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Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator

Boundary Value Problems20172017:172

https://doi.org/10.1186/s13661-017-0904-8

  • Received: 14 August 2017
  • Accepted: 12 November 2017
  • Published:

Abstract

The authors consider the impulsive differential equation with Monge-Ampère operator in the form of
$$\textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f (-u(t) ), \quad t\in(0,1), t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta (u' )^{n}|_{t=t_{k}}=\lambda I_{k} (-u(t_{k}) ), \quad k=1, 2, \ldots , m, \\ u'(0)=0, \quad\quad u(1)=0, \end{cases} $$
where λ is a nonnegative parameter and \(n\geq1\). We show the existence, uniqueness, and continuity results. Our approach is largely based on the eigenvalue theory and the theory of α-concave operators. The nonexistence result of a nontrivial convex solution is also studied by taking advantage of the internal geometric properties related to the problem.

Keywords

  • continuity on a parameter
  • existence of nontrivial convex solutions
  • Monge-Ampère operator
  • impulsive differential equation
  • geometric properties

1 Introduction

In natural sciences, there are various concrete problems involving the Monge-Ampère equation. For example, the Monge-Ampère equation can describe Weingarten curvature, or reflector shape design (see [1]). In recent years, increasing attention has been paid to the study of the Monge-Ampère equation by different methods (see [210]).

The typical model of the Monge-Ampère equation is
$$ \textstyle\begin{cases} \det (D^{2}u)=\lambda f(-u)\quad \text{in }B, \\ u=0 \quad \text{on }\partial B, \end{cases} $$
(1.1)
where \(B=\{x\in R^{n}: \vert x\vert <1\}\) is the unit ball in \(R^{n}\) and \(D^{2}u=(\frac{\partial^{2} u}{\partial x_{i}\partial x_{j}})\) is the Hessian of u, λ is a nonnegative parameter and \(f: R\rightarrow R\) is a continuous function.

The study of problem (1.1) in general domains of \(R^{n}\) may be found in [2, 3]. Kutev [4] investigated the existence of strictly convex radial solutions of problem (1.1) when \(f(u)=u^{p}\). Delanoë [5] treated the existence of convex radial solutions of problem (1.1) for a class of more general functions, namely \(\lambda \exp f(\vert x\vert , u, \vert \nabla u\vert )\). Recently, under the case \(f(u)=e^{\lambda u}\), Zhang and Wang [6] obtained some interesting results of problem (1.1). They got the local structure of the solutions near a degenerate point by using the Lyapunov-Schmidt reduction method and established the global structure by Leray-Schauder degree theory and bifurcation theory.

In [4], Kutev also pointed out that problem (1.1) can reduce to the following boundary value problem:
$$\textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f(-u(t)), \quad t\in(0,1), \\ u'(0)=0, \quad\quad u(1)=0. \end{cases} $$
For the case \(f\geq0\), Hu and Wang [8] proved the existence, multiplicity, and nonexistence of strictly convex solutions for the above problem by using fixed point index theory. However, the corresponding results for impulsive Monge-Ampère type equations have not been investigated until now, even for the unique solution \(u_{\lambda}\) of the above equation depending continuously on the parameter λ.

At the same time, we notice that a class of differential equations with impulsive effects appeared in biological systems, population dynamics, biotechnology, ecology, industrial robotic, and optimal control; for details and references, see [1114]. Recently, the existence of solutions to the impulsive differential equations has attracted the attention of many researchers; see Bai et al. [15], Agarwal et al. [16], Liu and Guo [17], Wang and Feng [18], Zhang and Tian [19], Karaca and Tokmak [20], Liu et al. [21], Zeng and Xie [22], and Zhang and Ge [23] and the references cited therein. However, it is not difficult to see that there is almost no paper addressing impulsive differential equations with fully nonlinear operator. It is well known that the Monge-Ampère operator is just fully nonlinear. This motivates us to study an impulsive differential equation with Monge-Ampère operator.

Consider the impulsive differential equation with Monge-Ampère operator
$$ \textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f(-u(t)), \quad t\in(0,1), t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta(u')^{n}|_{t=t_{k}}=\lambda I_{k}(-u(t_{k})), \quad k=1, 2, \ldots , m, \\ u'(0)=0, \quad\quad u(1)=0, \end{cases} $$
(1.2)
where λ is a nonnegative parameter and \(n\geq1\), \(t_{k}\) (\(k = 1, 2, \ldots, m\)) (here m is a fixed positive integer) are fixed points with \(0=t_{0} < t_{1} < t_{2} <\cdots< t_{k}< \cdots< t_{m}<t_{m+1}=1\), \(\Delta(u')^{n}|_{t=t_{k}}=[u'(t^{+}_{k})]^{n}-[u'(t ^{-}_{k})]^{n}\), where \(u'(t^{+}_{k})\) and \(u'(t^{-}_{k})\) represent the right-hand limit and left-hand limit of \(u'(t)\) at \(t=t_{k}\). In addition, f and \(I_{k}\) satisfy
\((H_{1})\)

\(f\in C(\mathcal{R}^{+}, \mathcal{R}^{+})\), \(I_{k}\in C( \mathcal{R}^{+}, \mathcal{R}^{+})\) with \(f(0)=0\) and \(I_{k}(0)=0\), where \(\mathcal{R}^{+}=[0,+\infty)\), \(k=1,2,\dots,m\).

Some special cases of problem (1.3) have been investigated. For example, when \(n=1\), problem (1.3) reduces to a second order impulsive boundary value problem (1.2), which has been studied in [24]. The authors obtained many existence results by means of the theory of fixed point index in cones. For other related results on problem (1.2), we refer the reader to the references [2540].

Here we point out that our problem is new in the sense of impulsive Monge-Ampère type equations introduced here. To the best of our knowledge, the existence of single or multiple positive solutions for impulsive Monge-Ampère type equation (1.2) has not yet been studied, especially for the unique solution \(u_{\lambda}\) of problem (1.2) depending continuously on the parameter λ. In consequence, our main results of the present work will be a useful contribution to the existing literature on the topic of impulsive Monge-Ampère type equations. The existence, uniqueness, and continuity of positive solutions for the given problem are new, though they are proved by applying the well-known method based on the eigenvalue theory and the theory of α-concave operators.

Remark 1.1

Very recently, Han et al. [41] also considered problem (1.2) under the case \(I_{k}=0\), \(k=1,2,\dots,m\). By the bifurcation theory, the authors investigated the existence of strictly convex or concave solutions of problem (1.2). Notice that differential equations with impulses are characterized by sudden changing of their states. This requires a complete different method from those used in [8, 41] to tackle problem (1.2).

Remark 1.2

On the nonexistence, the arguments that we present here are based in geometric properties of the super-sublinearity of f and I at zero and infinity which was not observed in [8, 41] (see Properties 1.1-1.2 below).

The following geometric Properties 1.1-1.2 will be very important in our arguments.

Property 1.1

If \(f_{0}=0\) and \(f_{\infty}=0\), then there exists \(R>0\) such that
$$ \frac{f(R)}{R^{n}}=\max_{u>0}\frac{f(u)}{u^{n}}. $$
(1.3)

Property 1.2

If \(I_{0}(k)=0\) and \(I_{\infty}(k)=0\), then there exists \(R_{k}>0\) such that
$$ \frac{I_{k}(R_{k})}{R_{k}^{n}}=\max_{u>0} \frac{I_{k}(u)}{u^{n}}, $$
(1.4)
where
$$I_{0}(k)=\lim_{u\rightarrow0} \frac{I_{k}(u)}{u^{n}},\quad\quad I_{\infty}(k)=\lim_{u\rightarrow\infty} \frac{I_{k}(u)}{u ^{n}}, \quad k=1, 2, \ldots, m. $$

Remark 1.3

Some ideas of Properties 1.1-1.2 are from [42].

Remark 1.4

There is almost no result except [8] studying the uniqueness of a nontrivial solution of problem (1.2). However, in [8], they did not obtain that the unique solution \(u_{\lambda}(t)\) of problem (1.2) depends continuously on the parameter λ.

2 Some lemmas

Let \(J=[0,1]\) and \(J'=J\setminus\{t_{1}, t_{2}, \ldots, t_{m}\}\), \(J _{0}=[t_{0}, t_{1}]\), \(J_{k}=(t_{k}, t_{k+1}]\), \(k=1,2, \ldots,m\), and
$$\mathit {PC}^{1}[0,1]= \bigl\{ v\in C[0,1]: v'\in C(t_{k}, t_{k+1}), v' \bigl(t_{k} ^{-} \bigr)=v'(t_{k}), \exists v' \bigl(t_{k}^{+} \bigr), k=1, 2, \ldots, m \bigr\} . $$
Then \(\mathit {PC}^{1}[0,1]\) is a real Banach space with the norm
$$\Vert v\Vert _{\mathit {PC}^{1}}=\max \bigl\{ \Vert v\Vert _{\infty}, \bigl\Vert v' \bigr\Vert _{\infty} \bigr\} , $$
where
$$\Vert v\Vert _{\infty}=\sup_{t\in J} \bigl\vert v(t) \bigr\vert , \quad\quad \bigl\Vert v' \bigr\Vert _{\infty}= \sup_{t\in J} \bigl\vert v'(t) \bigr\vert . $$
A function \(v\in \mathit {PC}^{1}[0,1]\cap C^{2}(J')\) is called a solution of problem (1.3) if it satisfies (1.3). A strictly convex solution of (1.3) is negative on \([0,1)\).
Let \(v=-u\). Then problem (1.3) is equivalent to the following problem defined on J:
$$ \textstyle\begin{cases} ( (-v'(t) )^{n} )'=\lambda nt^{n-1}f(v(t)), \quad 0< t< 1, t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta(-v')^{n}|_{t=t_{k}}=\lambda I_{k}(v(t_{k})), \quad k=1, 2, \ldots , m, \\ v'(0)=0, \quad\quad v(1)=0, \end{cases} $$
(2.1)
where \(\Delta(-v')^{n}|_{t=t_{k}}=[-v'(t^{+}_{k})]^{n}-[-v'(t^{-} _{k})]^{n}\), \(v'(t^{+}_{k})\) and \(v'(t^{-}_{k})\) represent the right-hand limit and left-hand limit of \(v'(t)\) at \(t=t_{k}\).

The following lemmas will be used in the proof of our main results.

Lemma 2.1

Assume that \((H_{1})\) holds. Then
  1. (i)

    If \(v(t)\) is a solution of problem (2.1) on J, then \(u(t)=-v(t)\) is a solution of problem (1.3) on J;

     
  2. (ii)

    If \(u(t)\) is a solution of problem (1.3) on J, then \(v(t)=-u(t)\) is a solution of problem (2.1) on J.

     

Therefore, throughout this paper we shall study positive concave classical solutions of problem (2.1).

Lemma 2.2

Assume that \((H_{1})\) holds. Then \(v\in \mathit {PC}^{1}[0,1] \cap C^{2}(J')\) is a solution of problem (2.1) if and only if \(v\in \mathit {PC}^{1}[0,1]\) is a solution of the following equation:
$$ v(t)= \int_{t}^{1} \biggl( \int_{0}^{\tau}\lambda n s^{n-1}f \bigl(v(s) \bigr)\,ds+ \lambda\sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{ \frac{1}{n}}\,d\tau $$
(2.2)
and
$$ \min_{t\in[0,\xi]}v(t)\geq\delta \Vert v\Vert _{\mathit {PC}^{1}}, $$
(2.3)
where \(\xi\in(0,1)\) and
$$ \delta=1-\xi. $$
(2.4)

Proof

If \(0\leq t< t_{1}\), it is easy to see by integration from 0 to t of problem (2.1) that
$$\bigl(-v'(t) \bigr)^{n}- \bigl(-v'(0) \bigr)^{n}= \int_{0}^{t}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr. $$
If \(t_{1}\leq t< t_{2}\), then we have
$$\begin{aligned}& \bigl(-v' \bigl(t_{1}^{-} \bigr) \bigr)^{n}- \bigl(-v'(0) \bigr)^{n}= \int_{0}^{t_{1}^{-}}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr, \\& \bigl(-v'(t) \bigr)^{n}- \bigl(-v' \bigl(t_{1}^{+} \bigr) \bigr)^{n}= \int_{t_{1}^{+}}^{t}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr. \end{aligned}$$
It follows that
$$\bigl(-v'(t) \bigr)^{n}- \bigl(-v'(0) \bigr)^{n}= \int_{0}^{t}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr+I_{1} \bigl(v(t _{1}) \bigr). $$
If \(t_{k}\leq t< t_{k+1}\), we have
$$\bigl(-v'(t) \bigr)^{n}- \bigl(-v'(0) \bigr)^{n}= \int_{0}^{t}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr+ \sum_{t_{k}\leq t}I_{k} \bigl(v(t_{k}) \bigr). $$
Then
$$-v'(t)= \biggl[ \int_{0}^{t}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr+\sum_{t_{k} \leq t}I_{k} \bigl(v(t_{k}) \bigr) \biggr]^{1/n}. $$
Integrating again, we obtain
$$v(t)= \int_{t}^{1} \biggl[ \int_{0}^{\tau}\lambda nr^{n-1}f \bigl(v(r) \bigr)\,dr+\sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr]^{1/n}\,d\tau. $$

Conversely, if \(v\in \mathit {PC}^{1}[0,1]\) is a solution of (2.2).

Direct differentiation of (2.2) implies
$$v'(t)=- \biggl( \int_{0}^{t}\lambda ns^{n-1}f \bigl(v(s) \bigr)\,ds+\sum_{t_{k}\leq t}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{1/n}. $$
Evidently,
$$\Delta \bigl(-v' \bigr)^{n}|_{t=t_{k}}=\lambda I_{k} \bigl(v(t_{k}) \bigr)\quad ( k=1, 2, \ldots , m), \quad\quad v'(0)=0, \quad\quad v(1)=0. $$
Finally, we show that (2.3) holds. It is clear that \(v'(t)=- ( \int_{0}^{t}\lambda ns^{n-1}f(v(s))\,ds+ \sum_{t_{k}\leq t}I_{k}(v(t _{k})) )^{1/n}\), which implies that
$$\Vert v\Vert _{\mathit {PC}^{1}}=v(0),\quad\quad \min_{t\in J}v(t)=v(1). $$

As we assume that \(f(v)\geq0\), we see that any nontrivial solution v of problem (2.1) is concave on J, i.e., \(v''\leq0\), and then we get \(v'(t)\) is nonincreasing on J.

So, for every \(t\in(0,\xi]\), we have
$$\frac{v(1)-v(0)}{1}\leq\frac{v(t)-v(0)}{t}, $$
i.e., \(v(t)-v(0)\geq t(v(1)-v(0))\), and then
$$v(t)\geq(1-t)v(0)\geq(1-\xi)v(0),\quad \forall t\in[0,\xi]. $$
This shows that (2.3) holds. The lemma is proved. □
To establish the existence of positive concave classical solutions in \(\mathit {PC}^{1}[0,1]\cap C^{2}(J^{\prime})\) of problem (2.1), we construct a cone K in \(\mathit {PC}^{1}[0,1]\) by
$$K= \Bigl\{ v\in \mathit {PC}^{1}[0,1]: v\geq0, \min_{t\in[0,\xi]}v(t) \geq\delta \Vert v\Vert _{\mathit {PC}^{1}} \Bigr\} , $$
where δ is defined in (2.4). It is easy to see that K is a closed convex cone of \(\mathit {PC}^{1}[0,1]\).

Remark 2.1

The definition of K is completely different from those of [8, 9].

Remark 2.2

K is a solid normal cone, and
$$K^{0}= \Bigl\{ v\in \mathit {PC}^{1}[0,1]: v> 0, \min _{t\in[0,\xi]}v(t) \geq\delta \Vert v\Vert _{\mathit {PC}^{1}} \Bigr\} . $$
Define \(T:K\rightarrow \mathit {PC}^{1}[0,1]\) by
$$ (T v) (t) = \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}d \tau. $$
(2.5)

From (2.5) and Lemma 2.2, it is easy to obtain the following result.

Lemma 2.3

Assume that \((H_{1})\) holds. Then \(T:K\rightarrow K\) is completely continuous.

Proof

Similar to the proof of Lemma 2.2, we can show that \(T:K\rightarrow K\). The complete continuity of T is well known. □

Lemma 2.4

Suppose that D is an open subset of an infinite-dimensional real Banach space \(E,\theta\in D\), and P is a cone of E. If the operator \(\Gamma: P\cap D\rightarrow P\) is completely continuous with \(\Gamma\theta=\theta\) and satisfies
$$\inf_{x\in P\cap\partial D}\Gamma x>0, $$
then Γ has a proper element on \(P\cap\partial D\) associated with a positive eigenvalue. That is, there exist \(x_{0}\in P\cap \partial D\) and \(\mu_{0}\) such that \(\Gamma x_{0}=\mu_{0}x_{0}\).

Lemma 2.5

Suppose that P is a normal cone of a real Banach space, \(A: P^{\circ}\rightarrow P^{\circ}\) is an α-concave increasing (orα-convex decreasing) operator. Then A has exactly one fixed point in \(P^{\circ}\).

3 Existence and nonexistence of nontrivial convex solutions on a parameter

In this section, we shall establish the existence and nonexistence results of nontrivial convex solutions on a parameter for problem (1.2). We now state and prove our main results.

Theorem 3.1

Assume that \((H_{1})\) holds. If \(0< f_{\infty}<+ \infty\), \(0< I_{\infty}(k)<+\infty\) (\(k=1, 2, \ldots, m\)), then there exists \(\beta_{0}>0\) such that, for every \(R>\beta_{0}\), problem (1.2) has a strictly convex solution \(u_{R}(t)\) satisfying \(\Vert u_{R}\Vert _{\mathit {PC}^{1}}=R\) for any
$$ \lambda=\lambda_{R}\in[\lambda_{0},\bar{ \lambda}_{0}], $$
(3.1)
where \(\lambda_{0}\) and \(\bar{\lambda}_{0}\) are two positive finite numbers.

Proof

It follows from \(0< f_{\infty}<+\infty\) and \(0< I_{ \infty}(k)<+\infty\) that there exist \(0< l_{1}< l_{2}\), \(\mu>0\) such that
$$l_{1} v^{n} < f(v)< l_{2} v^{n}, \quad \quad l_{1} v^{n} < I_{k}(v)< l_{2} v ^{n},\quad k=1, 2, \ldots, m, \forall v\geq\mu. $$
Now, we prove that \(\beta_{0}=\frac{\mu}{\delta}\) is required. Let
$$\Omega_{R}= \bigl\{ x\in \mathit {PC}^{1}[0,1]:\Vert x\Vert _{\mathit {PC}^{1}}< R \bigr\} . $$
Then \(\Omega_{R}\) is a bounded open subset of the Banach space \(\mathit {PC}^{1}[0,1]\) and \(\theta\in\Omega_{R}\). Together with Lemma 2.3, we have \(T:K\cap\bar{\Omega}_{R}\rightarrow K\) is completely continuous with \(T\theta=\theta\).
Noticing \(R>\beta_{0}\), for any \(v\in K\cap\partial\Omega_{R}\), we have
$$v(t)\geq\delta \Vert v\Vert _{\mathit {PC}^{1}}=\delta R,\quad t\in[0,\xi], $$
and then
$$v(t)\geq\delta \Vert v\Vert _{\mathit {PC}^{1}}>\delta\beta_{0}=\mu, \quad t\in[0, \xi]. $$
Therefore, for any \(v\in K\cap\partial\Omega_{R}\), we have
$$\begin{aligned} (Tv) (t)&= \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v(s) \bigr) \,ds \biggr)^{ \frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi}n s^{n-1}f \bigl(v(s) \bigr) \,ds \biggr)^{ \frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s^{n-1}l_{1}v^{n}(s) \,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s^{n-1}l_{1} \delta^{n} \Vert v \Vert _{\mathit {PC}^{1}}^{n}\,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & = \xi(1-\xi)l_{1}^{\frac{1}{n}}\delta R, \end{aligned}$$
which shows that
$$\inf_{v\in K\cap\partial\Omega_{R}}Tv\geq\xi(1-\xi)l_{1} ^{\frac{1}{n}} \delta R>0. $$
By Lemma 2.4, for any \(R>\beta_{0}\), the operator T has a proper element \(v_{R}\in K\) associated with the eigenvalue \(\mu_{R}>0\), further \(v_{R}\) satisfies \(\Vert v_{R}\Vert _{\mathit {PC}^{1}}=R\). Let \(\lambda_{R}=\frac{1}{ \mu_{R}^{n}}\). Then problem (2.1) has a positive solution \(v_{R}\) associated with \(\lambda_{R}\).
From the proof above, for any \(R>\beta_{0}\), there exists a positive solution \(v_{R}\in K\cap\partial\Omega_{R}\) associated with \(\lambda=\lambda_{R}>0\). Thus,
$$v_{R}(t) =\lambda_{R}^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v_{R}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau $$
with \(\Vert v_{R}\Vert =R\).
On the one hand,
$$\begin{aligned}& \begin{aligned} v_{R}(t)&=\lambda_{R}^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v_{R}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq\lambda_{R}^{\frac{1}{n}} \int_{0}^{1} \biggl( \int_{0}^{\tau}n s ^{n-1}f \bigl(v_{R}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq\lambda_{R}^{\frac{1}{n}} \int_{0}^{1} \biggl( \int_{0}^{\tau} n s ^{n-1}l_{2}v_{R}^{n}(s) \,ds+ \sum_{t_{k}\leq\tau}l_{2}v_{R}^{n}(s) \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq(\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} \int_{0}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}\,ds+m \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq(\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} \int_{0}^{1} \bigl(\tau^{n}+m \bigr)^{\frac{1}{n}}\,d\tau \\ & < (\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} (1+m )^{ \frac{1}{n}}, \end{aligned} \\& \begin{aligned} \bigl\vert v_{R}'(t) \bigr\vert &= \lambda_{R}^{\frac{1}{n}} \biggl( \int_{0}^{t}n s^{n-1}f \bigl(v _{R}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{ \frac{1}{n}} \\ & \leq\lambda_{R}^{\frac{1}{n}} \biggl( \int_{0}^{1}n s^{n-1}f \bigl(v_{R}(s) \bigr)\,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{\frac{1}{n}} \\ & \leq\lambda_{R}^{\frac{1}{n}} \biggl( \int_{0}^{1} n s^{n-1}l_{2}v_{R} ^{n}(s)\,ds+\sum_{t_{k}\leq\tau}l_{2}v_{R}^{n}(s) \biggr)^{ \frac{1}{n}} \\ & \leq(\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} \biggl( \int _{0}^{1}n s^{n-1}\,ds+m \biggr)^{\frac{1}{n}} \\ & \leq(\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}}, \end{aligned} \end{aligned}$$
which implies that
$$\Vert v_{R}\Vert _{\mathit {PC}^{1}}=R\leq(\lambda_{R} l_{2})^{\frac{1}{n}}\Vert v_{R}\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}}, $$
and hence,
$$\lambda_{R}\geq\frac{1}{l_{2}(1+m)}= \lambda_{0}. $$
On the other hand,
$$\begin{aligned} v_{R}(t)&=\lambda_{R}^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}f \bigl(v_{R}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{R}(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\lambda_{R}^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s ^{n-1}f \bigl(v_{R}(s) \bigr)\,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\lambda_{R}^{\frac{1}{n}} \int_{\xi}^{1} \biggl( \int_{0}^{\xi}n s ^{n-1}f \bigl(v_{R}(s) \bigr)\,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\lambda_{R}^{\frac{1}{n}} \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s ^{n-1}l_{1}v_{R}^{n}(s) \,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\lambda_{R}^{\frac{1}{n}} \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s ^{n-1}l_{1} \bigl(\delta \Vert v_{R}\Vert _{\mathit {PC}^{1}} \bigr)^{n} \,ds \biggr)^{\frac{1}{n}}d \tau \\ & =(\lambda_{R} l_{1})^{\frac{1}{n}}\delta \Vert v_{R}\Vert _{\mathit {PC}^{1}}\xi(1- \xi), \end{aligned}$$
which shows that
$$\Vert v_{R}\Vert _{\mathit {PC}^{1}}=R\geq(\lambda_{R} l_{1})^{\frac{1}{n}}\delta \Vert v _{R}\Vert _{\mathit {PC}^{1}}\xi(1-\xi), $$
and hence,
$$\lambda_{R}\leq\frac{1}{ (l_{1}\delta(1-\xi)\xi )^{n}}=\bar{ \lambda}_{0}. $$

In conclusion, \(\lambda_{R}\in[\lambda_{0},\bar{\lambda}_{0}]\). It follows from Lemma 2.1 that Theorem 3.1 holds. The proof is complete. □

Theorem 3.2

Assume that \((H_{1})\) holds. If \(0< f_{0}<+\infty\), \(0< I_{0}(k)<+\infty\) (\(k=1, 2, \ldots, m\)), then there exists \(\beta^{*}_{0}>0\) such that, for every \(0< r_{0}<\beta^{*}_{0}\), problem (1.2) has a strictly convex solution \(u_{r_{0}}(t)\) satisfying \(\Vert u_{r_{0}}\Vert _{\mathit {PC}^{1}}=r_{0}\) associated with
$$\lambda=\lambda_{r_{0}}\in \bigl[\lambda^{*}_{0}, \bar{\lambda}^{*}_{0} \bigr], $$
where \(\lambda^{*}_{0}\) and \(\bar{\lambda}^{*}_{0}\) are two positive finite numbers.

Proof

The proof is similar to that of Theorem 4.1, we omit it here. □

Theorem 3.3

Assume that \((H_{1})\) holds. If \(f_{\infty}=+ \infty\), \(I_{\infty}(k)=+\infty\) (\(k=1, 2, \ldots, m\)), then there exists \(\bar{\beta}_{0}>0\) such that, for every \(r_{*}>\bar{\beta} _{0}\), problem (1.2) has a strictly convex solution \(u_{r_{*}}(t)\) satisfying \(\Vert u_{r_{*}}\Vert _{\mathit {PC}^{1}}=r_{*}\) for any
$$ \lambda=\lambda_{r_{*}}\in(0,\lambda_{*}], $$
(3.2)
where \(\lambda_{*}\) is a positive finite number.

Proof

It follows from \(f_{\infty}=+\infty\) and \(I_{\infty}(k)=+ \infty\) that there exist \(l_{*}>0\), \(\mu^{*}>0\) such that
$$f(v)> l_{*}v^{n},\quad\quad I_{k}(v)>l_{*} v^{n},\quad k=1, 2, \ldots , m, \forall v\geq\mu^{*}. $$
Now, we prove that \(\bar{\beta}_{0}=\frac{\mu^{*}}{\delta}\) is required. Let
$$\Omega_{r}= \bigl\{ v\in \mathit {PC}^{1}[0,1]:\Vert v\Vert _{\mathit {PC}^{1}}< r \bigr\} . $$
Then \(\Omega_{r}\) is a bounded open subset of the Banach space \(\mathit {PC}^{1}[0,1]\) and \(0\in\Omega_{r}\). Together with Lemma 2.3, we have \(T:K\cap\bar{\Omega}_{r}\rightarrow K\) is completely continuous with \(T\theta=\theta\).
Noticing \(r>\bar{\beta}_{0}\), for any \(v\in K\cap\partial\Omega _{r}\), we have
$$v(t)\geq\delta \Vert v\Vert _{\mathit {PC}^{1}}=\delta r,\quad t\in[0,\xi], $$
and then
$$v(t)\geq\delta \Vert v\Vert _{\mathit {PC}^{1}}>\delta\bar{\beta}_{0}= \mu^{*},\quad t \in[0,\xi]. $$
Therefore, for any \(v\in K\cap\partial\Omega_{r}\), we have
$$\begin{aligned} (Tv) (t)&= \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq \int_{t}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}f \bigl(v(s) \bigr) \,ds \biggr)^{ \frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi}n s^{n-1}f \bigl(v(s) \bigr) \,ds \biggr)^{ \frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s^{n-1}l_{*}v^{n}(s) \,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq \int_{\xi}^{1} \biggl( \int_{0}^{\xi} n s^{n-1}l_{*} \bigl(\delta \Vert v\Vert _{\mathit {PC}^{1}} \bigr)^{n}(s)\,ds \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq l_{*}^{\frac{1}{n}}\delta\xi(1-\xi) r, \end{aligned}$$
which shows that
$$\inf_{v\in K\cap\partial\Omega_{r}}Tv\geq l_{*}^{\frac{1}{n}} \delta\xi(1- \xi) r>0. $$
By Lemma 2.4, for any \(r>\bar{\beta}_{0}\), the operator T has a proper element \(v_{r}\in K\) associated with the eigenvalue \(\mu_{r}>0\), further \(v_{r}\) satisfies \(\Vert v_{r}\Vert _{\mathit {PC}^{1}}=r\). Let \(\lambda_{r}=\frac{1}{ \mu_{r}^{n}}\) and follow the proof of Theorem 3.1, we complete the proof of Theorem 3.3. □

Theorem 3.4

Assume that \((H_{1})\) holds. If \(f_{0}=+\infty\), \(I _{0}(k)=+\infty\) (\(k=1, 2, \ldots, m\)), then there exists \(\beta_{1}>0\) such that, for any \(0< r^{*}<\beta_{1}\), problem (1.2) has a strictly convex solution \(u_{ r^{*}}(t)\) satisfying \(\Vert u_{ r^{*}}\Vert _{\mathit {PC}^{1}}=r ^{*}\) for any
$$\lambda=\lambda_{ r^{*}}\in(0,\lambda^{*}], $$
where \(\lambda^{*}\) is a positive finite number.

Proof

The proof is similar to that of Theorem 3.3, we omit it here. □

For ease of exposition, we set
$$\begin{aligned}& m_{f}(r_{**})=\min \biggl\{ \frac{f(u)}{r_{**}^{n}}: u\in [\delta r_{**},r_{**} ] \biggr\} , \\& m_{I_{k}}(r_{**})=\min \biggl\{ \frac{I_{k}(u)}{r_{**}^{n}}: u\in [\delta r_{**},r_{**} ] \biggr\} , \quad k=1,2,\ldots,m, \end{aligned}$$
where δ is defined in (2.4).

Theorem 3.5

Assume that \((H_{1})\) holds. If there exist \(r^{**}>0\) and \(\beta_{r^{**}}>0\) such that \(m_{f}(r^{**})\geq \beta_{r^{**}}\) and \(m_{I_{k}}(r^{**})\geq\beta_{r^{**}}\) (\(k=1, 2, \ldots, m\)), then problem (1.2) has a strictly convex solution \(u_{r^{**}}(t)\) satisfying \(\Vert u_{r^{**}}\Vert _{\mathit {PC}^{1}}=r^{**}\) for any
$$ \lambda=\lambda_{ r^{**}}\in(0,\lambda^{**}], $$
(3.3)
where \(\lambda^{**}\) is a positive finite number.

Proof

In fact, for any \(v\in K\cap\partial\Omega_{r_{**}}\), we have \(\delta r_{**} \leq v(t)\leq r_{**}\), \(t\in[0,\xi] \).

Noticing that \(m_{f}(r_{**})\geq\beta_{r_{**}}>0\) and \(m_{I_{k}}(r ^{**})\geq\beta_{r^{**}}>0\) (\(k=1, 2, \ldots, m\)), we have
$$f \bigl(v(t) \bigr)\geq m_{f}(r_{**})r_{**}^{n} \geq r_{**}^{n}\beta_{r_{**}} \geq\beta_{r_{**}}v^{n}(t), \quad\forall t\in[0,\xi], v\in [ \delta r_{**},r_{**} ], $$
and
$$I_{k}(v)\geq m_{I_{k}} \bigl(r^{**} \bigr)r_{**}^{n}\geq r_{**}^{n} \beta_{r_{**}} \geq\beta_{r_{**}}v^{n},\quad k=1,2,\ldots,m, v\in [\delta r_{**},r_{**} ]. $$
The following proof is similar to that of Theorem 3.3. This finishes the proof of Theorem 3.5. □

Theorem 3.6

Assume that \((H_{1})\) holds. If \(f_{0}=f_{\infty}=0\) and \(I_{0}(k)=I_{\infty}(k)=0\) (\(k=1, 2, \ldots, m\)), then there exists \(\underline{\lambda}>0\) such that problem (1.2) has no strictly convex solutions for \(\lambda\in(0,\underline{\lambda})\).

Proof

It follows from \(f_{0}=f_{\infty}=0\) and \(I_{0}(k)=I _{\infty}(k)=0\) (\(k=1, 2, \ldots, m\)), (1.3) and (1.4) that there exist \(\bar{v}_{0}>0\) and \(v_{k}>0\) such that
$$\frac{f(\bar{v}_{0})}{\bar{v}_{0}^{n}}=\max_{v>0}\frac{f(v)}{v ^{n}},\quad\quad \frac{I_{k}(\bar{v}_{k})}{v_{k}^{n}}=\max_{v>0}\frac{I _{k}(v)}{v^{n}},\quad k=1,2, \ldots,m. $$
Let
$$M=\max \biggl\{ \frac{f(\bar{v}_{0})}{\bar{v}_{0}^{n}},\frac{I_{1}( \bar{v}_{1})}{v_{1}^{n}}, \frac{I_{2}(\bar{v}_{2})}{v_{2}^{n}}, \ldots, \frac{I_{m}(\bar{v}_{m})}{v_{m}^{n}} \biggr\} +1. $$
Then \(M>0\) and
$$ f(v)\leq M v^{n},\quad\quad I_{k}(v)\leq Mv^{n}, \quad k=1,2,\ldots ,m, v>0. $$
(3.4)
Assume that \(v(t)\) is a strictly concave solution of problem (2.1). We will show that this leads to a contradiction for \(\lambda<\underline{ \lambda}\), where \(\underline{\lambda}= ( (M(1+m) )^{ \frac{1}{n}} )^{-1}\). Let \(\mu=\frac{1}{\lambda^{\frac{1}{n}}}\). Since \((Tv)(t)=\mu v(t)\) for \(t\in J\), it follows from (2.5) that
$$\begin{aligned}& \begin{aligned} v(t)&=\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}d \tau \\ & \leq\lambda^{\frac{1}{n}} \int_{0}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}d \tau \\ & \leq\lambda^{\frac{1}{n}} \int_{0}^{1} \biggl( \int_{0}^{\tau} n s^{n-1}M v^{n}(s) \,ds+\sum_{t_{k}\leq\tau}M v^{n}(s) \biggr)^{\frac{1}{n}}d \tau \\ & \leq\lambda^{\frac{1}{n}}M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} \int_{0}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}\,ds+m \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq\lambda^{\frac{1}{n}} M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} \int_{0}^{1} \bigl(\tau^{n}+m \bigr)^{\frac{1}{n}}\,d\tau \\ & \leq\lambda^{\frac{1}{n}} M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}}, \end{aligned} \\& \begin{aligned} \bigl\vert v'(t) \bigr\vert &= \lambda^{\frac{1}{n}} \biggl( \int_{0}^{t}n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}} \\ & \leq\lambda^{\frac{1}{n}} \biggl( \int_{0}^{1}n s^{n-1}f \bigl(v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl(v(t_{k}) \bigr) \biggr)^{\frac{1}{n}} \\ & \leq\lambda^{\frac{1}{n}} \biggl( \int_{0}^{1} n s^{n-1}Mv^{n}(s) \,ds+ \sum_{t_{k}\leq\tau}Mv^{n}(s) \biggr)^{\frac{1}{n}} \\ & \leq\lambda^{\frac{1}{n}} M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} \biggl( \int _{0}^{1}n s^{n-1}\,ds+m \biggr)^{\frac{1}{n}} \\ & \leq\lambda^{\frac{1}{n}}M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}}, \end{aligned} \end{aligned}$$
which shows that
$$\begin{aligned} \Vert v\Vert _{\mathit {PC}^{1}}&\leq\lambda^{\frac{1}{n}} M^{\frac{1}{n}} \Vert v\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}} \\ & < \underline{\lambda}^{\frac{1}{n}}M^{\frac{1}{n}}\Vert v\Vert _{\mathit {PC}^{1}} (1+m )^{\frac{1}{n}} \\ & =\Vert v\Vert _{\mathit {PC}^{1}}, \end{aligned}$$
which is a contradiction. This finishes the proof. □

Remark 3.1

The method to study the existence and nonexistence results of nontrivial convex solutions is completely different from those of Hu and Wang [8] and Han et al. [41].

Remark 3.2

Some ideas of the proof of Theorem 3.6 come from Theorems 1-2 in [42]. From the proof of the main results in [42], it is easy to see that \(f(u)>0\) for \(u>0\) is an important condition, although we consider the nonexistence of a nontrivial convex solution without using it; for details, see Theorem 3.6. Moreover, we introduce a new notation
$$M=\max \biggl\{ \frac{f(\bar{v}_{0})}{\bar{v}_{0}^{n}},\frac{I_{1}( \bar{v}_{1})}{v_{1}^{n}}, \frac{I_{2}(\bar{v}_{2})}{v_{2}^{n}}, \ldots, \frac{I_{m}(\bar{v}_{m})}{v_{m}^{n}} \biggr\} +1. $$

4 Uniqueness and continuity of nontrivial convex solutions on a parameter

In the previous section, we have established some existence and nonexistence criteria of nontrivial convex solutions for problem (1.2). Next we consider the uniqueness and continuity of nontrivial convex solutions on a parameter for problem (1.2).

Theorem 4.1

Suppose that \(f(u), I_{k}(u): [0,+\infty )\rightarrow [0,+\infty)\) are nondecreasing functions with \(f(u)>0\), \(I_{k}(u)>0\) (\(k=1, 2, \ldots, m\)) for \(u>0\) and satisfy \(f(\rho u)\geq\rho^{n\alpha}f(u)\), \(I_{k}(\rho u)\geq\rho^{n\alpha}I _{k}(u)\) (\(k=1, 2, \ldots, m\)) for any \(0<\rho<1\), where \(0\leq\alpha <1\). Then, for any \(\lambda\in(0,\infty)\), problem (1.2) has a unique nontrivial convex solution \(u_{\lambda}(t)\). Furthermore, such a solution \(u_{\lambda}(t)\) satisfies the following properties:
  1. (i)

    \(u_{\lambda}(t)\) is strongly decreasing in λ. That is, \(\lambda_{1}>\lambda_{2}>0\) implies \(u_{\lambda_{1}}(t)\ll u_{\lambda _{2}}(t)\) for \(t\in J\).

     
  2. (ii)

    \(\lim_{\lambda\rightarrow0^{+}}\Vert u_{\lambda} \Vert _{\mathit {PC}^{1}}=0\), \(\lim_{\lambda\rightarrow+\infty} \Vert u_{\lambda} \Vert _{\mathit {PC}^{1}}=+\infty\).

     
  3. (iii)

    \(u_{\lambda}(t)\) is continuous with respect to λ. That is, \(\lambda\rightarrow\lambda_{0}>0\) implies \(\Vert u_{\lambda}-u_{ \lambda_{0}}\Vert _{\mathit {PC}^{1}}\rightarrow0\).

     

Proof

Set \(\Psi=\lambda^{\frac{1}{n}} T\), and T be the same as in (2.5).

Let
$$K_{1}= \bigl\{ y(t)\in \mathit {PC}^{1}[0,1]: y(t)\geq0 \bigr\} . $$
It is easy to see that \(K_{1}\) is a normal solid cone of \(\mathit {PC}^{1}[0,1]\), and its interior \(K_{1}^{0}= \{y(t)\in \mathit {PC}^{1}[0,1]: y(t)>0 \}\). Being similar to Lemma 2.3, the operator Ψ maps \(K_{1}\) into \(K_{1}\). In view of \(f(u)>0\), \(I_{k}(u)>0\) (\(k=1, 2, \ldots , m\)) for \(u>0\), it is easy to see that \(\Psi: K_{1}^{0}\rightarrow K_{1}^{0}\). We assert that \(\Psi: K_{1}^{0}\rightarrow K_{1}^{0}\) is an α-concave increasing operator. Indeed
$$\begin{aligned} \Psi(\rho v)&=\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}f \bigl(\rho v(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(\rho v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s^{n-1} \rho^{n\alpha}f \bigl( v(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}\rho^{n\alpha }I_{k} \bigl( v(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \geq\rho^{\alpha}\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0} ^{\tau}n s^{n-1}f \bigl( v(s) \bigr) \,ds+ \sum_{t_{k}\leq\tau}I_{k} \bigl( v(t _{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & =\rho^{\alpha}\Psi(v), \quad\forall0< \rho< 1, \end{aligned}$$
where \(0\leq\alpha<1\). Since \(f(u)\) and \(I_{k}(u)\) (\(k=1, 2, \ldots, m\)) are nondecreasing, then
$$\begin{aligned} (\Psi v_{*}) (t)&=\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{ \tau}n s^{n-1}f \bigl(v_{*}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl( v _{*}(t_{k}) \bigr) \biggr)^{\frac{1}{n}}\,d\tau \\ & \leq\lambda^{\frac{1}{n}} \int_{t}^{1} \biggl( \int_{0}^{\tau}n s^{n-1}f \bigl(v _{**}(s) \bigr)\,ds+\sum_{t_{k}\leq\tau}I_{k} \bigl(v_{**}(t_{k}) \bigr) \biggr)^{ \frac{1}{n}}\,d\tau \\ & =(\Psi v_{**}) (t) \quad \text{for }v_{*}\leq v_{**}, v_{*}, v_{**}\in \mathit {PC}^{1}[0,1]. \end{aligned}$$
In view of Lemma 2.5, Ψ has a unique fixed point \(v_{\lambda} \in K_{1}^{0}\). This shows that problem (2.1) has a unique concave positive solution \(v_{\lambda}(t)\). It follows from Lemma 2.1 that problem (1.3) has a unique nontrivial convex solution \(u_{\lambda}(t)\).
Next, we give the proof for (i)-(iii). Let \(\gamma=\frac{1}{ \lambda^{\frac{1}{n}}}\), and denote \(\lambda^{\frac{1}{n}} Tv_{\lambda }=v_{\lambda}\) by \(Tv_{\gamma}=\gamma v_{\gamma}\). Assume \(0<\gamma_{1}<\gamma_{2}\). Then \(v_{\gamma_{1}}\geq v_{\gamma_{2}}\). Indeed, set
$$ \bar{\eta}=\sup \{\eta: v_{\gamma_{1}}\geq\eta v_{\gamma _{2}} \}. $$
(4.1)
We assert \(\bar{\eta}\geq1\). If it is not true, then \(0<\bar{\eta}<1\), and further
$$\gamma_{1}v_{\gamma_{1}}=Tv_{\gamma_{1}}\geq T(\bar{ \eta}v_{\gamma _{2}})\geq\bar{\eta}^{\alpha}Tv_{\gamma_{2}}=\bar{ \eta}^{\alpha} \gamma_{2}v_{\gamma_{2}}, $$
which imply
$$v_{\gamma_{1}}\geq\bar{\eta}^{\alpha}\frac{\gamma_{2}}{\gamma_{1}}v _{\gamma_{2}} \gg\bar{\eta}^{\alpha}v_{\gamma_{2}}\gg\bar{\eta}v _{\gamma_{2}}. $$
This is a contradiction to (4.1).
In view of the discussion above, we have
$$ v_{\gamma_{1}}=\frac{1}{\gamma_{1}}Tv_{\gamma_{1}}\geq\frac{1}{\gamma _{1}}Tv_{\gamma_{2}}= \frac{\gamma_{2}}{\gamma_{1}}v_{\gamma_{2}} \gg v_{\gamma_{2}}. $$
(4.2)
Hence, \(v_{\gamma}(t)\) is strongly decreasing in γ. Namely \(v_{\lambda}(t)\) is strongly increasing in \(\lambda^{\frac{1}{n}}\). By Lemma 2.1, (i) is proved.
Set \(\gamma_{2}=\gamma\) and fix \(\gamma_{1}\) in (4.2), we have \(v_{\gamma_{1}}\geq\frac{\gamma}{\gamma_{1}}v_{\gamma}\), for \(\gamma>\gamma_{1}\). Further
$$ \Vert v_{\gamma} \Vert _{\mathit {PC}^{1}}\leq\frac{\gamma_{1}N_{1}}{\gamma} \Vert v_{ \gamma_{1}}\Vert _{\mathit {PC}^{1}}, $$
(4.3)
where \(N_{1}>0\) is a normal constant. Note that \(\gamma=\frac{1}{ \lambda^{\frac{1}{n}}}\), we have \(\lim_{\lambda\rightarrow0^{+}} \Vert v_{\lambda} \Vert _{\mathit {PC}^{1}}=0\).

And then, it follows from Lemma 2.1 that \(\lim_{\lambda\rightarrow 0^{+}}\Vert u_{\lambda} \Vert _{\mathit {PC}^{1}}=0\).

Let \(\gamma_{1}=\gamma\), and fix \(\gamma_{2}\), again by (4.2) and normality of K, we have \(\lim_{\lambda\rightarrow+\infty} \Vert v_{\lambda} \Vert _{\mathit {PC}^{1}}=+\infty\).

And then, it follows from Lemma 2.1 that \(\lim_{\lambda\rightarrow +\infty} \Vert u_{\lambda} \Vert _{\mathit {PC}^{1}}=+ \infty\).

This gives the proof of (ii).

Next, we show the continuity of \(u_{\gamma}(t)\). For given \(\gamma_{0}>0\), by (i),
$$ v_{\gamma}\ll v_{\gamma_{0}}\quad \forall\gamma>\gamma_{0}. $$
(4.4)
Let
$$l_{\gamma}=\sup \{\nu>0 \mid v_{\gamma}\geq\nu v_{\gamma _{0}}, \gamma>\gamma_{0} \}. $$
Obviously, \(0< l_{\gamma}<1\) and \(v_{\gamma}\geq l_{\gamma}v_{\gamma _{0}}\). So, we have
$$\gamma v_{\gamma}=Tv_{\gamma}\geq T(l_{\gamma}v_{\gamma_{0}}) \geq l_{\gamma}^{\alpha}Tv_{\gamma_{0}}=l_{\gamma}^{\alpha} \gamma _{0}v_{\gamma_{0}}, $$
and further
$$v_{\gamma}\geq\frac{\gamma_{0}}{\gamma}l_{\gamma}^{\alpha}v_{ \gamma_{0}}. $$
By the definition of \(l_{\gamma}\),
$$\frac{\gamma_{0}}{\gamma}l_{\gamma}^{\alpha}\leq l_{\gamma} \quad \text{or}\quad l_{\gamma}\geq \biggl(\frac{\gamma_{0}}{\gamma} \biggr) ^{\frac{1}{1-\alpha}}. $$
Again, by the definition of \(l_{\gamma}\), we have
$$ v_{\gamma}\geq \biggl(\frac{\gamma_{0}}{\gamma} \biggr)^{\frac{1}{1-\alpha}}v _{\gamma_{0}},\quad \forall\gamma>\gamma_{0}. $$
(4.5)
Notice that \(K_{1}\) is a normal cone. In view of (4.4) and (4.5), we obtain
$$\Vert v_{\gamma_{0}}-v_{\gamma} \Vert _{\mathit {PC}^{1}}\leq N_{2} \biggl[1- \biggl(\frac{\gamma _{0}}{\gamma} \biggr)^{\frac{1}{1-\alpha}} \biggr] \Vert v_{\gamma_{0}}\Vert _{\mathit {PC}^{1}} \rightarrow0,\quad \gamma \rightarrow\gamma_{0}+0, $$
where \(N_{2}>0\) is a normal constant.
In the same way, we can prove
$$\Vert v_{\gamma}-v_{\gamma_{0}}\Vert _{\mathit {PC}^{1}}\rightarrow0, \quad \gamma \rightarrow\gamma_{0}-0. $$
Hence, \(v_{\gamma}\) is continuous at \(\gamma=\gamma_{0}\).
Therefore, by Lemma 2.1, we have
$$\Vert u_{\gamma_{0}}-u_{\gamma} \Vert _{\mathit {PC}^{1}}=\Vert v_{\gamma}-v_{\gamma_{0}} \Vert _{\mathit {PC}^{1}}\rightarrow0,\quad \gamma\rightarrow\gamma_{0}+0\ (\gamma \rightarrow\gamma_{0}-0). $$
Consequently, (iii) holds. The proof is complete. □

Remark 4.1

Some ideas of the proof of Theorem 4.1 come from Theorem 2.2.7 in [43] and Theorem 6 in [44].

Remark 4.2

In Theorem 4.1, even though we do not assume that T is completely continuous, even continuous, we can assert that \(u_{\lambda}\) depends continuously on λ.

Remark 4.3

If we replace \(K_{1}\), \(K_{1}^{0}\) by K, \(K^{0}\), respectively, then Theorem 4.1 also holds.

Remark 4.4

The function f and \(I_{k}\) (\(k=1, 2, \ldots, m\)) satisfying the conditions of Theorem 4.1 can be easily found. For example,
$$\begin{aligned}& f(u)=u^{n\alpha_{1}}+u^{n\alpha_{2}}+\cdots+u^{n\alpha_{s}}, \\& I_{k}(u)=u^{n\alpha_{1}}+u^{n\alpha_{2}}+\cdots+u^{n\alpha_{s}},\quad k=1, 2, \ldots, m, \end{aligned}$$
where \(\alpha_{j}>0\), \(\sup_{j}\alpha_{j}<1\), s is a positive integer.

5 Conclusion

Using the eigenvalue theory, we show the existence of a strictly convex solution for problem (1.2), which is a new problem in the sense of impulsive Monge-Ampère type equations introduced here. Further, we prove that problem (1.2) has no strictly convex solution for sufficiently small λ by means of the internal geometric properties related to the problem. Finally, by applying the theory of α-concave operators, we prove that the unique solution \(u_{\lambda}(t)\) of problem (1.2) is strongly increasing and depends continuously on the parameter λ. In consequence, our main results of the present work will be a useful contribution to the existing literature on the topic of impulsive Monge-Ampère type equations.

Declarations

Acknowledgements

This work is sponsored by the National Natural Science Foundation of China (11301178), the Beijing Natural Science Foundation of China (1163007), and the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (KM201611232019). The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

Availability of data and materials

Not applicable.

Funding

This work is sponsored by the National Natural Science Foundation of China (11301178), the Beijing Natural Science Foundation (1163007) and the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (KM201611232019).

Authors’ contributions

All results belong to XZ and MF. All authors read and approved the final manuscript.

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that there is no conflict of interest regarding the publication of this manuscript. The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China
(2)
School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, People’s Republic of China

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