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

Boundary Value Problems20132013:14

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

• Accepted: 18 January 2013
• Published:

## Abstract

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

$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),{u}^{″}\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \left[0,1\right],\hfill \\ {u}^{\left(i\right)}\left(0\right)={u}^{\left(i\right)}\left(1\right),\phantom{\rule{1em}{0ex}}i=0,1,2,3,\hfill \end{array}$

where $f:\left[0,1\right]×\mathbf{R}×\mathbf{R}\to \mathbf{R}$ 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

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
(1)
(2)

where $f:\left[0,1\right]×\mathbf{R}×\mathbf{R}\to \mathbf{R}$ 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 [26].

Throughout this paper, we denote that $I=\left[0,1\right]$, $\mathbf{R}=\left(-\mathrm{\infty },+\mathrm{\infty }\right)$, $\mathbf{Z}=\left\{\dots ,-2,-1,0,1,2,\dots \right\}$, $\mathbf{N}=\left\{1,2,\dots \right\}$, ${\mathbf{N}}^{\ast }=\mathbf{N}\cup \left\{0\right\}$. In [710], authors showed the existence of solutions to Eq. (1) under the boundary condition
$u\left(0\right)=u\left(1\right)={u}^{″}\left(0\right)={u}^{″}\left(1\right)=0.$
(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:
$|f\left(t,u,v\right)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}a,b,c>0,\frac{a}{{\pi }^{4}}+\frac{b}{{\pi }^{2}}<1,$

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 $\left(\alpha ,\beta \right)$ satisfies
$\frac{\alpha }{{\left(k\pi \right)}^{4}}+\frac{\beta }{{\left(k\pi \right)}^{2}}\ne 1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in \mathbf{N},$
(4)
and that there are positive constants a, b, and c such that
$a\underset{k\in \mathbf{N}}{max}\frac{1}{|{\left(k\pi \right)}^{4}-\alpha -\beta {\left(k\pi \right)}^{2}|}+b\underset{k\in \mathbf{N}}{max}\frac{{\left(k\pi \right)}^{2}}{|{\left(k\pi \right)}^{4}-\alpha -\beta {\left(k\pi \right)}^{2}|}<1,$
(5)
and f satisfies the growth condition
$|f\left(t,u,v\right)-\left(\alpha u-\beta v\right)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v,\in \mathbf{R}.$

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

Condition (4)-(5) trivially implies that
$\frac{a+b{\left(k\pi \right)}^{2}}{|{\left(k\pi \right)}^{4}-\alpha -\beta {\left(k\pi \right)}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in \mathbf{N}.$
(6)
It is easy to prove that condition (6) is equivalent to the fact that the rectangle
$R\left(\alpha ,\beta ;a,b\right)=\left[\alpha -a,\alpha +a\right]×\left[\beta -b,\beta +b\right]$

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 $\left(\alpha ,\beta \right)$ satisfies
$\alpha +\beta {\left(2k\pi \right)}^{2}\ne {\left(2k\pi \right)}^{4},\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast },$
(7)
and that there are positive constants a, b, and c such that
$a\underset{k\in {\mathbf{N}}^{\ast }}{max}\frac{1}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}+b\underset{k\in {\mathbf{N}}^{\ast }}{max}\frac{{\left(2k\pi \right)}^{2}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}<1,$
(8)
and f satisfies the growth condition
$|f\left(t,u,v\right)-\left(\alpha u-\beta v\right)|\le a|u|+b|v|+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v,\in \mathbf{R}.$
(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)
$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)+\beta {u}^{″}\left(t\right)-\alpha u\left(t\right)=0,\phantom{\rule{1em}{0ex}}t\in I,\hfill \\ {u}^{\left(i\right)}\left(0\right)={u}^{\left(i\right)}\left(1\right),\phantom{\rule{1em}{0ex}}i=0,1,2,3.\hfill \end{array}$
(10)
In [2], it has been proved that $\left(\alpha ,\beta \right)$ is an eigenvalue pair of LEVP (10) if and only if $\alpha +\beta {\left(2k\pi \right)}^{2}={\left(2k\pi \right)}^{4}$, $k\in {\mathbf{N}}^{\ast }$. Hence, for each $k\in {\mathbf{N}}^{\ast }$, the straight line
${\ell }_{k}=\left\{\left(\alpha ,\beta \right)|\alpha +\beta {\left(2k\pi \right)}^{2}={\left(2k\pi \right)}^{4}\right\}$
is called an eigenline of LEVP (10). Condition (7)-(8) trivially implies that
$\frac{a+b{\left(2k\pi \right)}^{2}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast }.$
(11)

It is easy to prove that condition (11) is equivalent to the fact that the rectangle $R\left(\alpha ,\beta ;a,b\right)$ 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 $\left(\alpha ,\beta \right)$ satisfies (7). If there exist positive constants a, b, and c such that (11) and
$|f\left(t,u,v\right)-\left(\alpha u-\beta v\right)|\le \sqrt{{a}^{2}{u}^{2}+{b}^{2}{v}^{2}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v\in \mathbf{R}$
(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
$\left({f}_{u}\left(t,u,v\right),-{f}_{v}\left(t,u,v\right)\right)\in E\left(\alpha ,\beta ;a,b\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,\sqrt{{u}^{2}+{v}^{2}}\ge {R}_{0},$
(13)
where $E\left(\alpha ,\beta ;a,b\right)=\left\{\left(x,y\right)|\frac{{\left(x-\alpha \right)}^{2}}{{a}^{2}}+\frac{{\left(y-\beta \right)}^{2}}{{b}^{2}}\le 1\right\}$ is a certain ellipse, and the corresponding close rectangle $R\left(\alpha ,\beta ;a,b\right)$ satisfies
$R\left(\alpha ,\beta ;a,b\right)\cap {\ell }_{k}=\mathrm{\varnothing },\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast },$
(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 $I×\mathbf{R}×\mathbf{R}$. If there exists an ellipse $E\left(\alpha ,\beta ;a,b\right)$ such that (13) holds for a positive real number ${R}_{0}$ large enough, and the corresponding close rectangle $R\left(\alpha ,\beta ;a,b\right)$ 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
$\frac{\sqrt{{a}^{2}+{b}^{2}{\left(2k\pi \right)}^{4}}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}<1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast }.$
(15)

In this case, we have the following results.

Theorem 2 Assume that the pair $\left(\alpha ,\beta \right)$ 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
$E\left(\alpha ,\beta ;a,b\right)\cap {\ell }_{k}=\mathrm{\varnothing },\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast }.$
(16)

Condition (16) indicates that the ellipse $E\left(\alpha ,\beta ;a,b\right)$ 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 $I×\mathbf{R}×\mathbf{R}$. If there exists an ellipse $E\left(\alpha ,\beta ;a,b\right)$ 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 $I×\mathbf{R}×\mathbf{R}$. If there exists an ellipse $E\left(\alpha ,\beta ;a,b\right)$ such that (16) and
$\left({f}_{u}\left(t,u,v\right),-{f}_{v}\left(t,u,v\right)\right)\in E\left(\alpha ,\beta ;a,b\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v\in \mathbf{R},$
(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 $E\left(\alpha ,\beta ;a,b\right)$ by a circle
$\overline{B}\left(\alpha ,\beta ;r\right)=\left\{\left(x,y\right)|{\left(x-\alpha \right)}^{2}+{\left(y-\beta \right)}^{2}\le {r}^{2}\right\},\phantom{\rule{1em}{0ex}}r>0,$

and obtain the following results.

Corollary 3 Assume that there exist a circle $\overline{B}\left(\alpha ,\beta ;r\right)$ and a positive constant c such that
$\overline{B}\left(\alpha ,\beta ;r\right)\cap {\ell }_{k}=\mathrm{\varnothing },\phantom{\rule{1em}{0ex}}\mathrm{\forall }k\in {\mathbf{N}}^{\ast },$
(18)
and f satisfies the growth condition
$|f\left(t,u,v\right)-\left(\alpha u-\beta v\right)|\le r\sqrt{{u}^{2}+{v}^{2}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v\in \mathbf{R}.$
(19)

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

Condition (18) indicates that the circle $\overline{B}\left(\alpha ,\beta ;r\right)$ 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 $I×\mathbf{R}×\mathbf{R}$. If there exists a circle $\overline{B}\left(\alpha ,\beta ;r\right)$ such that (18) and
$\left({f}_{u}\left(t,u,v\right),-{f}_{v}\left(t,u,v\right)\right)\in \overline{B}\left(\alpha ,\beta ;r\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,\sqrt{{u}^{2}+{v}^{2}}\ge {R}_{0}$
(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 $I×\mathbf{R}×\mathbf{R}$. If there exists a circle $\overline{B}\left(\alpha ,\beta ;r\right)$ such that (18) and
$\left({f}_{u}\left(t,u,v\right),-{f}_{v}\left(t,u,v\right)\right)\in \overline{B}\left(\alpha ,\beta ;r\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in I,u,v\in \mathbf{R}$
(21)

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

## 2 Preliminaries

Let $\left(\alpha ,\beta \right)$ be not eigenvalue pair of LEVP (10), i.e., $\left(\alpha ,\beta \right)\notin \mathcal{L}:={\bigcup }_{k=0}^{+\mathrm{\infty }}{\ell }_{k}$. For any $h\in {L}^{2}\left(I\right)$, we consider the linear periodic boundary value problem (LPBVP)
$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)+\beta {u}^{″}\left(t\right)-\alpha u\left(t\right)=h\left(t\right),\phantom{\rule{1em}{0ex}}t\in I,\hfill \\ {u}^{\left(i\right)}\left(0\right)={u}^{\left(i\right)}\left(1\right),\phantom{\rule{1em}{0ex}}i=0,1,2,3.\hfill \end{array}$
(22)
By the Fredholm alternative, LPBVP (22) has a unique solution $u\in {H}^{4}\left(I\right)$. If $h\in C\left(I\right)$, then the solution $u\in {C}^{4}\left(I\right)$. We define an operator T by
$Th=u,\phantom{\rule{1em}{0ex}}\mathrm{\forall }h\in {L}^{2}\left(I\right).$

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

Let $a,b>0$. We choose an equivalent norm in the Sobolev space ${H}^{2}\left(I\right)$ by
${\parallel u\parallel }_{{E}_{a,b}}=\sqrt{{a}^{2}{\parallel u\parallel }_{2}^{2}+{b}^{2}{\parallel {u}^{″}\parallel }_{2}^{2}}$

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

Lemma 1 Let $\left(\alpha ,\beta \right)\notin \mathcal{L}$. Then the solution operator of LPBVP (22) $T:{L}^{2}\left(I\right)\to {E}_{a,b}$ is a compact linear operator and its norm satisfies
${\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}\le \underset{k\in {\mathbf{N}}^{\ast }}{max}\frac{\sqrt{{a}^{2}+{b}^{2}{\left(2k\pi \right)}^{4}}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}.$
(23)

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

Since $\left\{{e}^{2k\pi it}|k\in \mathbf{Z}\right\}$ is a complete orthogonal system of ${L}^{2}\left(I\right)$, every $h\in {L}^{2}\left(I\right)$ can be expressed by the Fourier series expansion
$h\left(t\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{h}_{k}\cdot {e}^{2k\pi it},$
where ${h}_{k}={\int }_{0}^{1}h\left(s\right){e}^{2k\pi is}\phantom{\rule{0.2em}{0ex}}ds$, $k\in \mathbf{Z}$. By the Parseval equality, we have
${\parallel h\parallel }_{2}^{2}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{|{h}_{k}|}^{2},$
where ${\parallel \cdot \parallel }_{2}$ is the norm in ${L}^{2}\left(I\right)$. Now, by uniqueness of the Fourier series expansion, the solution $u=Th$ of LPBVP (22) has the Fourier series expansion
$u\left(t\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{h}_{k}}{{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}}\cdot {e}^{2k\pi it},$
and ${u}^{″}$ can be expressed by the Fourier series expansion
${u}^{″}\left(t\right)=-\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\left(2k\pi \right)}^{2}{h}_{k}}{{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}}\cdot {e}^{2k\pi it}.$
Hence, by the Parseval equality, we have
(24)
(25)
From (24) and (25), we have
$\begin{array}{rcl}{\parallel Th\parallel }_{{E}_{a,b}}^{2}& =& {\parallel u\parallel }_{{E}_{a,b}}^{2}={a}^{2}{\parallel u\parallel }_{2}^{2}+{b}^{2}{\parallel {u}^{″}\parallel }_{2}^{2}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\left({a}^{2}+{b}^{2}{\left(2k\pi \right)}^{4}\right){|{h}_{k}|}^{2}}{{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}^{2}}\\ \le & {\left(\underset{k\in {\mathbf{N}}^{\ast }}{max}\frac{\sqrt{{a}^{2}+{b}^{2}{\left(2k\pi \right)}^{4}}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}\right)}^{2}\cdot \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{|{h}_{k}|}^{2}\\ =& {\left(\underset{k\in {\mathbf{N}}^{\ast }}{max}\frac{\sqrt{{a}^{2}+{b}^{2}{\left(2k\pi \right)}^{4}}}{|{\left(2k\pi \right)}^{4}-\alpha -\beta {\left(2k\pi \right)}^{2}|}\right)}^{2}\cdot {\parallel h\parallel }_{2}^{2}.\end{array}$

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 $R\left(\alpha ,\beta ;a,b\right)$ satisfies condition (14) if and only if condition (11) holds.

Proof Condition (14) holds

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

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

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

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

The proof of Lemma 2 is completed. □

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

Proof Condition (16) holds

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

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

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

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

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

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

The proof of Lemma 3 is completed. □

## 3 Proof of the main results

Proof of Theorem 1 We define a mapping $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ by
$F\left(u\right)\left(t\right)=f\left(t,u\left(t\right),{u}^{″}\left(t\right)\right)-\alpha u\left(t\right)+\beta {u}^{″}\left(t\right).$
(26)
It follows from (12) that $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ is continuous and satisfies
${\parallel F\left(u\right)\parallel }_{2}\le {\parallel u\parallel }_{{E}_{a,b}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in {E}_{a,b}.$
(27)
Therefore, the mapping defined by
$Q=T\circ F:{E}_{a,b}\to {E}_{a,b}$
(28)

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 ${\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}<1$. We choose $R\ge \frac{c\cdot {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}}{1-{\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}}$. Let $\overline{B}\left(\theta ,R\right)=\left\{u\in {E}_{a,b}|{\parallel u\parallel }_{{E}_{a,b}}\le R\right\}$. Then for any $u\in \overline{B}\left(\theta ,R\right)$, from (27) and (28), we have
$\begin{array}{rcl}{\parallel Qu\parallel }_{{E}_{a,b}}& =& {\parallel T\left(F\left(u\right)\right)\parallel }_{{E}_{a,b}}\le {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}\cdot {\parallel F\left(u\right)\parallel }_{2}\\ \le & {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}\cdot \left({\parallel u\parallel }_{{E}_{a,b}}+c\right)\\ \le & {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}\cdot \left(R+c\right)\le R.\end{array}$

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

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

Corollary 6 Assume that the pair $\left(\alpha ,\beta \right)$ 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 $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ be a mapping defined by (26). Then it follows from (12) that $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ is continuous and satisfies
${\parallel F\left(u\right)\parallel }_{2}\le {\parallel u\parallel }_{{E}_{a,b}}+c,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in {E}_{a,b}.$

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 $\overline{B}\left(\theta ,R\right)$, which is the solution of PBVP (1)-(2). □

Proof of Theorem 3 Let $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ be defined by (26). Then $F:{E}_{a,b}\to {L}^{2}\left(I\right)$ is continuous. For any ${u}_{1},{u}_{2}\in {E}_{a,b}$, from (17), we have
$\begin{array}{rcl}|F\left({u}_{2}\right)-F\left({u}_{1}\right)|& =& |f\left(t,{u}_{2},{u}_{2}^{″}\right)-\alpha {u}_{2}+\beta {u}_{2}^{″}-\left[f\left(t,{u}_{1},{u}_{1}^{″}\right)-\alpha {u}_{1}+\beta {u}_{1}^{″}\right]|\\ =& |\left({f}_{u}-\alpha \right)\left({u}_{2}-{u}_{1}\right)+\left({f}_{v}+\beta \right)\left({u}_{2}^{″}-{u}_{1}^{″}\right)|\\ =& |\frac{{f}_{u}-\alpha }{a}\cdot a\left({u}_{2}-{u}_{1}\right)+\frac{{f}_{v}+\beta }{b}\cdot b\left({u}_{2}^{″}-{u}_{1}^{″}\right)|\\ \le & \sqrt{\frac{{\left({f}_{u}-\alpha \right)}^{2}}{{a}^{2}}+\frac{{\left({f}_{v}+\beta \right)}^{2}}{{b}^{2}}}\cdot \sqrt{{a}^{2}{\left({u}_{2}-{u}_{1}\right)}^{2}+{b}^{2}{\left({u}_{2}^{″}-{u}_{1}^{″}\right)}^{2}}\\ \le & \sqrt{{a}^{2}{\left({u}_{2}-{u}_{1}\right)}^{2}+{b}^{2}{\left({u}_{2}^{″}-{u}_{1}^{″}\right)}^{2}}.\end{array}$
It follows from the above that ${\parallel F\left({u}_{2}\right)-F\left({u}_{1}\right)\parallel }_{2}\le {\parallel {u}_{2}-{u}_{1}\parallel }_{{E}_{a,b}}$. Thus, $Q=T\circ F:{E}_{a,b}\to {E}_{a,b}$ is a continuous mapping and it satisfies
$\begin{array}{rcl}{\parallel Q\left({u}_{2}\right)-Q\left({u}_{1}\right)\parallel }_{a,b}& =& {\parallel T\left(F\left({u}_{2}\right)-F\left({u}_{1}\right)\right)\parallel }_{{E}_{a,b}}\\ \le & {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}\cdot {\parallel F\left({u}_{2}\right)-F\left({u}_{1}\right)\parallel }_{2}\\ \le & {\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}{\parallel {u}_{2}-{u}_{1}\parallel }_{{E}_{a,b}}.\end{array}$

It follows from (16) and Lemma 3 that (15) holds. By (15) and Lemma 1, it is easy to see that ${\parallel T\parallel }_{B\left({L}^{2}\left(I\right),{E}_{a,b}\right)}<1$. 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

(1)
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China
(2)
Science College, Gansu Agricultural University, Lanzhou, 730070, People’s Republic of China

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