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# Existence and uniqueness of solutions for fourth-order periodic boundary value problems under two-parameter nonresonance conditions

- He Yang
^{1}Email author, - Yue Liang
^{2}and - Pengyu Chen
^{1}

**2013**:14

https://doi.org/10.1186/1687-2770-2013-14

© Yang et al.; licensee Springer. 2013

**Received:**16 December 2012**Accepted:**18 January 2013**Published:**4 February 2013

## Abstract

This paper deals with the existence and uniqueness of solutions of the fourth-order periodic boundary value problem

where $f:[0,1]\times \mathbf{R}\times \mathbf{R}\to \mathbf{R}$ is continuous. Under two-parameter nonresonance conditions described by rectangle and ellipse, some existence and uniqueness results are obtained by using fixed point theorems. These results improve and extend some existing results.

**MSC:**34B15.

## Keywords

- existence
- uniqueness
- two-parameter nonresonance condition
- equivalent norm

## 1 Introduction and main results

where $f:[0,1]\times \mathbf{R}\times \mathbf{R}\to \mathbf{R}$ is continuous. PBVP (1)-(2) models the deformations of an elastic beam in equilibrium state with a periodic boundary condition. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors; see [2–6].

*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 A**

*Assume that the pair*$(\alpha ,\beta )$

*satisfies*

*and that there are positive constants*

*a*,

*b*,

*and*

*c*

*such that*

*and*

*f*

*satisfies the growth condition*

*Then BVP* (1)-(3) *possesses at least one solution*.

does not intersect any of the eigenlines of the two-parameter 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 B**

*Assume that the pair*$(\alpha ,\beta )$

*satisfies*

*and that there are positive constants*

*a*,

*b*,

*and*

*c*

*such that*

*and*

*f*

*satisfies the growth condition*

*Then PBVP* (1)-(2) *has at least one solution*.

It is easy to prove that condition (11) is equivalent to the fact that the rectangle $R(\alpha ,\beta ;a,b)$ does not intersect any of the eigenline ${\ell}_{k}$ of LEVP (10). Hence, we call (11) and (9) the two-parameter nonresonance condition described by rectangle, which is a direct extension from a single-parameter nonresonance condition to a two-parameter one.

The purpose of this paper is to improve and extend the above-mentioned results. Different from the two-parameter nonresonance condition described by rectangle, we will present new two-parameter 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 1**

*Assume that the pair*$(\alpha ,\beta )$

*satisfies*(7).

*If there exist positive constants*

*a*,

*b*,

*and*

*c*

*such that*(11)

*and*

*hold*, *then PBVP* (1)-(2) *has at least one solution*.

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 1** *Assume that the partial derivatives* ${f}_{u}$ *and* ${f}_{v}$ *exist in* $I\times \mathbf{R}\times \mathbf{R}$. *If there exists an ellipse* $E(\alpha ,\beta ;a,b)$ *such that* (13) *holds for a positive real number* ${R}_{0}$ *large enough*, *and the corresponding close rectangle* $R(\alpha ,\beta ;a,b)$ *satisfies* (14), *then PBVP* (1)-(2) *has at least one solution*.

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 this case, we have the following results.

**Theorem 2** *Assume that the pair* $(\alpha ,\beta )$ *satisfies* (7). *If there exist positive constants* *a*, *b*, *and* *c* *such that* (12) *and* (15) *hold*, *then PBVP* (1)-(2) *has at least one solution*.

Condition (16) indicates that the ellipse $E(\alpha ,\beta ;a,b)$ does not intersect any of the eigenline ${\ell}_{k}$ of LEVP (10). Hence, we call (15) and (12) the two-parameter nonresonance condition described by ellipse, which is another extension of a single-parameter nonresonance condition. Similar to Corollary 1, we have the following corollary.

**Corollary 2** *Assume that the partial derivatives* ${f}_{u}$ *and* ${f}_{v}$ *exist in* $I\times \mathbf{R}\times \mathbf{R}$. *If there exists an ellipse* $E(\alpha ,\beta ;a,b)$ *such that* (13) *and* (16) *hold for a positive real number* ${R}_{0}$ *large enough*, *then PBVP* (1)-(2) *has at least one solution*.

**Theorem 3**

*Assume that the partial derivatives*${f}_{u}$

*and*${f}_{v}$

*exist in*$I\times \mathbf{R}\times \mathbf{R}$.

*If there exists an ellipse*$E(\alpha ,\beta ;a,b)$

*such that*(16)

*and*

*hold*, *then PBVP* (1)-(2) *has a unique solution*.

and obtain the following results.

**Corollary 3**

*Assume that there exist a circle*$\overline{B}(\alpha ,\beta ;r)$

*and a positive constant*

*c*

*such that*

*and*

*f*

*satisfies the growth condition*

*Then PBVP* (1)-(2) *has at least one solution*.

Condition (18) indicates that the circle $\overline{B}(\alpha ,\beta ;r)$ does not intersect any of the eigenline ${\ell}_{k}$ of LEVP (10). Hence, we call condition (18)-(19) the two-parameter nonresonance condition described by circle, which is also an extension of a single-parameter nonresonance condition. Similarly to Corollary 2 and Theorem 3, we have the following corollaries.

**Corollary 4**

*Assume that the partial derivatives*${f}_{u}$

*and*${f}_{v}$

*exist in*$I\times \mathbf{R}\times \mathbf{R}$.

*If there exists a circle*$\overline{B}(\alpha ,\beta ;r)$

*such that*(18)

*and*

*hold for a positive real number* ${R}_{0}$ *large enough*, *then PBVP* (1)-(2) *has at least one solution*.

**Corollary 5**

*Assume that the partial derivatives*${f}_{u}$

*and*${f}_{v}$

*exist in*$I\times \mathbf{R}\times \mathbf{R}$.

*If there exists a circle*$\overline{B}(\alpha ,\beta ;r)$

*such that*(18)

*and*

*hold*, *then PBVP* (1)-(2) *has a unique solution*.

## 2 Preliminaries

*i.e.*, $(\alpha ,\beta )\notin \mathcal{L}:={\bigcup}_{k=0}^{+\mathrm{\infty}}{\ell}_{k}$. For any $h\in {L}^{2}(I)$, we consider the linear periodic boundary value problem (LPBVP)

Then $T:{L}^{2}(I)\to {H}^{4}(I)$ is a bounded linear operator, and we call it the solution operator of LPBVP (22). By compactness of the embedding ${H}^{4}(I)\hookrightarrow {H}^{2}(I)$, $T:{L}^{2}(I)\to {H}^{2}(I)$ is a compact linear operator.

and denote the Banach space ${H}^{2}(I)$ reendowed norm ${\parallel \cdot \parallel}_{{E}_{a,b}}$ by ${E}_{a,b}$.

**Lemma 1**

*Let*$(\alpha ,\beta )\notin \mathcal{L}$.

*Then the solution operator of LPBVP*(22) $T:{L}^{2}(I)\to {E}_{a,b}$

*is a compact linear operator and its norm satisfies*

*Proof* We only need to prove that (23) holds.

This implies that (23) holds. The proof of Lemma 1 is completed. □

**Lemma 2** *Let* $\alpha ,\beta \notin \mathcal{L}$ *and* $a,b>0$. *Then the rectangle* $R(\alpha ,\beta ;a,b)$ *satisfies condition* (14) *if and only if condition* (11) *holds*.

*Proof* Condition (14) holds

⇔ $(\alpha -a,\beta -b)$ and $(\alpha +a,\beta +b)$ on the same side of every eigenline ${\ell}_{k}$,

⇔ ${(2k\pi )}^{4}-(\alpha -a)-(\beta -b){(2k\pi )}^{2}$ and ${(2k\pi )}^{4}-(\alpha +a)-(\beta +b){(2k\pi )}^{2}$ have the same sign,

⇔ ${({(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2})}^{2}-{(a+b{(2k\pi )}^{2})}^{2}>0$,

⇔ $\frac{a+b{(2k\pi )}^{2}}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1$.

The proof of Lemma 2 is completed. □

**Lemma 3** *Let* $\alpha ,\beta \notin \mathcal{L}$ *and* $a,b>0$. *Then the ellipse* $E(\alpha ,\beta ;a,b)$ *satisfies condition* (16) *if and only if condition* (15) *holds*.

*Proof* Condition (16) holds

⇔ for $\mathrm{\forall}\theta \in [0,2\pi ]$, $(\alpha -acos\theta ,\beta -bsin\theta )$ and $(\alpha +acos\theta ,\beta +bsin\theta )$ on the same side of every eigenline ${\ell}_{k}$,

⇔ ${(2k\pi )}^{4}-(\alpha -acos\theta )-(\beta -bsin\theta ){(2k\pi )}^{2}$ and ${(2k\pi )}^{4}-(\alpha +acos\theta )-(\beta +bsin\theta ){(2k\pi )}^{2}$ have the same sign,

⇔ ${({(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2})}^{2}-{(acos\theta +bsin\theta {(2k\pi )}^{2})}^{2}>0$,

⇔ $\frac{|acos\theta +bsin\theta {(2k\pi )}^{2}|}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1$,

⇔ ${max}_{\theta \in [0,2\pi ]}\frac{|acos\theta +bsin\theta {(2k\pi )}^{2}|}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1$,

⇔ $\frac{\sqrt{{a}^{2}+{b}^{2}{(2k\pi )}^{4}}}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1$.

The proof of Lemma 3 is completed. □

## 3 Proof of the main results

*Proof of Theorem 1*We define a mapping $F:{E}_{a,b}\to {L}^{2}(I)$ 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*.

Therefore, $Q(\overline{B}(\theta ,R))\subset \overline{B}(\theta ,R)$. By the Schauder’s fixed point theorem, *Q* has at least one fixed point in $\overline{B}(\theta ,R)$, which is a solution of PBVP (1)-(2). □

By Lemma 2, we can obtain the following existence result:

**Corollary 6** *Assume that the pair* $(\alpha ,\beta )$ *satisfies* (7). *If there exist positive constants* *a*, *b*, *and* *c* *such that* (12) *and* (14) *hold*, *then PBVP* (1)-(2) *has at least one solution*.

*Proof of Theorem 2*Let $F:{E}_{a,b}\to {L}^{2}(I)$ be a mapping defined by (26). Then it follows from (12) that $F:{E}_{a,b}\to {L}^{2}(I)$ is continuous and satisfies

Thus, the mapping $Q=T\circ F:{E}_{a,b}\to {E}_{a,b}$ is completely continuous. By using (7), (15), and Lemma 1, a similar argument as in the proof of Theorem 1 shows that *Q* has at least one fixed point in $\overline{B}(\theta ,R)$, which is the solution of PBVP (1)-(2). □

*Proof of Theorem 3*Let $F:{E}_{a,b}\to {L}^{2}(I)$ be defined by (26). Then $F:{E}_{a,b}\to {L}^{2}(I)$ is continuous. For any ${u}_{1},{u}_{2}\in {E}_{a,b}$, from (17), we have

It follows from (16) and Lemma 3 that (15) holds. By (15) and Lemma 1, it is easy to see that ${\parallel T\parallel}_{B({L}^{2}(I),{E}_{a,b})}<1$. Hence, $Q:{E}_{a,b}\to {E}_{a,b}$ is a contraction mapping. By the Banach contraction mapping principle, *Q* has a unique fixed point, which is the unique solution of PBVP (1)-(2). □

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).

## Declarations

### Acknowledgements

Research supported by the NNSF of China (Grant No. 11261053), the Fundamental Research Funds for the Gansu Universities and the Project of NWNU-LKQN-11-3.

## Authors’ Affiliations

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