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# Existence of positive solutions for a kind of periodic boundary value problem at resonance

- Mirosława Zima
^{1}Email author and - Piotr Drygaś
^{1}

**2013**:19

https://doi.org/10.1186/1687-2770-2013-19

© Zima and Drygaś; licensee Springer. 2013

**Received: **20 December 2012

**Accepted: **21 January 2013

**Published: **11 February 2013

## Abstract

In the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.

## Keywords

## 1 Introduction

The same PBVP was studied by Wang, Zhang and Wang in [10]. Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for *A*-proper semilinear operators on cones developed by Cremins [11].

The goal of our paper is to provide sufficient conditions that ensure the existence of positive solutions of (1) with the function *h* positive on $[0,T]$. Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation $Lx=Nx$, where *L* is a Fredholm operator of index zero and *N* is a nonlinear operator, and to apply the Leggett-Williams norm-type theorem for coincidences obtained by O’Regan and Zima [12]. We would like to emphasize that the idea of results of [11] and [12], as well as these of [13–15], goes back to the celebrated Mawhin’s coincidence degree theory [16]. For more details on this significant tool, its modifications and wide applications, we refer the reader to [17–22] and references therein.

In this paper, for the first time, the existence theorem from [12] is used for studying the boundary value problem with the nonlinearity *f* depending also on the derivative. In general, the presence of ${x}^{\prime}$ in *f* makes the problem much harder to handle. We point out that, to the best of our knowledge, there are only a few papers on PBVPs that discuss such a nonlinearity; we refer the reader to [15, 23–25] for some results of that type. We also complement several results in the literature, for example, in [1, 26] and [27]. It is evident that the existence theorems for PBVP (1) can be established by the shift method used in [6], that is, one can employ the results of [1] to the periodic problem we study here. However, the conditions imposed on *f* in [1] are not comparable with ours.

## 2 Coincidence equation

For the convenience of the reader, we begin this section by providing some background on cone theory and Fredholm operators in Banach spaces.

**Definition 1**A nonempty subset

*C*, $C\ne \{0\}$, of a real Banach space

*X*is called a cone if

*C*is closed, convex and

- (i)
$\lambda x\in C$ for all $x\in C$ and $\lambda \ge 0$,

- (ii)
*x*, $-x\in C$ implies $x=0$.

*X*as follows: for $x,y\in X$, we say that

The following property holds for every cone in a Banach space.

**Lemma 1**[28]

*For every*$u\in C\setminus \{0\}$,

*there exists a positive number*$\sigma (u)$

*such that*

*for all* $x\in C$.

*X*and

*Y*are Banach spaces. If

*L*is a Fredholm operator of index zero, that is, Im

*L*is closed and $dimKerL=codimImL<\mathrm{\infty}$, then there exist continuous projections $P:X\to X$ and $Q:Y\to Y$ such that $ImP=KerL$ and $KerQ=ImL$ (see, for example, [14, 16]). Moreover, since $dimImQ=codimImL$, there exists an isomorphism $J:ImQ\to KerL$. Denote by ${L}_{P}$ the restriction of

*L*to $KerP\cap domL$. Then ${L}_{P}$ is an isomorphism from $KerP\cap domL$ to Im

*L*and its inverse

is defined.

Let ${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$ be open bounded subsets of *X* with ${\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}$ and $C\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\ne \mathrm{\varnothing}$. Assume that

1^{∘} *L* is a Fredholm operator of index zero,

2^{∘} $QN:X\to Y$ is continuous and bounded and ${K}_{P}(I-Q)N:X\to X$ is compact on every bounded subset of *X*,

3^{∘} $Lx\ne \lambda Nx$ for all $x\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domL$ and $\lambda \in (0,1)$,

4^{∘} *ρ* maps subsets of ${\overline{\mathrm{\Omega}}}_{2}$ into bounded subsets of *C*,

5^{∘} ${d}_{B}([I-(P+JQN)\rho ]{|}_{KerL},KerL\cap {\mathrm{\Omega}}_{2},0)\ne 0$, where ${d}_{B}$ stands for the Brouwer degree,

^{∘}there exists ${u}_{0}\in C\setminus \{0\}$ such that $\parallel x\parallel \le \sigma ({u}_{0})\parallel \mathrm{\Psi}x\parallel $ for $x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}$, where

and $\sigma ({u}_{0})$ is such that $\parallel x+{u}_{0}\parallel \ge \sigma ({u}_{0})\parallel x\parallel $ for every $x\in C$,

7^{∘} $(P+JQN)\rho (\partial {\mathrm{\Omega}}_{2})\subset C$ and ${\mathrm{\Psi}}_{\rho}({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\subset C$.

**Theorem 1** [12]

*Under the assumptions* 1^{∘}-7^{∘} *the equation* $Lx=Nx$ *has a solution in the set* $C\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})$.

In the next section, we use Theorem 1 to prove the existence of a positive solution for PBVP (1). For applications of Theorem 1 to nonlocal boundary value problems at resonance, we refer the reader to [22], [29] and [30].

## 3 Periodic boundary value problem

where *M* is a positive constant.

We assume that

(H1) $f:[0,T]\times [0,\mathrm{\infty})\times \mathbb{R}\to \mathbb{R}$ and $h:[0,T]\to (0,\mathrm{\infty})$ are continuous functions.

We also assume that there exist $R>0$, $0<\alpha \le \beta $, $0<M\le \frac{e(T)(1-e(T)){\int}_{0}^{T}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\tau}{\alpha T}$, $r\in (0,R)$, $m\in (0,1)$, $\eta \in [0,T]$ and a continuous function $g:[0,T]\to [0,\mathrm{\infty})$ such that

(H2) $f(t,x,y)>-\alpha x+\beta |y|$ for $(t,x,y)\in [0,T]\times [0,R]\times [-R,R]$,

(H3) $f(t,R,0)<0$ for $t\in [0,T]$,

(H4) $f(0,x,R)=f(T,x,R)$ and $f(0,x,-R)=f(T,x,-R)$ for $x\in [0,R]$,

(H5) $f(t,x,-R)\le h(t)R$ for $t\in [0,T]$ and $x\in [0,R)$,

(H6) $f(t,x,y)\ge g(t)(x+|y|)$ for $(t,x,y)\in [0,T]\times (0,r]\times [-r,r]$,

(H7) $\frac{1}{\alpha T}\ge K(t,s)\ge 0$ for $t,s\in [0,T]$ and $m{\int}_{0}^{T}K(\eta ,s)g(s)\phantom{\rule{0.2em}{0ex}}ds\ge 1$.

**Theorem 2** *Under the assumptions* (H1)-(H7), *PBVP* (1) *has a positive solution on* $[0,T]$.

*Proof* Let ${\parallel \cdot \parallel}_{\mathrm{\infty}}$ denote the supremum norm in the space $C[0,T]$, that is, ${\parallel x\parallel}_{\mathrm{\infty}}={sup}_{t\in [0,T]}|x(t)|$. Consider the Banach spaces $X={C}^{1}[0,T]$ with the norm $\parallel x\parallel =max\{{\parallel x\parallel}_{\mathrm{\infty}},{\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}\}$, and $Y=C[0,T]$ with the norm ${\parallel \cdot \parallel}_{\mathrm{\infty}}$.

where *ψ* is given by (5).

*L*is closed and $Y={Y}_{1}+ImL$ with

Since ${Y}_{1}\cap ImL=\{0\}$, we have $Y={Y}_{1}\oplus ImL$. Moreover, $dim{Y}_{1}=1$, which gives $codimImL=1$. Consequently, *L* is Fredholm of index zero, and the assumption 1^{∘} is satisfied.

*k*defined by (6). Clearly, the assumption 2

^{∘}is satisfied. For $y\in ImQ$, define

*J*is an isomorphism from Im

*Q*to Ker

*L*. Next, consider a cone

Obviously, ${\mathrm{\Omega}}_{1}$ and ${\mathrm{\Omega}}_{2}$ are open bounded subsets of *X*, and ${\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}$.

^{∘}, suppose that there exist ${x}_{0}\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domL$ and ${\lambda}_{0}\in (0,1)$ such that $L{x}_{0}={\lambda}_{0}N{x}_{0}$. Then $x(t)\ge 0$ on $[0,T]$, $\parallel {x}_{0}\parallel =R$,

- 1.
If $\parallel {x}_{0}\parallel ={\parallel {x}_{0}\parallel}_{\mathrm{\infty}}$, then there exists ${t}_{0}\in [0,T]$ such that $x({t}_{0})=R$. For ${t}_{0}\in (0,T)$, we get $0\le -{x}^{\u2033}({t}_{0})={\lambda}_{0}f({t}_{0},R,0)$, contrary to the assumption (H3). Similarly, if ${t}_{0}=0$ or ${t}_{0}=T$, BCs (9) imply ${x}^{\prime}(0)={x}^{\prime}(T)=0$. Hence, $0\le -{x}^{\u2033}({t}_{0})={\lambda}_{0}f({t}_{0},R,0)$ which contradicts (H3) again.

- 2.If $\parallel {x}_{0}\parallel ={\parallel {x}_{0}^{\prime}\parallel}_{\mathrm{\infty}}>{\parallel {x}_{0}\parallel}_{\mathrm{\infty}}$, then there exists ${t}_{0}\in [0,T]$ such that $|{x}^{\prime}({t}_{0})|=R$. Observe that (H2) implies $f(t,x,\pm R)>0$ for $t\in [0,T]$ and $x\in [0,R]$. Suppose that ${t}_{0}\in (0,T)$. If ${x}^{\prime}({t}_{0})=R$, we get from (8)$-h({t}_{0})R={\lambda}_{0}f({t}_{0},{x}_{0}({t}_{0}),R),$(10)

contrary to (H5). By similar arguments, if ${t}_{0}=0$ or ${t}_{0}=T$, BCs (9) and (H4) imply either (10) or (11). Thus, 3^{∘} is fulfilled.

Clearly, *ρ* is a retraction and maps subsets of ${\overline{\mathrm{\Omega}}}_{2}$ into bounded subsets of *C*, so 4^{∘} holds.

^{∘}, it is enough to consider, for $x\in KerL\cap {\mathrm{\Omega}}_{2}$ and $\lambda \in [0,1]$, the mapping

^{∘}holds. Let $x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}$. Then for $t\in [0,T]$, we have $r\ge x(t)\ge m{\parallel x\parallel}_{\mathrm{\infty}}>0$, $r\ge |{x}^{\prime}(t)|\ge {\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}$, and by (H6) and (H7), we obtain

This implies $\parallel x\parallel \le \parallel \mathrm{\Psi}x\parallel $ for $x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}$, so 6^{∘} is satisfied.

^{∘}holds. If $x\in \partial {\mathrm{\Omega}}_{2}$, then in view of (H2), we get

Thus, 7^{∘} is fulfilled and the assertion follows. □

We now give two examples illustrating Theorem 2. Some calculations have been made with *Mathematica*. In the first example, the function *h* is constant, while in the second $h(t)=1/(1+t)$ and *f* is independent of *t*.

**Example 1**

and the assumptions (H2)-(H7) are met with $R=20$, $\alpha =\frac{2}{9}$, $\beta =\frac{3}{4}$, $r=\frac{36}{53}$, $m\in [\frac{12(e-1)}{17+7e},1)$, $\eta =0$ and $g(t)=t(1-t)+1$. By Theorem 2, problem (12) has a positive solution.

**Example 2**

The assumptions of Theorem 2 are fulfilled with $M=1$, $R=10$, $\alpha =\frac{1}{3}$, $\beta =\frac{1}{2}$, $r=\frac{1}{100}$, $m=0.9$, $\eta =\frac{1}{4}$ and $g(t)=3$.

## Declarations

### Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

## Authors’ Affiliations

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