Solvability of multi-point boundary value problems on the half-line. J. Nonlinear Sci. Appl. 2012, 5(1):27-33. (Special issue)

    MathSciNet  Google Scholar 

  6. 6.

    Kim C-G: Existence and iteration of positive solutions for multi-point boundary value problems on a half-line. Comput. Math. Appl. 2011, 61(7):1898-1905. 10.1016/j.camwa.2011.02.023

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68(8):2158-2171. 10.1016/j.na.2007.01.038

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kosmatov N: Second order boundary value problems on an unbounded domain. Nonlinear Anal. 2008, 68(4):875-882. 10.1016/j.na.2006.11.043

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lian H, Ge W: Solvability for second-order three-point boundary value problems on a half-line. Appl. Math. Lett. 2006, 19(10):1000-1006. 10.1016/j.aml.2005.10.018

    MathSciNet  Article  Google Scholar 

  10. 10.

    Liang S, Zhang J: Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. Acta Appl. Math. 2010, 111(1):27-43. 10.1007/s10440-009-9528-z

    MathSciNet  Article  Google Scholar 

  11. 11.

    Liu L, Wang Y, Hao X, Wu Y: Positive solutions for nonlinear singular differential systems involving parameter on the half-line. Abstr. Appl. Anal. 2012., 2012:

    Google Scholar 

  12. 12.

    Ma R: Existence of positive solutions for second-order boundary value problems on infinity intervals. Appl. Math. Lett. 2003, 16(1):33-39. 10.1016/S0893-9659(02)00141-6

    MathSciNet  Article  Google Scholar 

  13. 13.

    Marcelli C: The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-252

    Google Scholar 

  14. 14.

    O’Regan D, Yan B, Agarwal RP: Nonlinear boundary value problems on semi-infinite intervals using weighted spaces: an upper and lower solution approach. Positivity 2007, 11(1):171-189. 10.1007/s11117-006-0050-5

    MathSciNet  Article  Google Scholar 

  15. 15.

    O’Regan D, Yan B, Agarwal RP: Solutions in weighted spaces of singular boundary value problems on the half-line. J. Comput. Appl. Math. 2007, 205(2):751-763. 10.1016/j.cam.2006.02.055

    MathSciNet  Article  Google Scholar 

  16. 16.

    Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.

    Google Scholar 

  17. 17.

    Cui Y: Solvability of second-order boundary-value problems at resonance involving integral conditions. Electron. J. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-45

    Google Scholar 

  18. 18.

    Feng W, Webb JRL: Solvability of three point boundary value problems at resonance. Proceedings of the Second World Congress of Nonlinear Analysts: Part 6 1997, 3227-3238. (Athens, 1996) doi: 10.1016/S0362-546X(96)00118-6

    Google Scholar 

  19. 19.

    Franco D, Infante G, Zima M: Second order nonlocal boundary value problems at resonance. Math. Nachr. 2011, 284(7):875-884. 10.1002/mana.200810841

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wang F, Zhang F: Existence of n positive solutions to second-order multi-point boundary value problem at resonance. Bull. Korean Math. Soc. 2012, 49(4):815-827. 10.4134/BKMS.2012.49.4.815

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zima M, Drygaś P: Existence of positive solutions for a kind of periodic boundary value problem at resonance. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-19

    Google Scholar 

  22. 22.

    Du Z, Meng F: Solutions to a second-order multi-point boundary value problem at resonance. Acta Math. Sci. 2010, 30(5):1567-1576. 10.1016/S0252-9602(10)60150-6

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kosmatov N: A multi-point boundary value problem with two critical conditions. Nonlinear Anal. 2006, 65(3):622-633. 10.1016/j.na.2005.09.042

    MathSciNet  Article  Google Scholar 

  24. 24.

    Meng F, Du Z: Solvability of a second-order multi-point boundary value problem at resonance. Appl. Math. Comput. 2009, 208(1):23-30. 10.1016/j.amc.2008.11.026

    MathSciNet  Article  Google Scholar 

  25. 25.

    Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 2009, 353(1):311-319. 10.1016/j.jmaa.2008.11.082

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This study was supported by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2014, Project No. RIBS-PNU-2014-101.

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Correspondence to Eun Kyoung Lee.

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Jeong, J., Kim, CG. & Lee, E.K. Solvability for nonlocal boundary value problems on a half line with dim(kerL)=2. Bound Value Probl 2014, 167 (2014). https://doi.org/10.1186/s13661-014-0167-6

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Keywords

  • resonance
  • nonlocal boundary condition
  • semilinear
  • a half line