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.
Keywords:fixed point index; order-type existence theorem; cone expansion and compression; positive solutions; periodic boundary value problems
1 Introduction and preliminaries
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 , 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 , Cremins established a norm-type existence theorem concerning cone expansion and compression in , which generalizes some corresponding results contained in .
In this paper, we will use the properties of the fixed point index in  and partial order to present a new order-type existence theorem concerning cone expansion and compression which extends the corresponding results in . 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 , 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 .
Proposition 1.1Let be open and bounded and . Assume that , mapsKtoK, and on .
(P1) (Existence property) If , then there exists such that .
(P2) (Normality) If , then , where and for every .
(P3) (Additivity) If for , where and are disjoint relatively open subsets of , then
with equality if either of indices on the right is a singleton.
(P4) (Homotopy invariance) If is an A-proper homotopy on for and and for , then is independent of , where .
Lemma 1.1If is Fredholm of index zero, Ω is an open bounded set and , . Let be A-proper for . Assume thatNis bounded and mapsKtoK. If on for , then
Lemma 1.2If is Fredholm of index zero, Ω is an open bounded set and . Let be A-proper for . Assume thatNis bounded and mapsKtoK. If there exists such that
for every and all , then
2 An abstract result
We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows.
Theorem 2.1If is Fredholm of index zero, let be A-proper for . Assume thatNis bounded and mapsKtoK. Suppose further that and are two bounded open sets inXsuch that , and . If one of the following two conditions is satisfied:
(C1) for all and for all ;
(C2) for all and for all .
Then there exists such that .
Proof We assume that (C1) is satisfied. First we show that
In fact, otherwise, there exist and such that
then we obtain
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
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
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
is given by
We can verify that
To state the existence result, we introduce two conditions:
(H1) for all ,
(H2) for all .
Theorem 3.1Assume that there exist two positive numbers such that (H1), (H2) and
(H3) for all
hold. Then (3.1) has at least one positive periodic solution with .
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 [, Lemma 2(a)].
For each , then by condition (H3),
Clearly, and are bounded open sets and
We now show that
In fact, if there exists such that
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 , 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 [, 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
Since , we have
One may easily see that there exists such that
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,
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
We can verify that
Theorem 3.2Assume that there exist two positive numbers such that (H1), (H2) and
(H4) for all
hold. Then (3.4) has at least one positive periodic solution with .
Proof It is again easy to show that is A-proper for by [, Lemma 2(a)].
For each , then by condition (H4),
Clearly, and are bounded and open sets and
Next, we show that
On the contrary, suppose that there exists such that
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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