Abstract
This paper deals with the existence and uniqueness of solutions of the fourthorder periodic boundary value problem
where
MSC: 34B15.
Keywords:
existence; uniqueness; twoparameter nonresonance condition; equivalent norm1 Introduction and main results
In mathematics, the equilibrium state of an elastic beam is described by fourthorder boundary value problems. According to the difference of supported condition on both ends, it brings out various fourthorder boundary value problems; see [1]. In this paper, we deal with the periodic boundary value problem (PBVP) of the fourthorder ordinary differential equation
where
Throughout this paper, we denote that
At first, the existence of a solution to twopoint boundary value problem (BVP) (1)(3) was studied by Aftabizadeh in [7] under the restriction that f is a bounded function. Then, under the following growth condition:
Yang in [[8], Theorem 1] extended Aftabizadeh’s result and showed the existence to BVP (1)(3). Later, Del Pino and Manasevich in [9] further extended the result of Aftabizadeh and Yang in [7,8] and obtained the following existence theorem.
Theorem AAssume that the pair
and that there are positive constantsa, b, andcsuch that
andfsatisfies the growth condition
Then BVP (1)(3) possesses at least one solution.
Condition (4)(5) trivially implies that
It is easy to prove that condition (6) is equivalent to the fact that the rectangle
does not intersect any of the eigenlines of the twoparameter linear eigenvalue problem corresponding to BVP (1)(3).
In [2], Ma applied Theorem A to PBVP (1)(2) successfully and obtained the following existence theorem.
Theorem BAssume that the pair
and that there are positive constantsa, b, andcsuch that
andfsatisfies the growth condition
Then PBVP (1)(2) has at least one solution.
Condition (7)(9) concerns a nonresonance condition involving the twoparameter linear eigenvalue problem (LEVP)
In [2], it has been proved that
is called an eigenline of LEVP (10). Condition (7)(8) trivially implies that
It is easy to prove that condition (11) is equivalent to the fact that the rectangle
The purpose of this paper is to improve and extend the abovementioned results. Different from the twoparameter nonresonance condition described by rectangle, we will present new twoparameter nonresonance conditions described by ellipse and circle. Under these nonresonance conditions, we obtain several existence and uniqueness theorems.
The main results are as follows.
Theorem 1Assume that the pair
hold, then PBVP (1)(2) has at least one solution.
When the partial derivatives
where
by the theorem of differential mean value, we easily see that (7), (11), and (12) hold. Hence, by Theorem 1, we have the following corollary.
Corollary 1Assume that the partial derivatives
Condition (11) is weaker than condition (8), but condition (12) is stronger than condition (9). Hence, Theorem 1 and Corollary 1 partly improve Theorem B.
In the nonresonance condition of Theorem 1, condition (11) can be weakened as
In this case, we have the following results.
Theorem 2Assume that the pair
Condition (15) is equivalent to the fact that
Condition (16) indicates that the ellipse
Corollary 2Assume that the partial derivatives
Theorem 3Assume that the partial derivatives
hold, then PBVP (1)(2) has a unique solution.
In Theorem 2, Theorem 3, and Corollary 2, we present a new twoparameter nonresonance
condition described by ellipse, which is another extension of a singleparameter nonresonance
condition. As a special case, we replace the ellipse
and obtain the following results.
Corollary 3Assume that there exist a circle
andfsatisfies the growth condition
Then PBVP (1)(2) has at least one solution.
Condition (18) indicates that the circle
Corollary 4Assume that the partial derivatives
hold for a positive real number
Corollary 5Assume that the partial derivatives
hold, then PBVP (1)(2) has a unique solution.
2 Preliminaries
Let
By the Fredholm alternative, LPBVP (22) has a unique solution
Then
Let
and denote the Banach space
Lemma 1Let
Proof We only need to prove that (23) holds.
Since
where
where
and
Hence, by the Parseval equality, we have
From (24) and (25), we have
This implies that (23) holds. The proof of Lemma 1 is completed. □
Lemma 2Let
Proof Condition (14) holds
⇔
⇔
⇔
⇔
The proof of Lemma 2 is completed. □
Lemma 3Let
Proof Condition (16) holds
⇔ for
⇔
⇔
⇔
⇔
⇔
The proof of Lemma 3 is completed. □
3 Proof of the main results
Proof of Theorem 1 We define a mapping
It follows from (12) that
Therefore, the mapping defined by
is a completely continuous mapping. By the definition of the operator T, the solution of PBVP (1)(2) is equivalent to the fixed point of the operator Q.
From (7), (11), and Lemma 1, it follows that
Therefore,
By Lemma 2, we can obtain the following existence result:
Corollary 6Assume that the pair
Proof of Theorem 2 Let
Thus, the mapping
Proof of Theorem 3 Let
It follows from the above that
It follows from (16) and Lemma 3 that (15) holds. By (15) and Lemma 1, it is easy
to see that
As in Corollary 6, in Theorem 2 we can use condition (16) to replace condition (15), and in Theorem 3, we use condition (15) to replace condition (16).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HY carried out the study of the twoparameter nonresonance conditions for periodic boundary value problems, participated in the proof of the main results and drafted the manuscript. YL participated in the design of the study and performed the coordination. PC participated in the proof of the main results. All authors read and approved the final manuscript.
Acknowledgements
Research supported by the NNSF of China (Grant No. 11261053), the Fundamental Research Funds for the Gansu Universities and the Project of NWNULKQN113.
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