Let E=C[0,1] be a Banach space with the maximum norm \parallel u\parallel ={max}_{0\le t\le 1}|u(t)| for u\in E. Define P=\{u\in E\mid u(t)\ge 0,t\in [0,1]\} and {B}_{r}=\{u\in E\mid \parallel u\parallel <r\}. Then *P* is a total cone in *E*, that is, E=\overline{P-P}. {P}^{\ast} denotes the dual cone of *P*, namely, {P}^{\ast}=\{g\in {E}^{\ast}\mid g(u)\ge 0,\text{for all}u\in P\}. Let {E}^{\ast} denote the dual space of *E*, then by Riesz representation theorem, {E}^{\ast} is given by

\begin{array}{rcl}{E}^{\ast}& =& \{v\mid v\text{is right continuous on}[0,1)\text{and is bounded variation on}[0,1]\\ \text{with}v(0)=0\}.\end{array}

We assume that the following condition holds throughout this paper.

(H_{1}) u(t)\equiv 0 *is the unique* {C}^{2} *solution of the linear boundary value problem*

\{\begin{array}{l}-(Lu)(t)=0,\phantom{\rule{1em}{0ex}}0<t<1,\\ (cos{\gamma}_{0})u(0)-(sin{\gamma}_{0}){u}^{\prime}(0)=0,\phantom{\rule{2em}{0ex}}(cos{\gamma}_{1})u(1)+(sin{\gamma}_{1}){u}^{\prime}(1)=0.\end{array}

Let \phi ,\psi \in {C}^{2}([0,1],{\mathbf{R}}^{+}) solve the following inhomogeneous boundary value problems, respectively:

\{\begin{array}{l}-(L\phi )(t)=0,\phantom{\rule{1em}{0ex}}0<t<1,\\ (cos{\gamma}_{0})\phi (0)-(sin{\gamma}_{0}){\phi}^{\prime}(0)=1,\\ (cos{\gamma}_{1})\phi (1)+(sin{\gamma}_{1}){\phi}^{\prime}(1)=0\end{array}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\{\begin{array}{l}-(L\psi )(t)=0,\phantom{\rule{1em}{0ex}}0<t<1,\\ (cos{\gamma}_{0})\psi (0)-(sin{\gamma}_{0}){\psi}^{\prime}(0)=0,\\ (cos{\gamma}_{1})\psi (1)+(sin{\gamma}_{1}){\psi}^{\prime}(1)=1.\end{array}

Let {\kappa}_{1}=1-{\int}_{0}^{1}\phi (\tau )\phantom{\rule{0.2em}{0ex}}d\alpha (\tau ), {\kappa}_{2}={\int}_{0}^{1}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\alpha (\tau ), {\kappa}_{3}={\int}_{0}^{1}\phi (\tau )\phantom{\rule{0.2em}{0ex}}d\beta (\tau ), {\kappa}_{4}=1-{\int}_{0}^{1}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\beta (\tau ).

(H_{2}) {\kappa}_{1}>0, {\kappa}_{4}>0, k={\kappa}_{1}{\kappa}_{4}-{\kappa}_{2}{\kappa}_{3}>0.

**Lemma 2.1** ([7])

*If* (H_{1}) *and* (H_{2}) *hold*, *then BVP* (1.1) *is equivalent to*

u(t)={\int}_{0}^{1}G(t,s)h(s)f(s,u(s))\phantom{\rule{0.2em}{0ex}}ds,

*where* G(t,s)\in C([0,1]\times [0,1],{\mathbf{R}}^{+}) *is the Green function for* (1.1).

Define an operator A:E\to E as follows:

(Au)(t)={\int}_{0}^{1}G(t,s)h(s)f(s,u(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}u\in E.

(2.1)

It is easy to show that A:E\to E is a completely continuous nonlinear operator, and if u\in E is a fixed point of *A*, then *u* is a solution of BVP (1.1) by Lemma 2.1.

For any u\in E, define a linear operator K:E\to E as follows:

(Ku)(t)={\int}_{0}^{1}G(t,s)h(s)u(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}u\in E.

(2.2)

It is easy to show that K:E\to E is a completely continuous nonlinear operator and K(P)\subset P holds. By [7], the spectral radius r(K) of *K* is positive. The Krein-Rutman theorem [21] asserts that there are \varphi \in P\setminus \{0\} and \omega \in {P}^{\ast}\setminus \{0\} corresponding to the first eigenvalue {\lambda}_{1}=1/r(K) of *K* such that

{\lambda}_{1}K\varphi =\varphi

(2.3)

and

{\lambda}_{1}{K}^{\ast}\omega =\omega ,\phantom{\rule{2em}{0ex}}\omega (1)=1.

(2.4)

Here {K}^{\ast}:{E}^{\ast}\to {E}^{\ast} is the dual operator of *K* given by:

\left({K}^{\ast}v\right)(s)={\int}_{0}^{s}{\int}_{0}^{1}G(t,\tau )h(\tau )\phantom{\rule{0.2em}{0ex}}dv(t)\phantom{\rule{0.2em}{0ex}}d\tau ,\phantom{\rule{1em}{0ex}}v\in {E}^{\ast}.

The representation of {K}^{\ast}, the continuity of *G* and the integrability of *h* imply that \omega \in {C}^{1}[0,1]. Let e(t):={\omega}^{\prime}(t). Then e\in P\setminus \{0\}, and (2.4) can be rewritten equivalently as

r(K)e(s)={\int}_{0}^{1}G(t,s)h(s)e(t)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{2em}{0ex}}{\int}_{0}^{1}e(t)\phantom{\rule{0.2em}{0ex}}dt=1.

(2.5)

**Lemma 2.2** ([7])

*If* (H_{1}) *holds*, *then there is* \delta >0 *such that* {P}_{0}=\{u\in P\mid {\int}_{0}^{1}u(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\ge \delta \parallel u\parallel \} *is a subcone of* *P* *and* K(P)\subset {P}_{0}.

**Lemma 2.3** ([22])

*Let* *E* *be a real Banach space and* \mathrm{\Omega}\subset E *be a bounded open set with* 0\in \mathrm{\Omega}. *Suppose that* A:\overline{\mathrm{\Omega}}\to E *is a completely continuous operator*. (1) *If there is* {y}_{0}\in E *with* {y}_{0}\ne 0 *such that* u\ne Au+\mu {y}_{0} *for all* u\in \partial \mathrm{\Omega} *and* \mu \ge 0, *then* deg(I-A,\mathrm{\Omega},0)=0. (2) *If* Au\ne \mu u *for all* u\in \partial \mathrm{\Omega} *and* \mu \ge 1, *then* deg(I-A,\mathrm{\Omega},0)=1. *Here* deg *stands for the Leray*-*Schauder topological degree in* *E*.

**Lemma 2.4** *Assume that* (H_{1}), (H_{2}) *and the following assumptions are satisfied*:

(C_{1}) *There exist* \varphi \in P\setminus \{0\}, \omega \in {P}^{\ast}\setminus \{0\} *and* \delta >0 *such that* (2.3), (2.4) *hold and* *K* *maps* *P* *into* {P}_{0}.

(C_{2}) *There exists a continuous operator* H:E\to P *such that*

\underset{\parallel u\parallel \to +\mathrm{\infty}}{lim}\frac{\parallel Hu\parallel}{\parallel u\parallel}=0.

(C_{3}) *There exist a bounded continuous operator* F:E\to E *and* {u}_{0}\in E *such that* Fu+{u}_{0}+Hu\in P *for all* u\in E.

(C_{4}) *There exist* {v}_{0}\in E *and* \zeta >0 *such that* KFu\ge {\lambda}_{1}(1+\zeta )Ku-KHu-{v}_{0} *for all* u\in E.

*Let* A=KF, *then there exists* R>0 *such that*

*where* {B}_{R}=\{u\in E\mid \parallel u\parallel <R\}.

*Proof* Choose a constant {L}_{0}={(\delta {\lambda}_{1})}^{-1}(1+{\zeta}^{-1})+\parallel K\parallel >0. From (C_{2}), for 0<{\epsilon}_{0}<{L}_{0}^{-1}, there exists {R}_{1}>0 such that \parallel u\parallel >{R}_{1} implies

\parallel Hu\parallel <{\epsilon}_{0}\parallel u\parallel .

(2.6)

Now we shall show

u\ne KFu+\mu \varphi \phantom{\rule{1em}{0ex}}\text{for any}u\in \partial {B}_{R}\text{and}\mu \ge 0,

(2.7)

provided that *R* is sufficiently large.

In fact, if (2.7) is not true, then there exist {u}_{1}\in \partial {B}_{R} and {\mu}_{1}\ge 0 satisfying

{u}_{1}=KF{u}_{1}+{\mu}_{1}\varphi .

(2.8)

Since \varphi \in P\setminus \{0\}, e(t)\in P\setminus \{0\}, {\int}_{0}^{1}\varphi (t)e(t)\phantom{\rule{0.2em}{0ex}}dt>0. Multiply (2.8) by e(t) on both sides and integrate on [0,1]. Then, by (C_{4}), (2.5), we get

\begin{array}{c}{\int}_{0}^{1}{u}_{1}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}={\int}_{0}^{1}(KF{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\mu}_{1}{\int}_{0}^{1}\varphi (t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\ge {\lambda}_{1}(1+\zeta ){\int}_{0}^{1}{\int}_{0}^{1}G(t,s)h(s){u}_{1}(s)\phantom{\rule{0.2em}{0ex}}dse(t)\phantom{\rule{0.2em}{0ex}}dt-{\int}_{0}^{1}(KH{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt-{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda}_{1}(1+\zeta ){\int}_{0}^{1}{\int}_{0}^{1}G(t,s)h(s){u}_{1}(s)e(t)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{2em}{0ex}}-{\int}_{0}^{1}{\int}_{0}^{1}G(t,s)h(s)(H{u}_{1})(s)e(t)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt-{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda}_{1}(1+\zeta ){\int}_{0}^{1}[{\int}_{0}^{1}G(t,s)h(s)e(t)\phantom{\rule{0.2em}{0ex}}dt]{u}_{1}(s)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int}_{0}^{1}[{\int}_{0}^{1}G(t,s)h(s)e(t)\phantom{\rule{0.2em}{0ex}}dt](H{u}_{1})(s)\phantom{\rule{0.2em}{0ex}}ds-{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda}_{1}(1+\zeta )r(K){\int}_{0}^{1}e(s){u}_{1}(s)\phantom{\rule{0.2em}{0ex}}ds-r(K){\int}_{0}^{1}(H{u}_{1})(s)e(s)\phantom{\rule{0.2em}{0ex}}ds-{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}=(1+\zeta ){\int}_{0}^{1}{u}_{1}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt-r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt-{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt.\hfill \end{array}

(2.9)

Thus,

{\int}_{0}^{1}{u}_{1}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\le {\zeta}^{-1}(r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt).

(2.10)

By (2.9), {\int}_{0}^{1}(KH{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt=r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt holds. Then (2.3), (2.6) and (2.10) imply

\begin{array}{c}{\int}_{0}^{1}({u}_{1}(t)+(KH{u}_{1})(t)+(K{u}_{0})(t))e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {\zeta}^{-1}(r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt)\hfill \\ \phantom{\rule{2em}{0ex}}+r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}(K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {\zeta}^{-1}(1+\zeta )r(K){\int}_{0}^{1}(H{u}_{1})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\zeta}^{-1}{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}(K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {\zeta}^{-1}(1+\zeta )r(K){\epsilon}_{0}\parallel {u}_{1}\parallel +{L}_{1},\hfill \end{array}

(2.11)

where {L}_{1}={\zeta}^{-1}{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}(K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt is a constant.

(C_{3}) shows F{u}_{1}+{u}_{0}+H{u}_{1}\in P and (C_{1}) implies {\mu}_{1}\varphi ={\mu}_{1}{\lambda}_{1}K{\phi}_{1}\in {P}_{0}. Then (C_{1}), (2.8) and Lemma 2.2 tell us that

{u}_{1}+KH{u}_{1}+K{u}_{0}=KF{u}_{1}+{\mu}_{1}\varphi +KH{u}_{1}+K{u}_{0}=K(F{u}_{1}+H{u}_{1}+{u}_{0})+{\mu}_{1}\varphi \in {P}_{0}.

The definition of {P}_{0} yields

\begin{array}{rcl}{\int}_{0}^{1}({u}_{1}+KH{u}_{1}+K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt& \ge & \delta \parallel {u}_{1}+KH{u}_{1}+K{u}_{0}\parallel \\ \ge & \delta \parallel {u}_{1}\parallel -\delta \parallel KH{u}_{1}\parallel -\delta \parallel K{u}_{0}\parallel .\end{array}

(2.12)

It follows from (2.6), (2.11) and (2.12) that

\begin{array}{rcl}\parallel {u}_{1}\parallel & =& {\delta}^{-1}{\int}_{0}^{1}({u}_{1}+KH{u}_{1}+K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+\parallel KH{u}_{1}\parallel +\parallel K{u}_{0}\parallel \\ \le & {\epsilon}_{0}{(\delta {\lambda}_{1})}^{-1}(1+{\zeta}^{-1})\parallel {u}_{1}\parallel +{L}_{1}{\delta}^{-1}+{\epsilon}_{0}\parallel K\parallel \cdot \parallel {u}_{1}\parallel +\parallel K{u}_{0}\parallel \\ =& {\epsilon}_{0}{L}_{0}\parallel {u}_{1}\parallel +{L}_{2},\end{array}

(2.13)

where {L}_{2}=\parallel K{u}_{0}\parallel +{L}_{1}{\delta}^{-1} is a constant.

Since 0<{\epsilon}_{0}{L}_{0}<1, then (2.13) deduces that (2.7) holds provided that *R* is sufficiently large such that R>max\{{L}_{2}/(1-{\epsilon}_{0}{L}_{0}),{R}_{1}\}. By (2.13) and Lemma 2.3, we have

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