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An order-type existence theorem and applications to periodic problems

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Abstract

Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established. As applications, we present sufficient existence conditions for the first- and second-order periodic problems.

MSC:34B15.

1 Introduction and preliminaries

Let X, Y be real Banach spaces. Consider a linear mapping L:domLXY and a nonlinear operator N:XY. Here we assume that L is a Fredholm operator of index zero, that is, ImL is closed and dim KerL=codim ImL<. Then the solvability of the operator equation

Lx=Nx

has been studied by many researchers in the literature; see [18] and the references therein. In [1], Cremins established a fixed point index for A-proper semilinear operators defined on cones which includes and improves the results in [5, 8, 9]. Using the fixed point index and the concept of a quasi-normal cone introduced in [10], Cremins established a norm-type existence theorem concerning cone expansion and compression in [11], which generalizes some corresponding results contained in [12].

In this paper, we will use the properties of the fixed point index in [1] and partial order to present a new order-type existence theorem concerning cone expansion and compression which extends the corresponding results in [12]. We recall that a partial order in X induced by a cone KX is defined by

xyyxK.

As applications, we study the first- and second-order periodic boundary problems and obtain new existence results. During the last few decades, periodic boundary value problems have been studied by many researchers in the literature; see, for example, [1319] and the references therein. Our new results improve those contained in [13, 18].

Next we recall some notations and results which will be needed in this paper. Let X and Y be Banach spaces, D be a linear subspace of X, { X n }D and { Y n }Y be the sequences of oriented finite dimensional subspaces such that Q n yy in Y for every y and dist(x, X n )0 for every xD, where Q n :Y Y n and P n :X X n are sequences of continuous linear projections. The projection scheme Γ={ X n , Y n , P n , Q n } is then said to be admissible for maps from DX to Y. A map T:DXY is called approximation-proper (abbreviated A-proper) at a point yY with respect to an admissible scheme Γ if T n Q n T | D X n is continuous for each nN and whenever { x n j : x n j D X n j } is bounded with T n j x n j y, then there exists a subsequence { x n j k } such that x n j k xD and Tx=y. T is simply called A-proper if it is A-proper at all points of Y. L:domLXY is a Fredholm operator of index zero if ImL is closed and dim  KerL=codim  ImL<. As a consequence of this property, X and Y may be expressed as direct sums; X= X 0 X 1 , Y= Y 0 Y 1 with continuous linear projections P:XKerL= X 0 and Q:Y Y 0 . The restriction of L to domL X 1 , denoted L 1 , is a bijection onto ImL= Y 1 with continuous inverse L 1 1 : Y 1 domL X 1 . Since X 0 and Y 0 have the same finite dimension, there exists a continuous bijection J: Y 0 X 0 . Let H=L+ J 1 P, then H:domLXY is a linear bijection with bounded inverse. Let K be a cone in a Banach space X. Then K 1 =H(KdomL) is a cone in Y. In [20], Petryshyn has shown that an admissible scheme Γ L can be constructed such that L is A-proper with respect to Γ L . The following properties of the fixed point index ind K and two lemmas can be found in [1].

Proposition 1.1 Let ΩX be open and bounded and Ω K =ΩK. Assume that Q n K 1 K 1 , P+JQN+ L 1 1 (IQ)N maps K to K, and LxNx on Ω K .

(P1) (Existence property) If ind K ([L,N],Ω){0}, then there exists x Ω K such that Lx=Nx.

(P2) (Normality) If x 0 Ω K , then ind K ([L, J 1 P+ y ˆ 0 ],Ω)={1}, where y ˆ 0 =H x 0 and y ˆ 0 (y)= y 0 for every yH Ω K .

(P3) (Additivity) If LxNx for x Ω ¯ K ( Ω 1 Ω 2 ), where Ω 1 and Ω 2 are disjoint relatively open subsets of Ω K , then

ind K ( [ L , N ] , Ω ) ind K ( [ L , N ] , Ω 1 ) + ind K ( [ L , N ] , Ω 2 )

with equality if either of indices on the right is a singleton.

(P4) (Homotopy invariance) If LN(λ,x) is an A-proper homotopy on Ω K for λ[0,1] and (N(λ,x)+ J 1 P) H 1 : K 1 K 1 and θ(LN(λ,x))(domL Ω K ) for λ[0,1], then ind K ([L,N(λ,x)],Ω)= ind K 1 ( T λ ,U) is independent of λ[0,1], where T λ =(N(λ,x)+ J 1 P) H 1 .

Lemma 1.1 If L:domLY is Fredholm of index zero, Ω is an open bounded set and Ω K domL, θΩX. Let LλN be A-proper for λ[0,1]. Assume that N is bounded and P+JQN+ L 1 1 (IQ)N maps K to K. If LxμNx(1μ) J 1 Px on Ω K for μ[0,1], then

ind K ( [ L , N ] , Ω ) ={1}.

Lemma 1.2 If L:domLY is Fredholm of index zero, Ω is an open bounded set and Ω K domL. Let LλN be A-proper for λ[0,1]. Assume that N is bounded and P+JQN+ L 1 1 (IQ)N maps K to K. If there exists e K 1 {θ} such that

LxNxμe,

for every x Ω K and all μ0, then

ind K ( [ L , N ] , Ω ) ={0}.

2 An abstract result

We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows.

Theorem 2.1 If L:domLY is Fredholm of index zero, let LλN be A-proper for λ[0,1]. Assume that N is bounded and P+JQN+ L 1 1 (IQ)N maps K to K. Suppose further that Ω 1 and Ω 2 are two bounded open sets in X such that θ Ω 1 Ω ¯ 1 Ω 2 , Ω 1 KdomL and Ω 2 KdomL. If one of the following two conditions is satisfied:

(C1) (P+JQN)x+ L 1 1 (IQ)Nxx for all x Ω 1 K and (P+JQN)x+ L 1 1 (IQ)Nxx for all x Ω 2 K;

(C2) (P+JQN)x+ L 1 1 (IQ)Nxx for all x Ω 1 K and (P+JQN)x+ L 1 1 (IQ)Nxx for all x Ω 2 K.

Then there exists x( Ω ¯ 2 Ω 1 )K such that Lx=Nx.

Proof We assume that (C1) is satisfied. First we show that

LxμNx(1μ) J 1 Px,for any x Ω 1 K,μ[0,1].
(2.1)

In fact, otherwise, there exist x 1 Ω 1 K and μ 1 [0,1] such that

L x 1 = μ 1 N x 1 (1 μ 1 ) J 1 P x 1 ,

then we obtain

( L + J 1 P ) x 1 = μ 1 ( N + J 1 P ) x 1 .

Therefore,

x 1 = μ 1 ( L + J 1 P ) 1 ( N + J 1 P ) x 1 = μ 1 [ ( P + J Q N ) x 1 + L 1 1 ( I Q ) N x 1 ] ( P + J Q N ) x 1 + L 1 1 ( I Q ) N x 1 ,

which contradicts condition (C1). From (2.1) and Lemma 1.1, we have

ind K ( [ L , N ] , Ω 1 ) ={1}.
(2.2)

Choosing an arbitrary e K 1 {θ}, next we prove that

LxNxμe.
(2.3)

In fact, otherwise, there exist x 2 Ω 2 K and μ 2 0 such that

L x 2 N x 2 = μ 2 e,

then we obtain

( L + J 1 P ) x 2 = ( N + J 1 P ) x 2 + μ 2 e 1 ( N + J 1 P ) x 2 ,

in which the partial order is induced by the cone K 1 in Y. So,

x 2 ( L + J 1 P ) 1 ( N + J 1 P ) x 2 =(P+JQN) x 2 + L 1 1 (IQ)N x 2 ,

which is a contradiction to condition (C1). Hence (2.3) holds, and then by Lemma 1.2, we have

ind K ( [ L , N ] , Ω 2 ) ={0}.
(2.4)

It follows therefore from (2.2), (2.4) and the additivity property (P3) of Proposition 1.1 that

ind K ( [ L , N ] , Ω 2 Ω 1 ) = ind K ( [ L , N ] , Ω 2 ) ind K ( [ L , N ] , Ω 1 ) = { 0 } { 1 } = { 1 } .
(2.5)

Since the index is nonzero, the existence property (P1) of Proposition 1.1 implies that there exists x( Ω ¯ 2 Ω 1 )K such that Lx=Nx.

Similarly, when (C2) is satisfied, instead of (2.2), (2.4) and (2.5), we have

ind K ( [ L , N ] , Ω 1 ) ={0}, ind K ( [ L , N ] , Ω 2 ) ={1},

and therefore

ind K ( [ L , N ] , Ω 2 Ω 1 ) ={1}.

Also, we can assert that there exists x( Ω ¯ 2 Ω 1 )K such that Lx=Nx. □

3 Applications

3.1 First-order periodic boundary value problems

We consider the following first-order periodic boundary value problem:

{ x ( t ) = f ( t , x ( t ) ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) ,
(3.1)

where f:[0,1]×[0,+)R is continuous and f(0,x)=f(1,x) for all xR.

Consider the Banach spaces X=Y=C[0,1] endowed with the norm x= max t [ 0 , 1 ] |x(t)|. Define the cone K in X by

K= { x X : x ( t ) 0 , t [ 0 , 1 ] } .

Let L be the linear operator from domLX to Y with

domL= { x X : x C [ 0 , 1 ] , x ( 0 ) = x ( 1 ) } ,

and

Lx(t)= x (t),xdomL,t[0,1].

Let us define N:XY by

Nx(t)=f ( t , x ( t ) ) ,t[0,1].

Then (3.1) is equivalent to the equation

Lx=Nx.

It is obvious that L is a Fredholm operator of index zero with

Next we define the projections P:XX, Q:YY by

and the isomorphism J:ImQImP as Jy=y. Note that for yImL, the inverse operator

L 1 1 :ImLdomLKerP

of

L | dom L Ker P :domLKerPImL

is given by

( L 1 1 y ) (t)= 0 1 K(t,s)y(s)ds,

where

K(t,s)={ s + 1 , 0 s < t 1 , s , 0 t s 1 .

Set

G(t,s)=1+K(t,s) 0 1 K(t,s)ds.

We can verify that

G(t,s)={ 3 2 ( t s ) , 0 s < t 1 , 1 2 + ( s t ) , 0 t s 1 ,

and

1 2 G(t,s) 3 2 ,t,s[0,1].

To state the existence result, we introduce two conditions:

(H1) f(t,b)<0 for all t[0,1],

(H2) f(t,x)>0 for all (t,x)[0,1]×[0,a].

Theorem 3.1 Assume that there exist two positive numbers 0<a<b such that (H1), (H2) and

(H3) f(t,x) 2 3 x for all (t,x)[0,1]×[0,b]

hold. Then (3.1) has at least one positive periodic solution x K with a x b.

Proof First, we note that L, as defined, is Fredholm of index zero, L 1 1 is compact by the Arzela-Ascoli theorem and thus LλN is A-proper for λ[0,1] by [[20], Lemma 2(a)].

For each xK, then by condition (H3),

Thus (P+JQN+ L 1 1 (IQ)N)(K)K.

Let

Ω 1 = { x X : x < a } , Ω 2 = { x X : x < b } .

Clearly, Ω 1 and Ω 2 are bounded open sets and

θ Ω 1 Ω ¯ 1 Ω 2 .

We now show that

(P+JQN)x+ L 1 1 (IQ)Nxxfor any x Ω 2 K.
(3.2)

In fact, if there exists x 3 Ω 2 K such that

(P+JQN) x 3 + L 1 1 (IQ)N x 3 x 3 .

Then

x 3 (t)f ( t , x 3 ( t ) ) ,t[0,1].

Let t 1 [0,1] be such that x 3 ( t 1 )=b. Clearly, the function x 3 2 attains a maximum on [0,1] at t= t 1 . Therefore 2 x 3 ( t 1 ) x 3 ( t 1 )=0. As a consequence,

0=2b x 3 ( t 1 )2bf ( t 1 , x 3 ( t 1 ) ) =2bf( t 1 ,b),

which is a contradiction to (H1). Therefore (3.2) holds.

On the other hand, we claim that

(P+JQN)x+ L 1 1 (IQ)Nxxfor any x Ω 1 K.
(3.3)

In fact, if not, there exists x 4 Ω 1 K such that

(P+JQN) x 4 + L 1 1 (IQ)N x 4 x 4 .

For any x 4 Ω 1 K, we have x 4 =a, then 0 x 4 (t)a for t[0,1]. By condition (H2), we have

x 4 ( t ) ( P + J Q N ) x 4 ( t ) + L 1 1 ( I Q ) N x 4 ( t ) = 0 1 x 4 ( s ) d s + 0 1 G ( t , s ) f ( s , x 4 ( s ) ) d s > 0 1 x 4 ( s ) d s , for any  t [ 0 , 1 ] ,

which is a contradiction. As a result, (3.3) is verified.

It follows from (3.2), (3.3) and Theorem 2.1 that there exists x K( Ω ¯ 2 Ω 1 ) such that L x =N x with a x b. □

Remark 3.1 In [18], the following condition is required instead of (H2):

(H) there exist a(0,b), t 0 [0,1], r(0,1], and continuous functions g:[0,1][0,), h:(0,a][0,) such that f(t,x)g(t)h(x) for all t[0,1] and x(0,a], h(x)/ x r is nonincreasing on (0,a] with

h ( a ) 2 r 1 0 1 G( t 0 ,s)g(s)dsa.

Obviously, our condition (H2) is much weaker and less strict compared with (H). Moreover, (H2) is easier to check than (H). So, our result generalizes and improves [[18], Theorem 5].

Remark 3.2 From the proof of Theorem 3.1, we can see that condition (H2) can be replaced by one of the following two relatively weaker conditions:

( H 2 ) f(t,x)0 for all (t,x)[0,1]×[0,a] and f(t,) is positive for almost everywhere on [0,a].

( H 2 ) lim x 0 + min t [ 0 , 1 ] f(t,x)>0.

Remark 3.3 Finally in this section, we note that conditions (H1) and (H2) can be replaced by the following asymptotic conditions:

( H 1 ) lim x + f ( t , x ) x <0 uniformly for t;

( H 2 ) lim x 0 + f ( t , x ) x >0 uniformly for t.

Example 3.1 Let the nonlinearity in (3.1) be

f(t,x)=c(t) x α +μd(t) x β kx,

where 0<α<1<β, c(t),d(t)C[0,1] are positive 1-periodic functions, k(0,2/3) and μ>0 is a positive parameter. Then (3.1) has at least one positive 1-periodic solution for each 0<μ< μ , here μ is some positive constant.

Proof We will apply Theorem 3.1 with f(t,x)=c(t) x α +μd(t) x β kx. Since k(0,2/3), it is easy to see that (H3) holds. Set

T(x)= k x c x α d x β ,

where

c = max t c(t), d = max t d(t).

Since 0<α<1<β, we have

T ( 0 + ) =,T(+)=0.

One may easily see that there exists b>0 such that

T(b)= k b c b α d b β = sup x > 0 T(x)>0.

Let

μ = k b c b α d b β .

Then, for each μ(0, μ ), we have

f ( t , b ) = c ( t ) b α + μ d ( t ) b β k b < c b α + μ d b β k b = 0 ,

which implies that (H1) holds.

On the other hand, we have

which implies that ( H 2 ) holds. Now we have the desired result. □

3.2 Second-order periodic boundary value problems

Let f:[0,1]×[0,+)R be continuous and f(0,x)=f(1,x) for all xR. We will discuss the existence of positive solutions of the second-order periodic boundary value problem

{ x ( t ) = f ( t , x ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) .
(3.4)

Since some parts of the proof are in the same line as that of Theorem 3.1, we will outline the proof with the emphasis on the difference.

Let X, Y be Banach spaces and the cone K be as in Section 3.1. In this case, we may define

domL= { x X : x C [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) } ,

and let the linear operator L:domLY be defined by

Lx= x ,for xdomL.

Then L is Fredholm of index zero,

KerL= { x dom L : x ( t ) constants } ,

and

ImL= { y Y : 0 1 y ( s ) d s = 0 } .

Define N:XY by

Nx(t)=f ( t , x ( t ) ) .

Thus it is clear that (3.4) is equivalent to

Lx=Nx.

We use the same projections P, Q as in Section 3.1 and define the isomorphism J:ImQImP as

Jy=βy,

where β= 1 6 . It is easy to verify that the inverse operator L 1 1 :ImLdomLKerP of L | dom L Ker P :domLKerPImL is

( L 1 1 y ) (t)= 0 1 Λ(t,s)y(s)ds,

where

Λ(t,s)={ s 2 ( 1 2 t + s ) , 0 s < t 1 , 1 2 ( 1 s ) ( 2 t s ) , 0 t s 1 .

Set

H(t,s)= 1 6 +Λ(t,s) 0 1 Λ(t,s)ds.

We can verify that

H(t,s)={ 1 4 + s 2 ( 1 2 t + s ) + t 2 2 t 2 , 0 s < t 1 , 1 4 + 1 2 ( 1 s ) ( 2 t s ) + t 2 2 + t 2 , 0 t s 1 ,

and

1 8 H(t,s) 1 4 ,t,s[0,1].

Theorem 3.2 Assume that there exist two positive numbers 0<a<b such that (H1), (H2) and

(H4) f(t,x)4x for all (t,x)[0,1]×[0,b]

hold. Then (3.4) has at least one positive periodic solution x K with a x b.

Proof It is again easy to show that LλN is A-proper for λ[0,1] by [[20], Lemma 2(a)].

For each xK, then by condition (H4),

Thus (P+JQN+ L 1 1 (IQ)N)(K)K.

Let

Ω 3 = { x X : x < a } , Ω 4 = { x X : x < b } .

Clearly, Ω 3 and Ω 4 are bounded and open sets and

θ Ω 3 Ω ¯ 3 Ω 4 .

Next, we show that

(P+JQN)x+ L 1 1 (IQ)Nxx,for any x Ω 4 K.
(3.5)

On the contrary, suppose that there exists x 5 Ω 4 K such that

(P+JQN) x 5 + L 1 1 (IQ)N x 5 x 5 .

Then

x 5 (t)f ( t , x 5 ( t ) ) ,t[0,1].

Let t 2 [0,1] such that x 5 ( t 2 )= max t [ 0 , 1 ] x 5 (t)=b. Using the boundary conditions, we have t 2 (0,1). In this case, x 5 ( t 2 )=0, x 5 ( t 2 )0. This gives

0 x 5 ( t 2 )f ( t 2 , x 5 ( t 2 ) ) =f( t 2 ,b),

which is a contradiction to condition (H1). Therefore (3.5) holds.

Finally, similar to the proof of (3.3), it follows from condition (H2) that

(P+JQN)x+ L 1 1 (IQ)Nxx,for any x Ω 3 K.

Consequently all conditions of Theorem 2.1 are satisfied. Therefore, there exists x K( Ω ¯ 4 Ω 3 ) such that L x =N x with x K and a x b and the assertion follows. □

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Acknowledgements

JC was supported by the National Natural Science Foundation of China (Grant No. 11171090, No. 11271333 and No. 11271078), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0325), China Postdoctoral Science Foundation funded project (Grant No. 2012T50431). FW was supported by the National Natural Science Foundation of China (Grant No. 10971179) and the Natural Science Foundation of Changzhou University (Grant No. JS201008).

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Keywords

  • fixed point index
  • order-type existence theorem
  • cone expansion and compression
  • positive solutions
  • periodic boundary value problems