- Open Access
An order-type existence theorem and applications to periodic problems
© Chu and Wang; licensee Springer. 2013
- Received: 8 November 2012
- Accepted: 5 February 2013
- Published: 21 February 2013
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.
- fixed point index
- order-type existence theorem
- cone expansion and compression
- positive solutions
- periodic boundary value problems
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 .
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 1.1 Let be open and bounded and . Assume that , maps K to K, and on .
(P1) (Existence property) If , then there exists such that .
(P2) (Normality) If , then , where and for every .
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 .
We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows.
Theorem 2.1 If is Fredholm of index zero, let be A-proper for . Assume that N is bounded and maps K to K. Suppose further that and are two bounded open sets in X such 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 .
Since the index is nonzero, the existence property (P1) of Proposition 1.1 implies that there exists such that .
Also, we can assert that there exists such that . □
3.1 First-order periodic boundary value problems
where is continuous and for all .
To state the existence result, we introduce two conditions:
(H1) for all ,
(H2) for all .
Theorem 3.1 Assume 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)].
which is a contradiction to (H1). Therefore (3.2) holds.
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):
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.
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.
which implies that (H1) holds.
which implies that () holds. Now we have the desired result. □
3.2 Second-order periodic boundary value problems
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.
Theorem 3.2 Assume 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)].
which is a contradiction to condition (H1). Therefore (3.5) holds.
Consequently all conditions of Theorem 2.1 are satisfied. Therefore, there exists such that with and and the assertion follows. □
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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