An order-type existence theorem and applications to periodic problems

Abstract

Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established. As applications, we present sufficient existence conditions for the first- and second-order periodic problems.

MSC:34B15.

1 Introduction and preliminaries

Let X, Y be real Banach spaces. Consider a linear mapping $L:domL\subset X\to Y$ and a nonlinear operator $N:X\to Y$. Here we assume that L is a Fredholm operator of index zero, that is, ImL is closed and $dim\hspace{0.17em}KerL=codim\hspace{0.17em}ImL<\mathrm{\infty }$. Then the solvability of the operator equation

$$Lx=Nx$$

has been studied by many researchers in the literature; see [1–8] and the references therein. In [1], Cremins established a fixed point index for A-proper semilinear operators defined on cones which includes and improves the results in [5, 8, 9]. Using the fixed point index and the concept of a quasi-normal cone introduced in [10], Cremins established a norm-type existence theorem concerning cone expansion and compression in [11], which generalizes some corresponding results contained in [12].

In this paper, we will use the properties of the fixed point index in [1] and partial order to present a new order-type existence theorem concerning cone expansion and compression which extends the corresponding results in [12]. We recall that a partial order in X induced by a cone $K\subset X$ is defined by

$$x\le y⟺y-x\in K.$$

As applications, we study the first- and second-order periodic boundary problems and obtain new existence results. During the last few decades, periodic boundary value problems have been studied by many researchers in the literature; see, for example, [13–19] and the references therein. Our new results improve those contained in [13, 18].

Next we recall some notations and results which will be needed in this paper. Let X and Y be Banach spaces, D be a linear subspace of X, $\{X_n\}\subset D$ and $\{Y_n\}\subset Y$ be the sequences of oriented finite dimensional subspaces such that $Q_{n}y\to y$ in Y for every y and dist$(x,X_n)\to 0$ for every $x\in D$, where $Q_n:Y\to Y_n$ and $P_n:X\to X_n$ are sequences of continuous linear projections. The projection scheme $\mathrm{\Gamma }=\{X_n,Y_n,P_n,Q_n\}$ is then said to be admissible for maps from $D\subset X$ to Y. A map $T:D\subset X\to Y$ is called approximation-proper (abbreviated A-proper) at a point $y\in Y$ with respect to an admissible scheme Γ if $T_n\equiv Q_{n}T{|}_{D\cap X_n}$ is continuous for each $n\in \mathbb{N}$ and whenever $\{x_{n_j}:x_{n_j}\in D\cap X_{n_j}\}$ is bounded with $T_{n_j}{x}_{n_j}\to y$, then there exists a subsequence $\{x_{n_{j_k}}\}$ such that $x_{n_{j_k}}\to x\in D$ and $Tx=y$. T is simply called A-proper if it is A-proper at all points of Y. $L:domL\subset X\to Y$ is a Fredholm operator of index zero if ImL is closed and $dim\hspace{0.17em}\; KerL=codim\; \hspace{0.17em}ImL<\mathrm{\infty }$. As a consequence of this property, X and Y may be expressed as direct sums; $X=X_0⨁X_1$, $Y=Y_0\oplus Y_1$ with continuous linear projections $P:X\to KerL=X_0$ and $Q:Y\to Y_0$. The restriction of L to $domL\cap X_1$, denoted $L_1$, is a bijection onto $ImL=Y_1$ with continuous inverse $L_1^{-1}:Y_1\to domL\cap X_1$. Since $X_0$ and $Y_0$ have the same finite dimension, there exists a continuous bijection $J:Y_0\to X_0$. Let $H=L+J^{-1}P$, then $H:domL\subset X\to Y$ is a linear bijection with bounded inverse. Let K be a cone in a Banach space X. Then $K_1=H(K\cap domL)$ is a cone in Y. In [20], Petryshyn has shown that an admissible scheme ${\mathrm{\Gamma }}_L$ can be constructed such that L is A-proper with respect to ${\mathrm{\Gamma }}_L$. The following properties of the fixed point index ${ind}_K$ and two lemmas can be found in [1].

Proposition 1.1 Let $\mathrm{\Omega }\subset X$ be open and bounded and $\partial {\mathrm{\Omega }}_K=\partial \mathrm{\Omega }\cap K$. Assume that $Q_n{K}_1\subset K_1$, $P+JQN+L_1^{-1}(I-Q)N$ maps K to K, and $Lx\ne Nx$ on $\partial {\mathrm{\Omega }}_K$.

(P₁) (Existence property) If ${ind}_K([L,N],\mathrm{\Omega })\ne \{0\}$, then there exists $x\in {\mathrm{\Omega }}_K$ such that $Lx=Nx$.

(P₂) (Normality) If $x_0\in {\mathrm{\Omega }}_K$, then ${ind}_K([L,-J^{-1}P+{\hat{y}}_0],\mathrm{\Omega })=\{1\}$, where ${\hat{y}}_0=H{x}_0$ and ${\hat{y}}_0(y)=y_0$ for every $y\in H{\mathrm{\Omega }}_K$.

(P₃) (Additivity) If $Lx\ne Nx$ for $x\in {\overline{\mathrm{\Omega }}}_K\mathrm{\setminus }({\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2)$, where ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are disjoint relatively open subsets of ${\mathrm{\Omega }}_K$, then

$${ind}_K([L,N],\mathrm{\Omega })\subseteq {ind}_K([L,N],{\mathrm{\Omega }}_1)+{ind}_K([L,N],{\mathrm{\Omega }}_2)$$

with equality if either of indices on the right is a singleton.

(P₄) (Homotopy invariance) If $L-N(\lambda ,x)$ is an A-proper homotopy on ${\mathrm{\Omega }}_K$ for $\lambda \in [0,1]$ and $(N(\lambda ,x)+J^{-1}P)H^{-1}:K_1\to K_1$ and $\theta \notin (L-N(\lambda ,x))(domL\cap \partial {\mathrm{\Omega }}_K)$ for $\lambda \in [0,1]$, then ${ind}_K([L,N(\lambda ,x)],\mathrm{\Omega })={ind}_{K_1}(T_{\lambda },U)$ is independent of $\lambda \in [0,1]$, where $T_{\lambda }=(N(\lambda ,x)+J^{-1}P)H^{-1}$.

Lemma 1.1 If $L:domL\to Y$ is Fredholm of index zero, Ω is an open bounded set and ${\mathrm{\Omega }}_K\cap domL\ne \mathrm{\varnothing }$, $\theta \in \mathrm{\Omega }\subset X$. Let $L-\lambda N$ be A-proper for $\lambda \in [0,1]$. Assume that N is bounded and $P+JQN+L_1^{-1}(I-Q)N$ maps K to K. If $Lx\ne \mu Nx-(1-\mu )J^{-1}Px$ on $\partial {\mathrm{\Omega }}_K$ for $\mu \in [0,1]$, then

$${ind}_K([L,N],\mathrm{\Omega })=\{1\}.$$

Lemma 1.2 If $L:domL\to Y$ is Fredholm of index zero, Ω is an open bounded set and ${\mathrm{\Omega }}_K\cap domL\ne \mathrm{\varnothing }$. Let $L-\lambda N$ be A-proper for $\lambda \in [0,1]$. Assume that N is bounded and $P+JQN+L_1^{-1}(I-Q)N$ maps K to K. If there exists $e\in K_1\mathrm{\setminus }\{\theta \}$ such that

$$Lx-Nx\ne \mu e,$$

for every $x\in \partial {\mathrm{\Omega }}_K$ and all $\mu \ge 0$, then

$${ind}_K([L,N],\mathrm{\Omega })=\{0\}.$$

2 An abstract result

We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows.

Theorem 2.1 If $L:domL\to Y$ is Fredholm of index zero, let $L-\lambda N$ be A-proper for $\lambda \in [0,1]$. Assume that N is bounded and $P+JQN+L_1^{-1}(I-Q)N$ maps K to K. Suppose further that ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are two bounded open sets in X such that $\theta \in {\mathrm{\Omega }}_1\subset {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2$, ${\mathrm{\Omega }}_1\cap K\cap domL\ne \mathrm{\varnothing }$ and ${\mathrm{\Omega }}_2\cap K\cap domL\ne \mathrm{\varnothing }$. If one of the following two conditions is satisfied:

(C₁) $(P+JQN)x+L_1^{-1}(I-Q)Nx\ngeqq x$ for all $x\in \partial {\mathrm{\Omega }}_1\cap K$ and $(P+JQN)x+L_1^{-1}(I-Q)Nx\nleqq x$ for all $x\in \partial {\mathrm{\Omega }}_2\cap K$;

(C₂) $(P+JQN)x+L_1^{-1}(I-Q)Nx\nleqq x$ for all $x\in \partial {\mathrm{\Omega }}_1\cap K$ and $(P+JQN)x+L_1^{-1}(I-Q)Nx\ngeqq x$ for all $x\in \partial {\mathrm{\Omega }}_2\cap K$.

Then there exists $x\in ({\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)\cap K$ such that $Lx=Nx$.

Proof We assume that (C₁) is satisfied. First we show that

$$Lx\ne \mu Nx-(1-\mu )J^{-1}Px,\text{for any }x\in \partial {\mathrm{\Omega }}_1\cap K,\mu \in [0,1].$$

(2.1)

In fact, otherwise, there exist $x_1\in \partial {\mathrm{\Omega }}_1\cap K$ and ${\mu }_1\in [0,1]$ such that

$$L{x}_1={\mu }_{1}N{x}_1-(1-{\mu }_1)J^{-1}P{x}_1,$$

then we obtain

$$(L+J^{-1}P)x_1={\mu }_1(N+J^{-1}P)x_1.$$

Therefore,

$$\begin{array}{rcl}{x}_1& =& {\mu }_1{(L+J^{-1}P)}^{-1}(N+J^{-1}P)x_1\\ =& {\mu }_1[(P+JQN)x_1+L_1^{-1}(I-Q)N{x}_1]\\ \le & (P+JQN)x_1+L_1^{-1}(I-Q)N{x}_1,\end{array}$$

which contradicts condition (C₁). From (2.1) and Lemma 1.1, we have

$${ind}_K([L,N],{\mathrm{\Omega }}_1)=\{1\}.$$

(2.2)

Choosing an arbitrary $e\in K_1\mathrm{\setminus }\{\theta \}$, next we prove that

$$Lx-Nx\ne \mu e.$$

(2.3)

In fact, otherwise, there exist $x_2\in \partial {\mathrm{\Omega }}_2\cap K$ and ${\mu }_2\ge 0$ such that

$$L{x}_2-N{x}_2={\mu }_{2}e,$$

then we obtain

$$(L+J^{-1}P)x_2=(N+J^{-1}P)x_2+{\mu }_{2}e{\ge }_1(N+J^{-1}P)x_2,$$

in which the partial order is induced by the cone $K_1$ in Y. So,

$$x_2\ge {(L+J^{-1}P)}^{-1}(N+J^{-1}P)x_2=(P+JQN)x_2+L_1^{-1}(I-Q)N{x}_2,$$

which is a contradiction to condition (C₁). Hence (2.3) holds, and then by Lemma 1.2, we have

$${ind}_K([L,N],{\mathrm{\Omega }}_2)=\{0\}.$$

(2.4)

It follows therefore from (2.2), (2.4) and the additivity property (P₃) of Proposition 1.1 that

$$\begin{array}{rcl}{ind}_K([L,N],{\mathrm{\Omega }}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)& =& {ind}_K([L,N],{\mathrm{\Omega }}_2)-{ind}_K([L,N],{\mathrm{\Omega }}_1)\\ =& \{0\}-\{1\}\\ =& \{-1\}.\end{array}$$

(2.5)

Since the index is nonzero, the existence property (P₁) of Proposition 1.1 implies that there exists $x\in ({\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)\cap K$ such that $Lx=Nx$.

Similarly, when (C₂) is satisfied, instead of (2.2), (2.4) and (2.5), we have

$${ind}_K([L,N],{\mathrm{\Omega }}_1)=\{0\},{ind}_K([L,N],{\mathrm{\Omega }}_2)=\{1\},$$

and therefore

$${ind}_K([L,N],{\mathrm{\Omega }}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)=\{1\}.$$

Also, we can assert that there exists $x\in ({\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)\cap K$ such that $Lx=Nx$. □

3 Applications

3.1 First-order periodic boundary value problems

We consider the following first-order periodic boundary value problem:

$$\{\begin{array}{c}{x}^{\prime }(t)=f(t,x(t)),t\in (0,1),\hfill \\ x(0)=x(1),\hfill \end{array}$$

(3.1)

where $f:[0,1]\times [0,+\mathrm{\infty })\to \mathbb{R}$ is continuous and $f(0,x)=f(1,x)$ for all $x\in \mathbb{R}$.

Consider the Banach spaces $X=Y=C[0,1]$ endowed with the norm $\parallel x\parallel ={max}_{t\in [0,1]}|x(t)|$. Define the cone K in X by

$$K=\{x\in X:x(t)\ge 0,t\in [0,1]\}.$$

Let L be the linear operator from $domL\subset X$ to Y with

$$domL=\{x\in X:x^{\prime }\in C[0,1],x(0)=x(1)\},$$

and

$$Lx(t)=x^{\prime }(t),x\in domL,t\in [0,1].$$

Let us define $N:X\to Y$ by

$$Nx(t)=f(t,x(t)),t\in [0,1].$$

Then (3.1) is equivalent to the equation

$$Lx=Nx.$$

It is obvious that L is a Fredholm operator of index zero with

Next we define the projections $P:X\to X$, $Q:Y\to Y$ by

and the isomorphism $J:ImQ\to ImP$ as $Jy=y$. Note that for $y\in ImL$, the inverse operator

$$L_1^{-1}:ImL\to domL\cap KerP$$

of

$$L{|}_{domL\cap KerP}:domL\cap KerP\to ImL$$

is given by

$$\left(L_1^{-1}y\right)(t)={\int }_0^{1}K(t,s)y(s)ds,$$

where

$$K(t,s)=\{\begin{array}{cc}s+1,\hfill & 0\le s<t\le 1,\hfill \\ s,\hfill & 0\le t\le s\le 1.\hfill \end{array}$$

Set

$$G(t,s)=1+K(t,s)-{\int }_0^{1}K(t,s)ds.$$

We can verify that

$$G(t,s)=\{\begin{array}{cc}\frac{3}{2}-(t-s),\hfill & 0\le s<t\le 1,\hfill \\ \frac{1}{2}+(s-t),\hfill & 0\le t\le s\le 1,\hfill \end{array}$$

and

$$\frac{1}{2}\le G(t,s)\le \frac{3}{2},t,s\in [0,1].$$

To state the existence result, we introduce two conditions:

(H₁) $f(t,b)<0$ for all $t\in [0,1]$,

(H₂) $f(t,x)>0$ for all $(t,x)\in [0,1]\times [0,a]$.

Theorem 3.1 Assume that there exist two positive numbers $0<a<b$ such that (H₁), (H₂) and

(H₃) $f(t,x)\ge -\frac{2}{3}x$ for all $(t,x)\in [0,1]\times [0,b]$

hold. Then (3.1) has at least one positive periodic solution $x^{\ast }\in K$ with $a\le \parallel x^{\ast }\parallel \le b$.

Proof First, we note that L, as defined, is Fredholm of index zero, $L_1^{-1}$ is compact by the Arzela-Ascoli theorem and thus $L-\lambda N$ is A-proper for $\lambda \in [0,1]$ by [[20], Lemma 2(a)].

For each $x\in K$, then by condition (H₃),

Thus $(P+JQN+L_1^{-1}(I-Q)N)(K)\subset K$.

Let

$${\mathrm{\Omega }}_1=\{x\in X:\parallel x\parallel <a\},{\mathrm{\Omega }}_2=\{x\in X:\parallel x\parallel <b\}.$$

Clearly, ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are bounded open sets and

$$\theta \in {\mathrm{\Omega }}_1\subset {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2.$$

We now show that

$$(P+JQN)x+L_1^{-1}(I-Q)Nx\ngeqq x\text{for any }x\in \partial {\mathrm{\Omega }}_2\cap K.$$

(3.2)

In fact, if there exists $x_3\in \partial {\mathrm{\Omega }}_2\cap K$ such that

$$(P+JQN)x_3+L_1^{-1}(I-Q)N{x}_3\ge x_3.$$

Then

$$x_3^{\prime }(t)\le f(t,x_3(t)),t\in [0,1].$$

Let $t_1\in [0,1]$ be such that $x_3(t_1)=b$. Clearly, the function $x_3^2$ attains a maximum on $[0,1]$ at $t=t_1$. Therefore $2x_3(t_1)x_3^{\prime }(t_1)=0$. As a consequence,

$$0=2b{x}_3^{\prime }(t_1)\le 2bf(t_1,x_3(t_1))=2bf(t_1,b),$$

which is a contradiction to (H₁). Therefore (3.2) holds.

On the other hand, we claim that

$$(P+JQN)x+L_1^{-1}(I-Q)Nx\nleqq x\text{for any }x\in \partial {\mathrm{\Omega }}_1\cap K.$$

(3.3)

In fact, if not, there exists $x_4\in \partial {\mathrm{\Omega }}_1\cap K$ such that

$$(P+JQN)x_4+L_1^{-1}(I-Q)N{x}_4\le x_4.$$

For any $x_4\in \partial {\mathrm{\Omega }}_1\cap K$, we have $\parallel x_4\parallel =a$, then $0\le x_4(t)\le a$ for $t\in [0,1]$. By condition (H₂), we have

$$\begin{array}{rcl}{x}_4(t)& \ge & (P+JQN)x_4(t)+L_1^{-1}(I-Q)N{x}_4(t)\\ =& {\int }_0^1{x}_4(s)ds+{\int }_0^{1}G(t,s)f(s,x_4(s))ds\\ >& {\int }_0^1{x}_4(s)ds,\text{for any }t\in [0,1],\end{array}$$

which is a contradiction. As a result, (3.3) is verified.

It follows from (3.2), (3.3) and Theorem 2.1 that there exists $x^{\ast }\in K\cap ({\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)$ such that $L{x}^{\ast }=N{x}^{\ast }$ with $a\le \parallel x^{\ast }\parallel \le b$. □

Remark 3.1 In [18], the following condition is required instead of (H₂):

(H^∗) there exist $a\in (0,b)$, $t_0\in [0,1]$, $r\in (0,1]$, and continuous functions $g:[0,1]\to [0,\mathrm{\infty })$, $h:(0,a]\to [0,\mathrm{\infty })$ such that $f(t,x)\ge g(t)h(x)$ for all $t\in [0,1]$ and $x\in (0,a]$, $h(x)/x^r$ is nonincreasing on $(0,a]$ with

$$\frac{h(a)}{2^{r-1}}{\int }_0^{1}G(t_0,s)g(s)ds\ge a.$$

Obviously, our condition (H₂) is much weaker and less strict compared with (H^∗). Moreover, (H₂) is easier to check than (H^∗). So, our result generalizes and improves [[18], Theorem 5].

Remark 3.2 From the proof of Theorem 3.1, we can see that condition (H₂) can be replaced by one of the following two relatively weaker conditions:

(${\mathrm{H}}_2^{\ast }$) $f(t,x)\ge 0$ for all $(t,x)\in [0,1]\times [0,a]$ and $f(t,\cdot )$ is positive for almost everywhere on $[0,a]$.

(${\mathrm{H}}_2^{\ast \ast }$) ${lim}_{x\to 0^{+}}{min}_{t\in [0,1]}f(t,x)>0$.

Remark 3.3 Finally in this section, we note that conditions (H₁) and (H₂) can be replaced by the following asymptotic conditions:

(${\mathrm{H}}_1^{\prime }$) ${lim}_{x\to +\mathrm{\infty }}\frac{f(t,x)}{x}<0$ uniformly for t;

(${\mathrm{H}}_2^{\prime }$) ${lim}_{x\to 0^{+}}\frac{f(t,x)}{x}>0$ uniformly for t.

Example 3.1 Let the nonlinearity in (3.1) be

$$f(t,x)=c(t)x^{\alpha }+\mu d(t)x^{\beta }-kx,$$

where $0<\alpha <1<\beta$, $c(t),d(t)\in C[0,1]$ are positive 1-periodic functions, $k\in (0,2/3)$ and $\mu >0$ is a positive parameter. Then (3.1) has at least one positive 1-periodic solution for each $0<\mu <{\mu }^{\ast }$, here ${\mu }^{\ast }$ is some positive constant.

Proof We will apply Theorem 3.1 with $f(t,x)=c(t)x^{\alpha }+\mu d(t)x^{\beta }-kx$. Since $k\in (0,2/3)$, it is easy to see that (H₃) holds. Set

$$T(x)=\frac{kx-c^{\ast }{x}^{\alpha }}{d^{\ast }{x}^{\beta }},$$

where

$$c^{\ast }=\underset{t}{max}c(t),d^{\ast }=\underset{t}{max}d(t).$$

Since $0<\alpha <1<\beta$, we have

$$T\left(0^{+}\right)=-\mathrm{\infty },T(+\mathrm{\infty })=0.$$

One may easily see that there exists $b>0$ such that

$$T(b)=\frac{kb-c^{\ast }{b}^{\alpha }}{d^{\ast }{b}^{\beta }}=\underset{x>0}{sup}T(x)>0.$$

Let

$${\mu }^{\ast }=\frac{kb-c^{\ast }{b}^{\alpha }}{d^{\ast }{b}^{\beta }}.$$

Then, for each $\mu \in (0,{\mu }^{\ast })$, we have

$$\begin{array}{rcl}f(t,b)& =& c(t)b^{\alpha }+\mu d(t)b^{\beta }-kb\\ <& c^{\ast }{b}^{\alpha }+{\mu }^{\ast }{d}^{\ast }{b}^{\beta }-kb\\ =& 0,\end{array}$$

which implies that (H₁) holds.

On the other hand, we have

which implies that (${\mathrm{H}}_2^{\prime }$) holds. Now we have the desired result. □

3.2 Second-order periodic boundary value problems

Let $f:[0,1]\times [0,+\mathrm{\infty })\to \mathbb{R}$ be continuous and $f(0,x)=f(1,x)$ for all $x\in \mathbb{R}$. We will discuss the existence of positive solutions of the second-order periodic boundary value problem

$$\{\begin{array}{c}-x^{″}(t)=f(t,x),t\in (0,1),\hfill \\ x(0)=x(1),x^{\prime }(0)=x^{\prime }(1).\hfill \end{array}$$

(3.4)

Since some parts of the proof are in the same line as that of Theorem 3.1, we will outline the proof with the emphasis on the difference.

Let X, Y be Banach spaces and the cone K be as in Section 3.1. In this case, we may define

$$domL=\{x\in X:x^{″}\in C[0,1],x(0)=x(1),x^{\prime }(0)=x^{\prime }(1)\},$$

and let the linear operator $L:domL\to Y$ be defined by

$$Lx=-x^{″},\text{for }x\in domL.$$

Then L is Fredholm of index zero,

$$KerL=\{x\in domL:x(t)\equiv \mathrm{constants}\},$$

and

$$ImL=\{y\in Y:{\int }_0^{1}y(s)ds=0\}.$$

Define $N:X\to Y$ by

$$Nx(t)=f(t,x(t)).$$

Thus it is clear that (3.4) is equivalent to

$$Lx=Nx.$$

We use the same projections P, Q as in Section 3.1 and define the isomorphism $J:ImQ\to ImP$ as

$$Jy=\beta y,$$

where $\beta =\frac{1}{6}$. It is easy to verify that the inverse operator $L_1^{-1}:ImL\to domL\cap KerP$ of $L{|}_{domL\cap KerP}:domL\cap KerP\to ImL$ is

$$\left(L_1^{-1}y\right)(t)={\int }_0^1\mathrm{\Lambda }(t,s)y(s)ds,$$

where

$$\mathrm{\Lambda }(t,s)=\{\begin{array}{cc}\frac{s}{2}(1-2t+s),\hfill & 0\le s<t\le 1,\hfill \\ \frac{1}{2}(1-s)(2t-s),\hfill & 0\le t\le s\le 1.\hfill \end{array}$$

Set

$$H(t,s)=\frac{1}{6}+\mathrm{\Lambda }(t,s)-{\int }_0^1\mathrm{\Lambda }(t,s)ds.$$

We can verify that

$$H(t,s)=\{\begin{array}{cc}\frac{1}{4}+\frac{s}{2}(1-2t+s)+\frac{t^2}{2}-\frac{t}{2},\hfill & 0\le s<t\le 1,\hfill \\ \frac{1}{4}+\frac{1}{2}(1-s)(2t-s)+\frac{t^2}{2}+\frac{t}{2},\hfill & 0\le t\le s\le 1,\hfill \end{array}$$

and

$$\frac{1}{8}\le H(t,s)\le \frac{1}{4},t,s\in [0,1].$$

Theorem 3.2 Assume that there exist two positive numbers $0<a<b$ such that (H₁), (H₂) and

(H₄) $f(t,x)\ge -4x$ for all $(t,x)\in [0,1]\times [0,b]$

hold. Then (3.4) has at least one positive periodic solution $x^{\ast }\in K$ with $a\le \parallel x^{\ast }\parallel \le b$.

Proof It is again easy to show that $L-\lambda N$ is A-proper for $\lambda \in [0,1]$ by [[20], Lemma 2(a)].

For each $x\in K$, then by condition (H₄),

Thus $(P+JQN+L_1^{-1}(I-Q)N)(K)\subset K$.

Let

$${\mathrm{\Omega }}_3=\{x\in X:\parallel x\parallel <a\},{\mathrm{\Omega }}_4=\{x\in X:\parallel x\parallel <b\}.$$

Clearly, ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_4$ are bounded and open sets and

$$\theta \in {\mathrm{\Omega }}_3\subset {\overline{\mathrm{\Omega }}}_3\subset {\mathrm{\Omega }}_4.$$

Next, we show that

$$(P+JQN)x+L_1^{-1}(I-Q)Nx\ngeqq x,\text{for any }x\in \partial {\mathrm{\Omega }}_4\cap K.$$

(3.5)

On the contrary, suppose that there exists $x_5\in \partial {\mathrm{\Omega }}_4\cap K$ such that

$$(P+JQN)x_5+L_1^{-1}(I-Q)N{x}_5\ge x_5.$$

Then

$$-x_5^{″}(t)\le f(t,x_5(t)),t\in [0,1].$$

Let $t_2\in [0,1]$ such that $x_5(t_2)={max}_{t\in [0,1]}{x}_5(t)=b$. Using the boundary conditions, we have $t_2\in (0,1)$. In this case, $x_5^{\prime }(t_2)=0$, $x_5^{″}(t_2)\le 0$. This gives

$$0\le -x_5^{″}(t_2)\le f(t_2,x_5(t_2))=f(t_2,b),$$

which is a contradiction to condition (H₁). Therefore (3.5) holds.

Finally, similar to the proof of (3.3), it follows from condition (H₂) that

$$(P+JQN)x+L_1^{-1}(I-Q)Nx\nleqq x,\text{for any }x\in \partial {\mathrm{\Omega }}_3\cap K.$$

Consequently all conditions of Theorem 2.1 are satisfied. Therefore, there exists $x^{\ast }\in K\cap ({\overline{\mathrm{\Omega }}}_4\mathrm{\setminus }{\mathrm{\Omega }}_3)$ such that $L{x}^{\ast }=N{x}^{\ast }$ with $x^{\ast }\in K$ and $a\le \parallel x^{\ast }\parallel \le b$ and the assertion follows. □