In mathematics, the equilibrium state of an elastic beam is described by fourth-order boundary value problems. According to the difference of supported condition on both ends, it brings out various fourth-order boundary value problems; see [

1]. In this paper, we deal with the periodic boundary value problem (PBVP) of the fourth-order ordinary differential equation

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].

Throughout this paper, we denote that

$I=[0,1]$,

$\mathbf{R}=(-\mathrm{\infty},+\mathrm{\infty})$,

$\mathbf{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$,

$\mathbf{N}=\{1,2,\dots \}$,

${\mathbf{N}}^{\ast}=\mathbf{N}\cup \{0\}$. In [

7–

10], authors showed the existence of solutions to Eq. (

1) under the boundary condition

$u(0)=u(1)={u}^{\u2033}(0)={u}^{\u2033}(1)=0.$

(3)

At first, the existence of a solution to two-point 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:

$|f(t,u,v)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}a,b,c>0,\frac{a}{{\pi}^{4}}+\frac{b}{{\pi}^{2}}<1,$

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*
$\frac{\alpha}{{(k\pi )}^{4}}+\frac{\beta}{{(k\pi )}^{2}}\ne 1,\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in \mathbf{N},$

(4)

*and that there are positive constants* *a*,

*b*,

*and* *c* *such that* $a\underset{k\in \mathbf{N}}{max}\frac{1}{|{(k\pi )}^{4}-\alpha -\beta {(k\pi )}^{2}|}+b\underset{k\in \mathbf{N}}{max}\frac{{(k\pi )}^{2}}{|{(k\pi )}^{4}-\alpha -\beta {(k\pi )}^{2}|}<1,$

(5)

*and*
*f*
*satisfies the growth condition*
$|f(t,u,v)-(\alpha u-\beta v)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v,\in \mathbf{R}.$

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

Condition (4)-(5) trivially implies that

$\frac{a+b{(k\pi )}^{2}}{|{(k\pi )}^{4}-\alpha -\beta {(k\pi )}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in \mathbf{N}.$

(6)

It is easy to prove that condition (6) is equivalent to the fact that the rectangle

$R(\alpha ,\beta ;a,b)=[\alpha -a,\alpha +a]\times [\beta -b,\beta +b]$

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*
$\alpha +\beta {(2k\pi )}^{2}\ne {(2k\pi )}^{4},\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast},$

(7)

*and that there are positive constants* *a*,

*b*,

*and* *c* *such that* $a\underset{k\in {\mathbf{N}}^{\ast}}{max}\frac{1}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}+b\underset{k\in {\mathbf{N}}^{\ast}}{max}\frac{{(2k\pi )}^{2}}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1,$

(8)

*and*
*f*
*satisfies the growth condition*
$|f(t,u,v)-(\alpha u-\beta v)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v,\in \mathbf{R}.$

(9)

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

Condition (7)-(9) concerns a nonresonance condition involving the two-parameter linear eigenvalue problem (LEVP)

$\{\begin{array}{c}{u}^{(4)}(t)+\beta {u}^{\u2033}(t)-\alpha u(t)=0,\phantom{\rule{1em}{0ex}}t\in I,\hfill \\ {u}^{(i)}(0)={u}^{(i)}(1),\phantom{\rule{1em}{0ex}}i=0,1,2,3.\hfill \end{array}$

(10)

In [

2], it has been proved that

$(\alpha ,\beta )$ is an eigenvalue pair of LEVP (10) if and only if

$\alpha +\beta {(2k\pi )}^{2}={(2k\pi )}^{4}$,

$k\in {\mathbf{N}}^{\ast}$. Hence, for each

$k\in {\mathbf{N}}^{\ast}$, the straight line

${\ell}_{k}=\{(\alpha ,\beta )|\alpha +\beta {(2k\pi )}^{2}={(2k\pi )}^{4}\}$

is called an eigenline of LEVP (10). Condition (7)-(8) trivially implies that

$\frac{a+b{(2k\pi )}^{2}}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast}.$

(11)

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* $|f(t,u,v)-(\alpha u-\beta v)|\le \sqrt{{a}^{2}{u}^{2}+{b}^{2}{v}^{2}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v\in \mathbf{R}$

(12)

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

When the partial derivatives

${f}_{u}$ and

${f}_{v}$ exist, if

$\sqrt{{u}^{2}+{v}^{2}}$ is large enough such that

$({f}_{u}(t,u,v),-{f}_{v}(t,u,v))\in E(\alpha ,\beta ;a,b),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,\sqrt{{u}^{2}+{v}^{2}}\ge {R}_{0},$

(13)

where

$E(\alpha ,\beta ;a,b)=\{(x,y)|\frac{{(x-\alpha )}^{2}}{{a}^{2}}+\frac{{(y-\beta )}^{2}}{{b}^{2}}\le 1\}$ is a certain ellipse, and the corresponding close rectangle

$R(\alpha ,\beta ;a,b)$ satisfies

$R(\alpha ,\beta ;a,b)\cap {\ell}_{k}=\mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast},$

(14)

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 the nonresonance condition of Theorem 1, condition (11) can be weakened as

$\frac{\sqrt{{a}^{2}+{b}^{2}{(2k\pi )}^{4}}}{|{(2k\pi )}^{4}-\alpha -\beta {(2k\pi )}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast}.$

(15)

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 (15) is equivalent to the fact that

$E(\alpha ,\beta ;a,b)\cap {\ell}_{k}=\mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast}.$

(16)

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* $({f}_{u}(t,u,v),-{f}_{v}(t,u,v))\in E(\alpha ,\beta ;a,b),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v\in \mathbf{R},$

(17)

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

In Theorem 2, Theorem 3, and Corollary 2, we present a new two-parameter nonresonance condition described by ellipse, which is another extension of a single-parameter nonresonance condition. As a special case, we replace the ellipse

$E(\alpha ,\beta ;a,b)$ by a circle

$\overline{B}(\alpha ,\beta ;r)=\{(x,y)|{(x-\alpha )}^{2}+{(y-\beta )}^{2}\le {r}^{2}\},\phantom{\rule{1em}{0ex}}r>0,$

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*
$\overline{B}(\alpha ,\beta ;r)\cap {\ell}_{k}=\mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in {\mathbf{N}}^{\ast},$

(18)

*and*
*f*
*satisfies the growth condition*
$|f(t,u,v)-(\alpha u-\beta v)|\le r\sqrt{{u}^{2}+{v}^{2}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v\in \mathbf{R}.$

(19)

*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* $({f}_{u}(t,u,v),-{f}_{v}(t,u,v))\in \overline{B}(\alpha ,\beta ;r),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,\sqrt{{u}^{2}+{v}^{2}}\ge {R}_{0}$

(20)

*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* $({f}_{u}(t,u,v),-{f}_{v}(t,u,v))\in \overline{B}(\alpha ,\beta ;r),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in I,u,v\in \mathbf{R}$

(21)

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