© Wuhan University 2025

0 Introduction

The boundary value problem in which the boundary conditions involve parameters is one of the most significant problems in the mathematical theory. In 1994, Binding was the first to propose the Sturm Liouville problems with boundary conditions dependent on eigenparameters^[1]. In 1999, Hai^[2] used the prior estimation method to prove the existence of solutions for boundary value problem

$\{\begin{array}{l}{u}^{″}(t)+a(t)u^{\text{'}}(t)+f(u(t))=\mathrm{0},t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u(\mathrm{1})=\lambda \end{array}$

with boundray conditions having parameters. Since then, many different problems with parameters under boundary conditions have been studied respectively by Marinets et al^[3], and a great deal of fruitful results have been achieved, which can be found in Refs. [4-7]. Fonseka et al^[6-7] investigated the positive solution of the boundary value problem with two boundary conditions having parameters by employing the upper and lower solution method. They proved the existence and multiplicity of the positive solutions and obtained the bifurcation diagram of the above positive solutions by using the time mapping method. Particularly, Fonseka et al^[6] discussed the existence and multiplicity results of the steady-state reaction diffusion equation

$\{\begin{array}{l}-u^{″}(t)=\lambda h(t)f(u(t)),t\in (\mathrm{0,1}),\\ -\mathrm{d}{u}^{\text{'}}(\mathrm{0})+\mu (\lambda )u(\mathrm{0})=\mathrm{0},\\ u^{\text{'}}(\mathrm{1})+\mu (\lambda )u(\mathrm{1})=\mathrm{0}\end{array}$

via the upper and lower solution method in three cases of $f(\mathrm{0})=\mathrm{0}$, $f(\mathrm{0})>\mathrm{0}$ and $f(\mathrm{0})<\mathrm{0}$. Further, they established a unique result for $\lambda \approx \mathrm{0}$ and $\lambda \gg \mathrm{1}$. In the above equation, $\lambda >\mathrm{0}$ is a parameter, $f\in C^{\mathrm{2}}([\mathrm{0},\mathrm{\infty }),\mathbb{R})$ is an increasing function which is sublinear at infinity, $h\in$ $C^{\mathrm{1}}([\mathrm{0,1}],(\mathrm{0},\mathrm{\infty }))$ is a nonin-creasing function with $h_{\mathrm{1}}:=h(\mathrm{1})>\mathrm{0}$ and there exist constants $d_{\mathrm{0}}>\mathrm{0}$, $\alpha \in [\mathrm{0,1}]$ such that $h(t)\le \frac{d_{\mathrm{0}}}{t^{\alpha }}$ for all $t\in (\mathrm{0,1}]$, and $\mu \in C([\mathrm{0},\mathrm{\infty }),[\mathrm{0},\mathrm{\infty }))$ is an increasing function such that $\mu (\mathrm{0})\ge \mathrm{0}$.

The prescribed mean curvature equation addressed in this paper is regarded as a significant problem in partial differential equations and possesses extensive applications in other domains like physics and biology, such as the shape of the human cornea^[8-9], drops of capillary droopiness^[10], micro electronic mechanical systems, and corresponding models of large spatial gradients^[11]. Specifically, the Dirichlet problem with the Minkowski-curvature operator

$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u(t)}{\sqrt[]{\mathrm{1}-(\mathrm{1}-\nabla u{(t))}^{\mathrm{2}}}}\right)=f(t,u(t)),t\in \mathrm{\Omega },\\ u(\mathrm{0})=\mathrm{0},t\in \partial \mathrm{\Omega }\end{array}$

has drawn the attention of numerous scholars. Especially, the existence and multiplicity of positive solutions for the Dirichlet problem with the one-dimensional Minkowski-curvature operator

$\{\begin{array}{l}-{\left(\frac{u^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}=\lambda f(u(t)),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=u(\mathrm{1})=\mathrm{0}\end{array}$

have been extensively studied by employing the variational method, the upper and lower solution method, and the time mapping method. See Refs. [12-20], where $\lambda >\mathrm{0}$ is parameter, $f\in C([\mathrm{0},\mathrm{\infty }),[\mathrm{0},\mathrm{\infty }))$ and $f(u)>\mathrm{0}(u>\mathrm{0})$. It is obvious that the problems in the above references merely study the positive solutions of the Dirichlet problem with the mean curvature operator in Minkowski space in the case where the nonlinear term has a parameter. However, the prescribed mean curvature boundary value problem where both the nonlinear term and the boundary conditions have parameters must be further considered to accurately describe the corresponding physical phenomena.

Inspired by the above-mentioned papers, we explore the existence and multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem in Minkowski space

$\{\begin{array}{l}-{\left(\frac{u^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}=\lambda h(t)f(u(t)),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0},\end{array}$(1)

with boundary conditions having parameter in two cases $f(\mathrm{0})=\mathrm{0}$ and $f(\mathrm{0})>\mathrm{0}$ by using upper and lower solution method, where $\lambda >\mathrm{0}$ is a parameter, $f\in C^{\mathrm{2}}([\mathrm{0},\mathrm{\infty }),\mathbb{R})$ is monotonically increasing and $\underset{u\to {\mathrm{1}}^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u)}{\mathrm{1}-u}=\mathrm{0}$,$h\in$ $C^{\mathrm{1}}([\mathrm{0,1}],(\mathrm{0},\mathrm{\infty }))$ is a nonincreasing function and $h(t)>\mathrm{1}$.

From the conclusion of Ref. [15], it follows that problem (1) has a positive solution $u$ if and only if the problem

$\{\begin{array}{l}-u^{″}(t)=\lambda (\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}{)}^{\frac{\mathrm{3}}{\mathrm{2}}}h(t)f(u(t)),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0}\end{array}$(2)

has a positive solution, and $\left|u^{\text{'}}\right|<\mathrm{1}$, ${\Vert u\Vert }_{\mathrm{\infty }}<\mathrm{1}$.

Let $\varphi :(-\mathrm{1,1})\to \mathbb{R}$ is monotone increasing homeomorphism defined by $\varphi (s)=\frac{s}{\sqrt[]{\mathrm{1}-s^{\mathrm{2}}}}$, then $\varphi (\mathrm{0})=\mathrm{0}$ and ${\varphi }^{-\mathrm{1}}(s)=\frac{s}{\sqrt[]{\mathrm{1}+s^{\mathrm{2}}}}$ is also monotone increasing homeomorphism and bounded. Let us consider the eigenvalue problem

$\{\begin{array}{l}-u^{″}(t)=\lambda u(t),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0}.\end{array}$(3)

Based on the results of Ref. [6], the problem (3) has principal eigenvalue ${\lambda }_{\mathrm{1}}>\mathrm{0}$, and the eigencurve $B_{\mathrm{1}}(\lambda )$ is Lipschitz continuous, strictly decreasing and convex. Further $\underset{\lambda \to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\lambda }_{\mathrm{1}}={\lambda }_{\mathrm{1}D}$, where ${\lambda }_{\mathrm{1}D}>\mathrm{0}$ is the principal eigenvalue of

$\{\begin{array}{l}-u^{″}(t)={\lambda }_{\mathrm{1}D}u(t),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0}=u(\mathrm{1}).\end{array}$

Throughout the paper we will assume that $f$ satisfies: $({\mathrm{C}}_{\mathrm{1}})$$f^{\text{'}}(\mathrm{0})=\mathrm{1}$; $({\mathrm{C}}_{\mathrm{2}})$$f^{″}(s)<\mathrm{0},s\in [\mathrm{0},{\sigma }^{\mathrm{*}})$ for some ${\sigma }^{\mathrm{*}}>\mathrm{0}$.

Let $a>\mathrm{0},b>\mathrm{0}$ and $M^{\mathrm{*}}>\mathrm{0}$. Define

$L(a,b)=\frac{\mathrm{4}b/(\tau (\frac{\mathrm{1}}{\mathrm{4}})f(b))}{\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{a}{{\Vert v_h\Vert }_{\mathrm{\infty }}f(a)},\frac{\mathrm{8}{M}^{\mathrm{*}}}{f(b)}\right\}},$

where $v_h$ is the solution of the problem

$\{\begin{array}{l}-v_h^{″}(t)=h(t),t\in (\mathrm{0,1}),\\ v_h(\mathrm{0})=\mathrm{0},v_h^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }{v}_h^{\text{'}}(\mathrm{1})=\mathrm{0}.\end{array}$(4)

First, we state the main results for the case: $f(\mathrm{0})=\mathrm{0}$.

Theorem 1   Assume $({\mathrm{C}}_{\mathrm{1}})$ and $f(\mathrm{0})=\mathrm{0}$ hold. Then problem (1) has no positive solution for $\lambda \approx \mathrm{0}$ and has at least one positive solution for $\lambda >{\lambda }_{\mathrm{1}}$. Further, if there exist $a>\mathrm{0},b>\mathrm{0},M^{\mathrm{*}}>\mathrm{0}$ such that $a\in (\mathrm{0},b),$ $M^{\mathrm{*}}>\mathrm{2}b,$ $L(a,b)<\mathrm{1}$ and $\frac{\mathrm{4}b}{\tau (\frac{\mathrm{1}}{\mathrm{4}})f(b)}>{\lambda }_{\mathrm{1}}$. Then problem (1) has at least three positive solutions for $\lambda \in \left(\frac{\mathrm{4}b}{\tau (\frac{\mathrm{1}}{\mathrm{4}})f(b)},\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{a}{{\Vert v_h\Vert }_{\mathrm{\infty }}f(a)},\frac{\mathrm{8}{M}^{\mathrm{*}}}{f(b)}\right\}\right)$.

Theorem 2   Assume $({\mathrm{C}}_{\mathrm{1}})$,$({\mathrm{C}}_{\mathrm{2}})$ and $f(\mathrm{0})=\mathrm{0}$ hold. Then problem (1) has least one positive solution $u_{\lambda }$ for $\lambda >{\lambda }_{\mathrm{1}}$ such that ${\Vert u_{\lambda }\Vert }_{\mathrm{\infty }}\to \mathrm{0}$.

Next, we state the main results for case: $f(\mathrm{0})>\mathrm{0}$.

Theorem 3   Assume $f(\mathrm{0})>\mathrm{0}$. Then problem (1) as at least one positive solution for $\lambda >\mathrm{0}$. Further, if there exist $a>\mathrm{0}$, $b>\mathrm{0}$ and $M^{\mathrm{*}}>\mathrm{0}$ such that $a\in (\mathrm{0},b)$, $M^{\mathrm{*}}>\mathrm{2}b$ and $L(a,b)<\mathrm{1}$. Then (1) has at least three positive solutions for $\lambda \in \left(\frac{\mathrm{4}b}{\tau (\frac{\mathrm{1}}{\mathrm{4}})f(b)},\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{a}{{\Vert v_h\Vert }_{\mathrm{\infty }}f(a)},\frac{\mathrm{8}{M}^{\mathrm{*}}}{f(b)}\right\}\right).$

Note that the connected component of positive solutions of (1) is the $\mathrm{\Sigma }$ shaped under the assumptions of Theorem 1 or Theorem 3. Figure 1(a) illustrated the main results of Theorem 1, Fig.1 (b) illustrated the main results of Theorem 3. The rest of the paper is organized as follows. In Section 1, we introduce some lemmas needed to prove the main theorems. In Section 2, we use the time-mapping method to prove that problem $(\mathrm{1})$ has at least one positive solution when $f(u)=u,h(t)=\mathrm{1}$, which will help to construct the subsolution of $(\mathrm{1})$. In Section 3, we give proofs of the main results. In Section 4, we show some examples.

Fig. 1 Bifurcation diagram for problem (1)

1 Preliminaries

Let $C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]=\left\{u\in C^{\mathrm{1}}[\mathrm{0,1}]|u(\mathrm{0})=\mathrm{0}\right\}$, it is not hard to verify that $C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]$ endowed with the norm $\Vert u\Vert ={\Vert u\Vert }_{\mathrm{\infty }}+{\Vert u\text{'}\Vert }_{\mathrm{\infty }}$ is Banach space.

We donote by $P,Q:C[\mathrm{0,1}]\to C[\mathrm{0,1}]$, and the continuous projects defined by $Pu(t)=u^{\text{'}}(\mathrm{0}),$ $Qu(t)={\int }_{\mathrm{0}}^{\mathrm{1}}u(t)\mathrm{d}t$, $t\in (\mathrm{0,1})$ and define the continuous linear operator $H:C[\mathrm{0,1}]\to C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]$, $Hu(t)={\int }_{\mathrm{0}}^{t}u(s)\mathrm{d}s$, $t\in (\mathrm{0,1})$. Integration of both sides of the equation in problem (1) from $\mathrm{0}$ to $t$ implies that

$\frac{u^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}}}=\frac{u^{\text{'}}(\mathrm{0})}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(\mathrm{0}))}^{\mathrm{2}}}}-\lambda {\int }_{\mathrm{0}}^{\mathrm{1}}h(s)f(u(s))\mathrm{d}s,$

i.e.

$\varphi (u^{\text{'}}(t))=\varphi (u^{\text{'}}(\mathrm{0}))-\lambda {\int }_{\mathrm{0}}^{t}h(s)f(u(s))\mathrm{d}s.$

Applying ${\varphi }^{-\mathrm{1}}$ to the above equation, we get that $u^{\text{'}}(t)={\varphi }^{-\mathrm{1}}[\varphi (u^{\text{'}}(\mathrm{0}))-\lambda {\int }_{\mathrm{0}}^{t}h(s)f(u(s))ds]$. Furthermore, we integrate the above equation from $\mathrm{0}$ to $t$, it follows that

$u(t)={\int }_{\mathrm{0}}^t{\varphi }^{-\mathrm{1}}[\varphi (u^{\text{'}}(\mathrm{0}))-\lambda {\int }_{\mathrm{0}}^{t}h(s)f(u(\tau ))\mathrm{d}\tau ]\mathrm{d}s.$(5)

Combining $u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0}$, we obtain that $u^{\text{'}}(\mathrm{0})$ satisfies

${\varphi }^{-\mathrm{1}}[\varphi (u^{\text{'}}(\mathrm{0}))-\lambda {\int }_{\mathrm{0}}^{t}h(s)f(u(s))ds]+\sqrt[]{\lambda }{\int }_{\mathrm{0}}^{\mathrm{1}}{\varphi }^{-\mathrm{1}}[\varphi (u^{\text{'}}(\mathrm{0}))-\lambda {\int }_{\mathrm{0}}^{t}h(s)f(u(\tau ))\mathrm{d}\tau ]\mathrm{d}s=\mathrm{0}$(6)

which yields that

$\begin{array}{l}{\varphi }^{-\mathrm{1}}[\varphi (Pu(t))-\lambda Q(h(t)f(u(t)))]\\ +\sqrt[]{\lambda }Q[{\varphi }^{-\mathrm{1}}(\varphi (Pu(t))-\lambda H(h(t)f(u(t))))]=\mathrm{0}\end{array}$(7)

Lemma 1   For $h\in C[\mathrm{0,1}]$, there is a unique $\gamma :=u^{\text{'}}(\mathrm{0})$ such that ${\varphi }^{-\mathrm{1}}[\varphi (\gamma )-\lambda Q(h(t)f(u(t)))]+\sqrt[]{\lambda }Q[{\varphi }^{-\mathrm{1}}(\varphi (\gamma )$ $-\lambda H(h(t)f(u(t))))]=\mathrm{0}$ and $\gamma$ is continuous.

Proof   For any given $\lambda >\mathrm{0}$, define the function $g:(-\mathrm{1,1})\to \mathbb{R}$ as follows:

$\begin{array}{l}g(\gamma )={\varphi }^{-\mathrm{1}}[\varphi (\gamma )-\lambda Q(h(t)f(u(t)))]\\ +\sqrt[]{\lambda }Q[{\varphi }^{-\mathrm{1}}(\varphi (\gamma )-\lambda H(h(t)f(u(t))))]\end{array}$

There are ${\gamma }_{\mathrm{1}},{\gamma }_{\mathrm{2}}\in (-\mathrm{1,1})$ such that

$\begin{array}{l}g({\gamma }_{\mathrm{1}})=\underset{\gamma \in (\mathrm{0,1})}{\mathrm{m}\mathrm{i}\mathrm{n}}g(\gamma )=-\mathrm{1}-\sqrt[]{\lambda }<\mathrm{0},\\ g({\gamma }_{\mathrm{2}})=\underset{\gamma \in (\mathrm{0,1})}{\mathrm{m}\mathrm{i}\mathrm{n}}g(\gamma )=\mathrm{1}+\sqrt[]{\lambda }>\mathrm{0},\end{array}$

then there exists $\gamma \in ({\gamma }_{\mathrm{1}},{\gamma }_{\mathrm{2}})$ such that $g(\gamma )=\mathrm{0}$.

Next we prove uniqueness of $\gamma$. Let ${\gamma }_{\mathrm{*}},{\gamma }^{\mathrm{*}}\in (-\mathrm{1,1})$ such that $g({\gamma }_{\mathrm{*}})=g({\gamma }^{\mathrm{*}})=\mathrm{0}$, hence there exists $t_{\mathrm{0}}\in (\mathrm{0,1})$ such that

$\begin{array}{l}{\varphi }^{-\mathrm{1}}[\varphi ({\gamma }_{\mathrm{*}})-\lambda Q(h(t_{\mathrm{0}})f(u(t_{\mathrm{0}})))]\\ +\sqrt[]{\lambda }Q[{\varphi }^{-\mathrm{1}}(\varphi ({\gamma }_{\mathrm{*}})-\lambda H(h(t_{\mathrm{0}})f(u(t_{\mathrm{0}}))))]\\ ={\varphi }^{-\mathrm{1}}[\varphi ({\gamma }^{\mathrm{*}})-\lambda Q(h(t_{\mathrm{0}})f(u(t_{\mathrm{0}})))]\\ +\sqrt[]{\lambda }Q[{\varphi }^{-\mathrm{1}}(\varphi ({\gamma }^{\mathrm{*}})-\lambda H(h(t_{\mathrm{0}})f(u(t_{\mathrm{0}}))))].\end{array}$

Since ${\varphi }^{-\mathrm{1}}$ and $Q$ is bijection, ${\gamma }_{\mathrm{*}}={\gamma }^{\mathrm{*}}$. Finally, from the continuity of $h$, $\varphi$ and ${\varphi }^{-\mathrm{1}}$, we can conclude that $\gamma$ is continuous.

Lemma 2   $u\in C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]$ being a solution to problem (1) is equivalent to $u\in C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]$ being a fixed point of

$A_f(u):=H{\varphi }^{-\mathrm{1}}[\varphi (Pu)-\lambda H(hf(u))]$

Proof   It follows from Lemma 1 that there exists a unique $u^{\text{'}}(\mathrm{0})$ such that (7) holds, so the operator equation of problem (1) is $A_f(u):=H{\varphi }^{-\mathrm{1}}[\varphi (Pu)-\lambda H(hf(u))]$. It is obvious that $\varphi :(-\mathrm{1,1})\to \mathbb{R}$, hence its inverse mapping ${\varphi }^{-\mathrm{1}}$ is a bounded operator, and it follows from (5) that ${\Vert u\Vert }_{\mathrm{\infty }}<\mathrm{1}$.

Next, we introduce definitions of a (strict) subsolution and a (strict) supersolution of problem (1) and establish the upper and lower solution theorem that is used to prove existence and multiplicity results of (1).

By a supersolution of the problem (1) we define $\alpha \in C^{\mathrm{2}}(\mathrm{0,1})\bigcap C^{\mathrm{1}}[\mathrm{0,1}]$ that satisfies

$\{\begin{array}{l}-{\left(\frac{{\alpha }^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-({\alpha }^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}\le \lambda h(t)f(\alpha (t)),t\in (\mathrm{0,1}),\\ \alpha (\mathrm{0})\le \mathrm{0},{\alpha }^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }\alpha (\mathrm{1})\le \mathrm{0}.\end{array}$(8)

By a supersolution of the problem (1), we define $\beta \in$ $C^{\mathrm{2}}(\mathrm{0,1})\bigcap C^{\mathrm{1}}[\mathrm{0,1}]$ that satisfies

$\{\begin{array}{l}-{\left(\frac{{\beta }^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-({\beta }^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}\ge \lambda h(t)f(\beta (t)),t\in (\mathrm{0,1}),\\ \beta (\mathrm{0})\ge \mathrm{0},{\beta }^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }\beta (\mathrm{1})\ge \mathrm{0}.\end{array}$(9)

By a strict subsolution (supersolution) of (1) we mean a subsolution (supersolution) which is not a solution of (1).

Lemma 3   Let $M>\mathrm{0}$ and $v_{\mathrm{1}},v_{\mathrm{2}}\in C^{\mathrm{1}}[\mathrm{0,1}]$ such that $\varphi (v_i^{\text{'}})\in C^{\mathrm{1}}[\mathrm{0,1}](i=\mathrm{1,2})$. If

$-(\varphi (v_{\mathrm{1}}^{\text{'}}{(t)))}^{\text{'}}+M{v}_{\mathrm{1}}(t)\le -(\varphi (v_{\mathrm{2}}^{\text{'}}{(t)))}^{\text{'}}+M{v}_{\mathrm{2}}(t),\forall t\in (\mathrm{0,1}),$(10)

and $v_{\mathrm{1}}(\mathrm{0})=\mathrm{0}$, $v_{\mathrm{2}}(\mathrm{0})=\mathrm{0}$, then $v_{\mathrm{1}}(t)\le v_{\mathrm{2}}(t)$.

Proof   Suppose on the contrary that there exists $t_{\mathrm{*}}\in (\mathrm{0,1})$ such that $\underset{t\in (\mathrm{0,1})}{\mathrm{m}\mathrm{a}\mathrm{x}}(v_{\mathrm{1}}(t)-v_{\mathrm{2}}(t))=v_{\mathrm{1}}(t_{\mathrm{*}})-v_{\mathrm{2}}(t_{\mathrm{*}})>\mathrm{0}$, then $v_{\mathrm{1}}^{\text{'}}(t_{\mathrm{*}})=v_{\mathrm{2}}^{\text{'}}(t_{\mathrm{*}})$, and there exists a sequence $\left\{t_k\right\}\to t_{\mathrm{*}}$ on $(\mathrm{0},t_{\mathrm{*}})$ such that $v_{\mathrm{1}}^{\text{'}}(t_k)-v_{\mathrm{2}}^{\text{'}}(t_k)\ge \mathrm{0}$. The fact that $\varphi$ is monotone increasing homeomorphism implies that $\varphi (v_{\mathrm{1}}^{\text{'}}(t_k))-\varphi (v_{\mathrm{1}}^{\text{'}}(t_{\mathrm{*}}))\ge \varphi (v_{\mathrm{2}}^{\text{'}}(t_k))-\varphi (v_{\mathrm{2}}^{\text{'}}(t_{\mathrm{*}}))$.

By the definition of the derivative we get that

$(\varphi (v_{\mathrm{1}}^{\text{'}}(t_k{)))}_{t=t_{\mathrm{*}}}^{\text{'}}\le (\varphi (v_{\mathrm{1}}^{\text{'}}(t_{\mathrm{*}}{)))}_{t=t_{\mathrm{*}}}^{\text{'}}.$(11)

This together with (10) and (11) concludes that $\mathrm{0}<M(v_{\mathrm{1}}(t_{\mathrm{*}})-v_{\mathrm{2}}(t_{\mathrm{*}}))\le (\varphi (v_{\mathrm{1}}^{\text{'}}{(t)))}_{t=t_{\mathrm{*}}}^{\text{'}}-(\varphi (v_{\mathrm{2}}^{\text{'}}{(t)))}_{t=t_{\mathrm{*}}}^{\text{'}}\le \mathrm{0}$, which is contradictory. Therefore, $v_{\mathrm{1}}(t)\le v_{\mathrm{2}}(t)$ for all $t\in (\mathrm{0,1})$.

Corollary 1   Let $v_{\mathrm{1}},v_{\mathrm{2}}\in C^{\mathrm{1}}[\mathrm{0,1}]$ such that $\varphi (v_i)\in$ $C^{\mathrm{1}}[\mathrm{0,1}](i=\mathrm{1,2})$.If $-(\varphi (v_{\mathrm{1}}^{\text{'}}{(t)))}^{\text{'}}<-(\varphi (v_{\mathrm{2}}^{\text{'}}{(t)))}^{\text{'}}$ for any $t\in (\mathrm{0,1})$, then $v_{\mathrm{1}}(t)<v_{\mathrm{2}}(t)$.

Lemma 4   Let $\alpha$ and $\beta$ be a subsolution and a supersolution of (1), respectively, such that $\alpha \le \beta$, then (1) has a solution $u\in C^{\mathrm{2}}(\mathrm{0,1})\bigcap C^{\mathrm{1}}[\mathrm{0,1}]$ such that $u\in [\alpha ,\beta ]$.

Proof   Let $\chi :\mathbb{R}\to \mathbb{R}$ be the continuous function which is defined by

$\chi (u)=\{\begin{array}{l}\alpha (t),u(t)<\alpha (t),\\ u(t),\alpha (t)\le u(t)\le \beta (t),\\ \beta (t),u(t)>\beta (t),\end{array}$

and define $F:[\mathrm{0,1}]\times \mathbb{R}\to \mathbb{R}$ by $F(t,u)=-\lambda h(t)f(\chi (u))$. Obviously, $F$ is continuous and bounded.

Next we consider the auxiliary problem

$\{\begin{array}{l}(\varphi (u^{\text{'}}{(t)))}^{\text{'}}=F(t,u(t))+u(t)-\chi (u(t)),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0}.\end{array}$(12)

First, the problem (12) has at least one solution $u$ by the Schauder fixed theorem. Next, we only show that $\alpha (t)\le u(t)\le \beta (t)$, $t\in (\mathrm{0,1})$, so $u$ is a solution of (1).

Suppose on the contrary that there exists $\overline{t}\in (\mathrm{0,1})$ such that $\underset{t\in (\mathrm{0,1})}{\mathrm{m}\mathrm{a}\mathrm{x}}[\alpha (t)-u(t)]=\alpha (\overline{t})-u(\overline{t})$>0. Because ${\alpha }^{\text{'}}(\overline{t})-u^{\text{'}}(\overline{t})=\mathrm{0}$, there exist two sequence $\left\{t_k\right\}\in [\overline{t}-\epsilon ,\overline{t})$ and $\left\{{\tilde{t}}_k\right\}\in [\overline{t},\overline{t}+\epsilon )$ such that ${\alpha }^{\text{'}}(t_k)-u^{\text{'}}(t_k)\ge \mathrm{0}$ and ${\alpha }^{\text{'}}({\tilde{t}}_k)-u^{\text{'}}({\tilde{t}}_k)\le \mathrm{0}$. Without loss of generality, it follows from ${\alpha }^{\text{'}}(t_k)-u^{\text{'}}(t_k)\ge \mathrm{0}$ that ${\alpha }^{\text{'}}(t_k)\ge u^{\text{'}}(t_k)$. By $\varphi$ is an increasing homeomorphism, we get that $\varphi ({\alpha }^{\text{'}}(t_k))\ge \varphi (u^{\text{'}}(t_k))$.

$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\varphi ({\alpha }^{\text{'}}(\overline{t}))=\varphi (u^{\text{'}}(\overline{t})),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$

$\frac{\varphi ({\alpha }^{\text{'}}(t_k))-\varphi ({\alpha }^{\text{'}}(t))}{t_k-\overline{t}}\le \frac{\varphi (u^{\text{'}}(t_k))-\varphi (u^{\text{'}}(\overline{t}))}{t_k-\overline{t}}(t_k<\overline{t}),\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}(\varphi ({\alpha }^{\text{'}}{(\overline{t})))}^{\text{'}}\le (\varphi (u^{\text{'}}{(\overline{t})))}^{\text{'}}.$

Moreover, $\alpha$ is a subsolution of (1), it yields that

$\begin{array}{l}(\varphi ({\alpha }^{\text{'}}{(\overline{t})))}^{\text{'}}\le (\varphi (u^{\text{'}}{(\overline{t})))}^{\text{'}}=F(\overline{t},u(\overline{t}))+u(\overline{t})+\alpha (\overline{t})\\ <F(\overline{t},u(\overline{t}))=-\lambda h(t)f(\chi (u(\overline{t})))=-\lambda h(t)f(\alpha (\overline{t}))\le (\varphi ({\alpha }^{\text{'}}{(\overline{t})))}^{\text{'}}.\end{array}$

This is a contradiction, thus $\alpha (t)\le u(t)$. In addition, by the similar arguments, it follows that $u(t)\le \beta (t)$. Therefore $u\in [\alpha ,\beta ]$ is a solution of problem (1).

Lemma 5   Let ${\alpha }_{\mathrm{1}}$ and ${\beta }_{\mathrm{1}}$ be a subsolution and supersolution of (1) respectively such that ${\alpha }_{\mathrm{1}}\le {\beta }_{\mathrm{1}}$. Let ${\alpha }_{\mathrm{2}}$ and ${\beta }_{\mathrm{2}}$ be a strict subsoltion and a strict supersolution of (1) respectively satisfying ${\alpha }_{\mathrm{2}},{\beta }_{\mathrm{2}}\in [{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}]$ and ${\alpha }_{\mathrm{2}}\cancel{\le }{\beta }_{\mathrm{2}}$. Then (1) has at least three solutions $u_{\mathrm{1}}$, $u_{\mathrm{2}}$ and $u_{\mathrm{3}}$, where $u_{\mathrm{1}}\in [{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{2}}]$, $u_{\mathrm{2}}\in [{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}}]$ and $u_{\mathrm{3}}\in [{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}]\backslash ([{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{2}}]\bigcup [{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}}])$.

Proof   Let $e\in C^{\mathrm{2}}[\mathrm{0,1}]$ denote the the unique positive solution of

$\{\begin{array}{l}-{\left(\frac{e^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(e^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}=\mathrm{1},t\in (\mathrm{0,1}),\\ e(\mathrm{0})=e(\mathrm{1})=\mathrm{0}.\end{array}$

Let $I=[{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}]$, $I_{\mathrm{1}}=[{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{2}}]$, $I_{\mathrm{3}}=[{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}}]$, $C_e[\mathrm{0,1}]=\left\{u\in C[\mathrm{0,1}]|-te\le u\le te\right\}$. It is not difficult to verify that $C_e[\mathrm{0,1}]$ endowed with the norm ${\Vert u\Vert }_e=\mathrm{i}\mathrm{n}\mathrm{f}\left\{t>\mathrm{0}|-te\le u\le te\right\}$ is Banach space, then $I$, $I_{\mathrm{1}}$ and $I_{\mathrm{2}}$ are non-empty closed convex subsets of Banach space $C_e[\mathrm{0,1}]$. In the following, we will prove that $A_f(u):C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]\to C_{\mathrm{0}}^{\mathrm{1}}[\mathrm{0,1}]$ is completely continuous and strongly increasing.

In fact, $I_{\mathrm{1}}$ and $I_{\mathrm{2}}$ are disjoint subsets of $I$. The map $A_f(u)$ is restricted to $I$. Since $A_f(u)$ is increasing and ${\alpha }_{\mathrm{1}}$, ${\beta }_{\mathrm{2}}$ are subsolution and supersolution of (1) respectively (${\alpha }_{\mathrm{1}}\le {\beta }_{\mathrm{2}}$), we have that ${\alpha }_{\mathrm{1}}\le A_f({\alpha }_{\mathrm{1}})\le A_f({\beta }_{\mathrm{2}})\le {\beta }_{\mathrm{2}}$, i.e. $A_f(u)\subset I$. Similarly, $A_f(I_k)\subset I_k,k=\mathrm{1,2}$. Since ${\beta }_{\mathrm{2}}$ is a strict supersolution of (2), we have that $A_f({\beta }_{\mathrm{2}})<{\beta }_{\mathrm{2}}$. By Ref. [21], Corollary 6.2, $A_f(u)$ has a maximal fixed point in $u_{\mathrm{1}}\in I_{\mathrm{1}}$ and ${\alpha }_{\mathrm{1}}\le u_{\mathrm{1}}<{\beta }_{\mathrm{2}}$. Similarly, $A_f(u)$ has a minimal fixed point in $u_{\mathrm{2}}\in I_{\mathrm{2}}$ and ${\alpha }_{\mathrm{2}}<u_{\mathrm{2}}\le {\beta }_{\mathrm{1}}$, because ${\alpha }_{\mathrm{2}}$ is a strict subsoluton of (1).

Notice that

$-(\varphi (u_{\mathrm{1}}^{\text{'}}{(t)))}^{\text{'}}+M{u}_{\mathrm{1}}(t)\le -(\varphi ({\beta }_{\mathrm{2}}^{\text{'}}{(t)))}^{\text{'}}+M{\beta }_{\mathrm{2}}(t)(M>\mathrm{0})$

it follows from Lemma 3 that $u_{\mathrm{1}}\le {\beta }_{\mathrm{2}}$, hence there exists a constant $c_{\mathrm{1}}>\mathrm{0}$ such that ${\beta }_{\mathrm{2}}-u_{\mathrm{1}}\ge c_{\mathrm{1}}e(t)$. Similarly there exists a constant $c_{\mathrm{2}}>\mathrm{0}$ such that $u_{\mathrm{2}}-{\alpha }_{\mathrm{2}}\ge c_{\mathrm{2}}e(t)$. Define

$J_k=I\bigcap \left\{z\in C_e[\mathrm{0,1}]|{\Vert z-u_k\Vert }_e<t_k,k=\mathrm{1,2}\right\}$

then $J_k\subset I_k$ is open set, $k=\mathrm{1,2}$. Hence $I_k$ has non-empty interior. Let $J_k$ be the largest open set in $I_k$, which contain $u_k$ such that $A_f(u)$ has no fixed point in $I_k\backslash J_k$. Finally by Ref. [22], Lemma 3.8, $A_f(u)$ has a third fixed point $u_{\mathrm{3}}\in I\backslash (I_{\mathrm{1}}\bigcup I_{\mathrm{2}})$. Therefore, problem (1) has at least three solutions $u_{\mathrm{1}}\in [{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{2}}]$, $u_{\mathrm{2}}\in [{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}}]$ and $u_{\mathrm{3}}\in [{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}]\backslash ([{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{2}}]\bigcup [{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}}])$.

2 The Autonomous Case of $\mathit{f}(\mathit{u})=\mathit{u}$, $\mathit{h}(\mathit{t})=\mathrm{1}$