An order-type existence theorem and applications to periodic problems

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.

Let X, Y be real Banach spaces. Consider a linear mapping and a nonlinear operator . Here we assume that L is a Fredholm operator of index zero, that is, ImL is closed and . Then the solvability of the operator equation

has been studied by many researchers in the literature; see [1-8] 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 is defined by

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, [13-19] 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, and be the sequences of oriented finite dimensional subspaces such that in Y for every y and dist for every , where and are sequences of continuous linear projections. The projection scheme is then said to be admissible for maps from to Y. A map is called approximation-proper (abbreviated A-proper) at a point with respect to an admissible scheme Γ if is continuous for each and whenever is bounded with , then there exists a subsequence such that and . T is simply called A-proper if it is A-proper at all points of Y. is a Fredholm operator of index zero if ImL is closed and . As a consequence of this property, X and Y may be expressed as direct sums; , with continuous linear projections and . The restriction of L to , denoted , is a bijection onto with continuous inverse . Since and have the same finite dimension, there exists a continuous bijection . Let , then is a linear bijection with bounded inverse. Let K be a cone in a Banach space X. Then is a cone in Y. In [20], Petryshyn has shown that an admissible scheme can be constructed such that L is A-proper with respect to . The following properties of the fixed point index and two lemmas can be found in [1].

Proposition 1.1Letbe open and bounded and. Assume that, mapsKtoK, andon.

(P₁) (Existence property) If, then there existssuch that.

(P₂) (Normality) If, then, whereandfor every.

(P₃) (Additivity) Iffor, whereandare disjoint relatively open subsets of, then

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

(P₄) (Homotopy invariance) Ifis an A-proper homotopy onforandandfor, thenis independent of, where.

Lemma 1.1Ifis Fredholm of index zero, Ω is an open bounded set and, . Letbe A-proper for. Assume thatNis bounded andmapsKtoK. Ifonfor, then

Lemma 1.2Ifis Fredholm of index zero, Ω is an open bounded set and. Letbe A-proper for. Assume thatNis bounded andmapsKtoK. If there existssuch that

for everyand all, then

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

Theorem 2.1Ifis Fredholm of index zero, letbe A-proper for. Assume thatNis bounded andmapsKtoK. Suppose further thatandare two bounded open sets inXsuch that, and. If one of the following two conditions is satisfied:

(C₁) for allandfor all;

(C₂) for allandfor all.

Then there existssuch that.

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

In fact, otherwise, there exist and such that

then we obtain

Therefore,

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

Choosing an arbitrary , next we prove that

In fact, otherwise, there exist and such that

then we obtain

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

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

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

Since the index is nonzero, the existence property (P₁) of Proposition 1.1 implies that there exists such that .

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

and therefore

Also, we can assert that there exists such that . □

3.1 First-order periodic boundary value problems

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

where is continuous and for all .

Consider the Banach spaces endowed with the norm . Define the cone K in X by

Let L be the linear operator from to Y with

and

Let us define by

Then (3.1) is equivalent to the equation

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

Next we define the projections , by

and the isomorphism as . Note that for , the inverse operator

of

is given by

where

Set

We can verify that

and

To state the existence result, we introduce two conditions:

(H₁) for all ,

(H₂) for all .

Theorem 3.1Assume that there exist two positive numberssuch that (H₁), (H₂) and

(H₃) for all

hold. Then (3.1) has at least one positive periodic solutionwith.

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

For each , then by condition (H₃),

Thus .

Let

Clearly, and are bounded open sets and

We now show that

In fact, if there exists such that

Then

Let be such that . Clearly, the function attains a maximum on at . Therefore . As a consequence,

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

On the other hand, we claim that

In fact, if not, there exists such that

For any , we have , then for . By condition (H₂), we have

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 such that with . □

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

(H^∗) there exist , , , and continuous functions , such that for all and , is nonincreasing on with

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:

() for all and is positive for almost everywhere on .

() .

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

() uniformly for t;

() uniformly for t.

Example 3.1 Let the nonlinearity in (3.1) be

where , are positive 1-periodic functions, and is a positive parameter. Then (3.1) has at least one positive 1-periodic solution for each , here is some positive constant.

Proof We will apply Theorem 3.1 with . Since , it is easy to see that (H₃) holds. Set

where

Since , we have

One may easily see that there exists such that

Let

Then, for each , we have

which implies that (H₁) holds.

On the other hand, we have

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

3.2 Second-order periodic boundary value problems

Let be continuous and for all . We will discuss the existence of positive solutions of the second-order periodic boundary value problem

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

and let the linear operator be defined by

Then L is Fredholm of index zero,

and

Define by

Thus it is clear that (3.4) is equivalent to

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

where . It is easy to verify that the inverse operator of is

where

Set

We can verify that

and

Theorem 3.2Assume that there exist two positive numberssuch that (H₁), (H₂) and

(H₄) for all

hold. Then (3.4) has at least one positive periodic solutionwith.

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

For each , then by condition (H₄),

Thus .

Let

Clearly, and are bounded and open sets and

Next, we show that

On the contrary, suppose that there exists such that

Then

Let such that . Using the boundary conditions, we have . In this case, , . This gives

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

Consequently all conditions of Theorem 2.1 are satisfied. Therefore, there exists such that with and and the assertion follows. □

The authors declare that they have no competing interests.

Both authors read and approved the final manuscript.

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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