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Positive solution of a system of integral equations with applications to boundary value problems of differential equations

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Abstract

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations.

Introduction

We consider the existence of eigenvalues yielding positive solutions to the system of integral equations

$$(P_{\lambda, \mu, \zeta}) \quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(t)=\lambda \int^{1}_{0}k_{1}(t, s)f(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,}\\ v(t)=\mu \int^{1}_{0}k_{2}(t, s)g(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,}\\ w(t)=\zeta \int^{1}_{0}k_{3}(t, s)h(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,} \end{array}\displaystyle \right . $$

where \(\lambda, \mu, \zeta\) are positive numbers and

  1. (H1)

    \(f, g, h\in C([0, 1]\times R^{+}\times R^{+}\times R^{+}, R^{+})\).

  2. (H2)

    \(k_{1}, k_{2}, k_{3}:[0, 1]\times[0, 1]\rightarrow R^{+}\) are continuous functions and there exist an interval \([\xi, \eta]\subset[0, 1]\), positive constants \(\gamma_{1}, \gamma _{2}, \gamma_{3}\), and functions \(\Phi_{1}, \Phi_{2}, \Phi_{3}\in C([0, 1], R^{+})\) such that

    $$k_{i}(t, s)\leq\Phi_{i}(s),\quad \mbox{for }(t, s)\in[0, 1] \times[0, 1], i=1,2,3, $$

    and

    $$k_{i}(t, s)\geq\gamma_{i}\Phi_{i}(s), \quad \mbox{for }(t, s)\in[\xi, \eta ]\times[0, 1], i=1,2,3. $$

    Here we denote \(\gamma=\min\{\gamma_{1}, \gamma_{2}, \gamma_{3}\}\).

Systems of differential equations or integral equations containing three equations have gained considerable popularity and importance due mainly to their demonstrated applications in widespread fields of science and technology. For example, to describe the development of an infectious disease, compartmental models have been given to separate a population into various classes based on the stages of inflection [1]. The classical SIR model is described by partitioning the population into susceptible, infectious, and recovered individuals, denoted by \(S, I, R\), respectively. Assume that the disease incubation period is negligible so that each susceptible individual becomes infectious and later recovers with a permanently or temporarily acquired immunity, then the SIR model is governed by the following system of differential equations:

$$\left \{ \textstyle\begin{array}{@{}l} \frac{dS}{dt}=-\beta S(t)I(t)-\mu_{1}S(t)+b,\\ \frac{dI}{dt}=\beta S(t)I(t)-\mu_{2}I(t)-\alpha I(t),\\ \frac{dR}{dt}=\alpha I(t)-\mu_{3}R(t), \end{array}\displaystyle \right . $$

where the total population size has been normalized to one and the influx of the susceptible comes from a constant recruitment rate b. The death rates for the \(S, I, R\) classes are given by \(\mu_{1}, \mu_{2}, \mu_{3}\), respectively.

It is well known that the predator-prey model, which was proposed by Volterra in 1926, is one of the basic and important models for the interacting species in both ecology and mathematical ecology due to the fact that the predator-prey interaction is the fundamental structure in population dynamic. Since then, various types of predator-prey models described by differential systems have been proposed and the dynamics of these systems has been considered. For example, Song and Chen [2] proposed the following predator-prey system with stage structure:

$$\left \{ \textstyle\begin{array}{@{}l} \frac{du_{1}}{dt}=\alpha u_{2}(t)-ru_{1}(t)-ae^{-t\tau}u_{2}(t-\tau),\\ \frac{du_{2}}{dt}=ae^{-t\tau}u_{2}(t-\tau )-mu^{2}_{2}(t)-a_{1}u_{2}(t)v(t),\\ \frac{dv}{dt}=v(t)[r_{1}+a_{2}u_{2}(t)-bv(t)], \end{array}\displaystyle \right . $$

where \(u_{1}(t), u_{2}(t)\) represent the densities of immature and mature population of the prey species, respectively, \(v(t)\) represent the density of the predator.

Positive solutions of a n-dimensional differential equation system or fractional differential equation system with some boundary conditions have received wide attention due to its distinguished applications in engineering, science, mathematical biology and other fields. For \(n=1\), see, for example, [38] (ordinary differential equations), [915] (differential equations on time scales), and [1620] (fractional differential equations). For \(n=2\), see [2125] (differential equations on time scales), [2636] (ordinary differential equations), and [3754] (fractional differential equations) and references along this line. A considerable number of these problems can be formulated as integral equation or integral equation system usually by finding the corresponding Green’s function of these problems. Thus the integral equation system can be seen naturally as an effective generalization of these types of boundary value problems. The advantage of studying the integral equation system is that we can avoid considering various boundary value problems of differential equations ad hoc.

The aim of this paper is to give a general approach of positive solutions to cover various systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations in a unified way, which avoids treating these problems on a case-by-case basis. We consider the existence and nonexistence of positive solutions of integral equation system \((P_{\lambda, \mu, \zeta})\) under the conditions (H1)-(H2) and so the results obtained in this paper may include some known results as a special cases and can be applied to unconsidered boundary value problems which can be formulated as a system of integral equations like \((P_{\lambda, \mu, \zeta})\).

Motivated by Webb and Infante [3, 5] and Webb and Lan [4], who established new existence results of positive solutions of a Hammerstein integral equation in an unified way, under some growth condition imposed on the nonlinear term, we obtain explicit ranges of values of \(\lambda, \mu\), and ζ with which the problem \((P_{\lambda, \mu, \zeta})\) has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to obtain eigenvalues yielding the existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations.

The main tool used is the following fixed point theorem by Guo and Krasnoselskii [55].

Lemma 1.1

[55]

Let E be a Banach space and \(K\subset E\) be a cone. Assume \(\Omega_{1},\Omega_{2}\) are open bounded subsets of E with \(0\in\Omega_{1}\subset\overline{\Omega}_{1}\subset\Omega _{2}\), and let

$$A:K\cap(\Omega_{2}\setminus\overline{\Omega}_{1})\rightarrow K $$

be a completely continuous operator such that

$$\Vert Au\Vert \leq \Vert u\Vert ,\quad u\in K\cap\partial \Omega_{1}, \quad\textit{and}\quad \Vert Au\Vert \geq \Vert u\Vert ,\quad u\in K\cap\partial\Omega_{2} $$

or

$$\Vert Au\Vert \geq \Vert u\Vert ,\quad u\in K\cap\partial \Omega_{1},\quad\textit{and}\quad \|Au\| \leq \|u\|,\quad u\in K\cap \partial\Omega_{2}, $$

then A has a fixed point in \(K\cap(\Omega_{2}\setminus\overline {\Omega}_{1})\).

Existence results of positive solutions

In this section we shall consider sufficient conditions on \(\lambda, \mu,\zeta, f, g\), and h such that a positive solution with respect to a cone for the problem \((P_{\lambda, \mu, \zeta})\) exists.

Let the Banach space \(X=\{u\in C[0, 1]\}\) be endowed with the norm

$$\|u\|= \sup_{0\leq t\leq1}\bigl\vert u(t)\bigr\vert ,\quad u\in X $$

and the Banach space \(Y=X\times X\times X\) with the norm

$$\bigl\Vert (u, v, w)\bigr\Vert _{Y}=\Vert u\Vert +\|v\|+\|w \|. $$

We define the cone \(P\subset Y\) by

$$P=\Bigl\{ u\in E \big| u(t)\geq0, v(t)\geq0, w(t)\geq0, \inf_{\xi\leq t\leq\eta} \bigl(u(t)+v(t)+w(t)\bigr)\geq\gamma\bigl\Vert (u, v, w)\bigr\Vert _{Y}\Bigr\} . $$

Define the operators \(T_{1}:Y\rightarrow X, T_{2}: Y\rightarrow X, T_{3}: Y\rightarrow X\), and \(T: Y\rightarrow Y\) by

$$\begin{aligned}& T_{1}\bigl(u(t), v(t), w(t)\bigr):=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds, \\& T_{2}\bigl(u(t), v(t), w(t)\bigr):=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s),v(s), w(s)\bigr)\,ds, \\& T_{3}\bigl(u(t), v(t), w(t)\bigr):=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s),v(s), w(s)\bigr)\,ds, \end{aligned}$$

and

$$T(u, v, w)=\bigl(T_{1}(u, v, w), T_{2}(u, v, w),T_{2}(u, v, w)\bigr). $$

It is obvious that the fixed points of the operator T are the positive solutions of the problem \((P_{\lambda, \mu, \zeta})\).

Lemma 2.1

\(T: P\rightarrow P\) is completely continuous.

Proof

The operator \(T: P\rightarrow Y\) is nonnegative and equicontinuous in view of the non-negativeness and continuity of functions \(k_{1}(t, s), k_{2}(t, s), k_{3}(t, s)\) and \(f(t, u, v, w)\), \(g(t, u, v, w), h(t, u, v, w)\).

Let \(\Omega\subset P\) be bounded. Then there exists a constant \(R_{0}>0\) such that \(\Vert (u, v,w)\Vert _{Y}\leq R_{0}, (u, v, w)\in\Omega\). Denote

$$\begin{aligned} R =&\max\Bigl\{ \max_{0\leq t\leq1, (u, v, w)\in\Omega}\bigl\vert f(t, u, v,w)\bigr\vert , \max_{0\leq t\leq1, (u, v, w)\in\Omega}\bigl\vert g(t, u, v, w)\bigr\vert , \\ &{}\max_{0\leq t\leq1, (u, v, w)\in\Omega}\bigl\vert h(t, u, v, w)\bigr\vert \Bigr\} +1. \end{aligned}$$

Then for \((u, v, w)\in\Omega\), we have

$$\begin{aligned}& \bigl\vert T_{1}(u, v, w)\bigr\vert \leq\lambda \int^{1}_{0}k_{1}(t, s)\bigl\vert f \bigl(s, u(s), v(s), w(s)\bigr)\bigr\vert \,ds\leq\lambda R \int^{1}_{0}\Phi_{1}(s)\,ds, \\& \bigl\vert T_{2}(u, v, w)\bigr\vert \leq\mu \int^{1}_{0}k_{2}(t, s)\bigl\vert f \bigl(s, u(s), v(s), w(s)\bigr)\bigr\vert \,ds\leq\mu R \int^{1}_{0}\Phi_{2}(s)\,ds, \\& \bigl\vert T_{3}(u, v, w)\bigr\vert \leq\zeta \int^{1}_{0}k_{3}(t, s)\bigl\vert f \bigl(s, u(s), v(s), w(s)\bigr)\bigr\vert \,ds\leq\zeta R \int^{1}_{0}\Phi_{3}(s)\,ds. \end{aligned}$$

Hence \(T(\Omega)\) is bounded.

By means of the Arzela-Ascoli theorem, we see that T is completely continuous. Furthermore, considering

$$\begin{aligned}& \inf_{\xi\leq t\leq\eta}T_{1}(u, v, w) (t)\geq\gamma _{1}\sup_{t'\in[0,1]}T_{1}(u, v, w) \bigl(t'\bigr), \\& \inf_{\xi\leq t\leq\eta}T_{2}(u, v, w) (t)\geq\gamma _{2}\sup_{t'\in[0,1]}T_{2}(u, v, w) \bigl(t'\bigr), \\& \inf_{\xi\leq t\leq\eta}T_{3}(u, v, w) (t)\geq\gamma _{3}\sup_{t'\in[0,1]}T_{3}(u, v, w) \bigl(t'\bigr). \end{aligned}$$

Hence

$$\begin{aligned} &\inf_{\xi\leq t\leq\eta}\bigl\vert T_{1}(u, v, w) (t)+T_{2}(u, v, w) (t)+T_{3}(u, v, w) (t)\bigr\vert \\ &\quad\geq\inf_{\xi\leq t\leq\eta}T_{1}(u, v, w) (t)+\inf _{\xi \leq t\leq\eta}T_{2}(u, v, w) +\inf_{\xi\leq t\leq\eta}T_{3}(u, v, w) (t) \\ &\quad\geq\gamma_{1}\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\gamma_{2}\bigl\Vert T_{2}(u, v, w)\bigr\Vert + \gamma_{3}\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &\quad\geq\gamma\bigl\Vert T(u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Thus, we show that \(T: P\rightarrow P\) is a completely continuous operator.

Here we introduce the following extreme limits:

$$\begin{aligned}& f^{s}_{0}= \lim_{u+v+w\rightarrow0^{+}}\sup\max _{t\in[0, 1]}\frac {f(t, u, v, w)}{u+v+w},\qquad f^{i}_{0}= \lim_{u+v+w\rightarrow 0^{+}}\inf\min_{t\in[\xi, \eta]}\frac{f(t, u, v, w)}{u+v+w}, \\& f^{s}_{\infty}= \lim_{u+v+w\rightarrow\infty}\sup\max _{t\in[0, 1]}\frac{f(t, u, v, w)}{u+v+w},\qquad f^{i}_{\infty}= \lim_{u+v+w\rightarrow\infty}\inf\min_{t\in[\xi, \eta]}\frac{f(t, u, v, w)}{u+v+w}, \\& g^{s}_{0}= \lim_{u+v+w\rightarrow0^{+}}\sup\max _{t\in [0, 1]}\frac{g(t, u, v, w)}{u+v+w},\qquad g^{i}_{0}= \lim_{u+v+w\rightarrow0^{+}}\inf\min_{t\in[\xi, \eta}\frac{g(t, u, v, w)}{u+v+w}, \\& g^{s}_{\infty}= \lim_{u+v+w\rightarrow\infty}\sup\max _{t\in[0, 1]}\frac{g(t, u, v, w)}{u+v+w}, \qquad g^{i}_{\infty}= \lim_{u+v+w\rightarrow\infty}\inf\min_{t\in[\xi, \eta]}\frac{g(t, u, v, w)}{u+v+w}. \\& h^{s}_{0}= \lim_{u+v+w\rightarrow0^{+}}\sup\max _{t\in [0, 1]}\frac{h(t, u, v, w)}{u+v+w},\qquad h^{i}_{0}= \lim_{u+v+w\rightarrow0^{+}}\inf\min_{t\in[\xi, \eta]}\frac{h(t, u, v, w)}{u+v+w}, \\& h^{s}_{\infty}= \lim_{u+v+w\rightarrow\infty}\sup\max _{t\in[0, 1]}\frac{h(t, u, v, w)}{u+v+w},\qquad h^{i}_{\infty}= \lim_{u+v+w\rightarrow\infty}\inf\min_{t\in[\xi, \eta]}\frac{h(t, u, v, w)}{u+v+w}. \end{aligned}$$

Denote the positive constants

$$\begin{aligned}& K_{1}=\frac{b}{\gamma\tilde{A}f^{i}_{\infty}},\qquad K_{2}=\frac{c}{Af^{s}_{0}}, \qquad K_{3}=\frac{a}{\gamma\tilde {B}g^{i}_{\infty}}, \\& K_{4}=\frac{d}{Bg^{s}_{0}},\qquad K_{5}=\frac{1-a-b}{\gamma\tilde {C}h^{i}_{\infty}}, \qquad K_{6}=\frac{1-c-d}{Ch^{s}_{0}}, \end{aligned}$$

where \(f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in[0, 1], d\in(0, 1)\), and

$$\begin{aligned}& A= \int^{1}_{0}\Phi_{1}(s)\,ds,\qquad B= \int^{1}_{0}\Phi_{2}(s)\,ds,\qquad C= \int ^{1}_{0}\Phi_{3}(s)\,ds,\qquad \tilde{A}= \gamma \int^{\eta}_{\xi}\Phi_{1}(s)\,ds,\\& \tilde {B}= \gamma \int ^{\eta}_{\xi} \Phi_{2}(s)\,ds,\qquad \tilde{C}= \gamma \int^{\eta}_{\xi}\Phi_{3}(s)\,ds. \end{aligned}$$

 □

Theorem 2.1

Assume that (H1)-(H2) hold. \(f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in [0, 1], b\in(0, 1), c\in[0, 1], d\in(0, 1)\), \(K_{1}< K_{2}, K_{3}< K_{4}\), and \(K_{5}< K_{6}\), then for \(\lambda\in (K_{1}, K_{2}), \mu\in(K_{3}, K_{4})\), and \(\zeta\in(K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Proof

Let \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, K_{4}), \zeta\in(K_{5}, K_{6})\), and ε be a positive number such that \(f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty }>\varepsilon\), and

$$\begin{aligned}& \frac{b}{\gamma\tilde{A}(f^{i}_{\infty}-\varepsilon)}< \lambda < \frac {c}{A(f^{s}_{0}+\varepsilon)}, \\& \frac{a}{\gamma\tilde{B}(g^{i}_{\infty}-\varepsilon)}< \mu< \frac {d}{B(g^{s}_{0}+\varepsilon)}, \\& \frac{1-a-b}{\gamma\tilde{C}(h^{i}_{\infty}-\varepsilon)}< \zeta < \frac {1-c-d}{C(h^{s}_{0}+\varepsilon)}. \end{aligned}$$

By condition (H1), there exists \(R_{1}>0\) such that for \(t\in[0, 1], u(t)\geq0, v(t)\geq 0, w(t)\geq0 \) and \(u(t)+v(t)+w(t)\leq R_{1}\),

$$\begin{aligned}& f\bigl(t, u(t), v(t), w(t)\bigr)\leq\bigl(f^{s}_{0}+ \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& g\bigl(t, u(t), v(t), w(t)\bigr)\leq\bigl(g^{s}_{0}+ \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& h\bigl(t, u(t), v(t), w(t)\bigr)\leq\bigl(h^{s}_{0}+ \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr). \end{aligned}$$

We define the set

$$\Omega_{1}=\bigl\{ \bigl(u(t), v(t), w(t)\bigr)\in Y, \bigl\Vert (u, v, w)\bigr\Vert _{Y}< R_{1}\bigr\} . $$

Let \((u, v, w)\in P\cap\partial\Omega_{1}\),

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\lambda \int^{1}_{0}k_{1}(t, s) \bigl(f^{s}_{0}+\varepsilon \bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\leq\lambda\bigl(f^{s}_{0}+\varepsilon\bigr) \int^{1}_{0}k_{1}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\lambda A\bigl(f^{s}_{0}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq c\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\mu \int^{1}_{0}k_{2}(t, s) \bigl(g^{s}_{0}+\varepsilon \bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\leq\mu\bigl(g^{s}_{0}+\varepsilon\bigr) \int^{1}_{0}k_{2}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\mu B\bigl(g^{s}_{0}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq d\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\zeta \int^{1}_{0}k_{3}(t, s) \bigl(h^{s}_{0}+\varepsilon \bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\leq\zeta\bigl(h^{s}_{0}+\varepsilon\bigr) \int^{1}_{0}k_{3}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\zeta C\bigl(h^{s}_{0}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Then for \((u, v, w)\in P\cap\partial\Omega_{1}\),

$$\begin{aligned} \bigl\Vert T(u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &\leq c\bigl\Vert (u, v, w)\bigr\Vert _{Y}+d\bigl\Vert (u, v, w)\bigr\Vert _{Y}+(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

On the other side, by condition (H1) and the definition of \(f^{i}_{\infty}, g^{i}_{\infty}\), and \(h^{i}_{\infty}\), there exists \(\overline{R}_{2}>0\) such that for \(t\in [\xi, \eta], u(t)\geq0, v(t)\geq0, w(t)\geq0\), and \(u(t)+v(t)+w(t)\geq\overline{R}_{2}\),

$$\begin{aligned}& f\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(f^{i}_{\infty}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& g\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(g^{i}_{\infty}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& h\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(h^{i}_{\infty}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr). \end{aligned}$$

We consider \(R_{2}=\max\{2R_{1}, \overline{R}_{2}/\gamma\} \), and we define the set

$$\Omega_{2}=\bigl\{ \bigl(u(t), v(t), w(t)\bigr)\in Y, \bigl\Vert (u, v, w)\bigr\Vert _{Y}< R_{2}\bigr\} . $$

Let \((u, v, w)\in P\cap\Omega_{2}\), then for \((u, v, w)\in P\) with \(\Vert (u, v, w )\Vert =R_{2}\), we have

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s) \bigl(f^{i}_{\infty }-\varepsilon\bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\lambda\gamma \int^{\eta}_{\xi}k_{1}(t, s) \bigl(f^{i}_{\infty}-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\lambda\gamma\tilde{A}\bigl(f^{i}_{\infty}-\varepsilon \bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq b\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s) \bigl(g^{i}_{\infty }-\varepsilon\bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\mu\gamma \int^{\eta}_{\xi}k_{2}(t, s) \bigl(g^{i}_{\infty }-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\mu\gamma\tilde{B}\bigl(g^{i}_{\infty}-\varepsilon\bigr) \bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq a\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s) \bigl(h^{i}_{\infty }-\varepsilon\bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\zeta\gamma \int^{\eta}_{\xi}k_{3}(t, s) \bigl(h^{i}_{\infty }-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\zeta\gamma\tilde{C}\bigl(h^{i}_{\infty}-\varepsilon\bigr) \bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq(1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Thus,

$$\begin{aligned} \bigl\Vert T(u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &\geq b\bigl\Vert (u, v, w)\bigr\Vert _{Y}+a\bigl\Vert (u, v, w)\bigr\Vert _{Y}+(1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

By using Lemma 1.1, T has a fixed point \((u, v, w)\in P\cap(\Omega _{2}\setminus\overline{\Omega}_{1})\). □

By a similar analysis, we can consider the case that the above limits reach 0 or ∞. We give the main results here and the proofs are omitted.

Theorem 2.2

Assume that (H1)-(H2) hold. If \(f^{s}_{0}=0, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{3}< K_{4}\), and \(K_{5}< K_{6}\), then for \(\lambda\in(K_{1}, \infty), \mu\in(K_{3}, K_{4})\), and \(\zeta \in(K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.3

Assume that (H1)-(H2) hold. If \(g^{s}_{0}=0, f^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), and \(K_{5}< K_{6}\), then for \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, \infty)\), and \(\zeta \in(K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.4

Assume that (H1)-(H2) hold. If \(h^{s}_{0}=0, f^{s}_{0}, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), and \(K_{3}< K_{4}\), then for \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, K_{4})\), and \(\zeta \in(K_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.5

Assume that (H1)-(H2) hold. If \(f^{s}_{0}=g^{s}_{0}=0, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{5}< K_{6}\), then for \(\lambda\in (K_{1}, \infty), \mu\in(K_{3}, \infty)\), and \(\zeta\in(K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.6

Assume that (H1)-(H2) hold. If \(f^{s}_{0}=h^{s}_{0}=0, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{3}< K_{4}\), then for \(\lambda\in (K_{1}, \infty), \mu\in(K_{3}, K_{4})\) and \(\zeta\in(K_{5}, \infty )\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.7

Assume that (H1)-(H2) hold. If \(g^{s}_{0}=h^{s}_{0}=0, f^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), then for \(\lambda\in (K_{1}, K_{2}), \mu\in(K_{3}, \infty)\) and \(\zeta\in(K_{5}, \infty )\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.8

Assume that (H1)-(H2) hold. If \(f^{s}_{0}=g^{s}_{0}=h^{s}_{0}=0, f^{i}_{\infty}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), then for \(\lambda\in(K_{1}, \infty), \mu\in(K_{3}, \infty)\), and \(\zeta\in(K_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.9

Assume that (H1)-(H2) hold. If \(f^{i}_{\infty }=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, g^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{3}< K_{4}\), and \(K_{5}< K_{6}\), then for \(\lambda\in(0, K_{2}), \mu\in(K_{3}, K_{4})\), and \(\zeta\in (K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.10

Assume that (H1)-(H2) hold. If \(g^{i}_{\infty }=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), and \(K_{5}< K_{6}\), then for \(\lambda\in(K_{1}, K_{2}), \mu\in(0, K_{4})\), and \(\zeta\in (K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.11

Assume that (H1)-(H2) hold. If \(h^{i}_{\infty }=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), and \(K_{3}< K_{4}\), then for \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, K_{4})\), and \(\zeta \in(0, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.12

Assume that (H1)-(H2) hold. If \(f^{i}_{\infty }=g^{i}_{\infty}=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, h^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{5}< K_{6}\), then for \(\lambda\in (0, K_{2}), \mu\in(0, K_{4})\), and \(\zeta\in(K_{5}, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.13

Assume that (H1)-(H2) hold. If \(f^{i}_{\infty }=h^{i}_{\infty}=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, g^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{3}< K_{4}\), then for \(\lambda\in (0, K_{2}), \mu\in(K_{3}, K_{4})\), and \(\zeta\in(0, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.14

Assume that (H1)-(H2) hold. If \(g^{i}_{\infty }=h^{i}_{\infty}=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}, f^{i}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), \(K_{1}< K_{2}\), then for \(\lambda\in (K_{1}, K_{2}), \mu\in(0, K_{4})\), and \(\zeta\in(0, K_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.15

Assume that (H1)-(H2) hold. If \(f^{i}_{\infty }=g^{i}_{\infty}=h^{i}_{\infty}=\infty, f^{s}_{0}, g^{s}_{0}, h^{s}_{0}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in[0, 1], d\in(0, 1)\), then for \(\lambda\in(0, K_{2}), \mu\in (0, K_{4})\), and \(\zeta\in(0, K_{6})\), the problem \((P_{\lambda, \mu , \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Denote the positive constants

$$\begin{aligned}& L_{1}=\frac{b}{\gamma\tilde{A}f^{i}_{0}},\qquad L_{2}=\frac {c}{Af^{s}_{\infty }}, \qquad L_{3}=\frac{a}{\gamma\tilde{B}g^{i}_{0}}, \\& L_{4}=\frac{d}{Bg^{s}_{\infty}},\qquad L_{5}=\frac{1-a-b}{\gamma\tilde {C}h^{i}_{0}}, \qquad L_{6}=\frac{1-c-d}{Ch^{s}_{\infty}}. \end{aligned}$$

Theorem 2.16

Assume that (H1)-(H2) hold, \(f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in (0, 1), c\in[0, 1], d\in(0, 1), L_{1}< L_{2}, L_{3}< L_{4}\), and \(L_{5}< L_{6}\). Then for \(\lambda\in(L_{1}, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in(L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Proof

Let \(\lambda\in(L_{1}, L_{2}), \mu\in(L_{3}, L_{4}),\zeta\in(L_{5}, L_{6})\), \(\varepsilon>0\) satisfying \(f^{i}_{0}>\varepsilon, g^{i}_{0}>\varepsilon, h^{i}_{0}>\varepsilon \), and

$$\begin{aligned}& \frac{b}{\gamma\tilde{A}(f^{i}_{0}-\varepsilon)}< \lambda< \frac {c}{A(f^{s}_{\infty}+\varepsilon)}, \\& \frac{a}{\gamma\tilde{B}(g^{i}_{0}-\varepsilon)}< \mu< \frac {d}{B(g^{s}_{\infty}+\varepsilon)}, \\& \frac{1-a-b}{\gamma\tilde{C}(h^{i}_{0}-\varepsilon)}< \zeta< \frac {1-c-d}{C(h^{s}_{\infty}+\varepsilon)}. \end{aligned}$$

By condition (H1), there exists \(R_{3}>0\) such that for \(t\in[\xi, \eta], u(t), v(t), w(t)\geq0\) and \(u(t)+v(t)+w(t)\leq R_{3}\),

$$\begin{aligned}& f\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(f^{i}_{0}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& g\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(g^{i}_{0}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr), \\& h\bigl(t, u(t), v(t), w(t)\bigr)\geq\bigl(h^{i}_{0}- \varepsilon\bigr) \bigl(u(t)+v(t)+w(t)\bigr). \end{aligned}$$

We define the set

$$\Omega_{3}=\bigl\{ \bigl(u(t), v(t), w(t)\bigr)\in Y, \bigl\Vert (u, v, w)\bigr\Vert _{Y}< R_{3}\bigr\} . $$

Let \((u, v, w)\in P\cap \partial\Omega_{3}\),

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s) \bigl(f^{i}_{0}-\varepsilon\bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\lambda\gamma \int^{\eta}_{\xi}k_{1}(t, s) \bigl(f^{i}_{0}-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\lambda\gamma\tilde{A}\bigl(f^{i}_{0}-\varepsilon \bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq b\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s) \bigl(g^{i}_{0}-\varepsilon \bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\mu\gamma \int^{\eta}_{\xi}k_{2}(t, s) \bigl(g^{i}_{0}-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\mu\gamma\tilde{B}\bigl(g^{i}_{0}-\varepsilon\bigr) \bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq a\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s) \bigl(h^{i}_{0}-\varepsilon\bigr) \bigl(u(s)+v(s)+w(s)\bigr) \,ds \\ &\geq\zeta\gamma \int^{\eta}_{\xi}k_{3}(t, s) \bigl(h^{i}_{0}-\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\zeta\gamma\tilde{C}\bigl(h^{i}_{0}-\varepsilon\bigr) \bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq(1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Then for \((u, v, w)\in P\cap\partial\Omega_{3}\),

$$\begin{aligned} \bigl\Vert T(u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &\geq b\bigl\Vert (u, v, w)\bigr\Vert _{Y}+a\bigl\Vert (u, v, w)\bigr\Vert _{Y}+(1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

On the other side, we define the functions \(f^{\ast}, g^{\ast}, h^{\ast} [0, 1]\times R^{+}\longrightarrow R^{+}\),

$$\begin{aligned}& f^{\ast}(t, x)= \max_{0\leq u+v+w\leq x}f(t, u, v, w),\quad t\in [0, 1], x\geq0, \\& g^{\ast}(t, x)= \max_{0\leq u+v+w\leq x}g(t, u, v, w),\quad t\in [0, 1], x\geq0. \\& h^{\ast}(t, x)= \max_{0\leq u+v+w\leq x}h(t, u, v, w),\quad t\in [0, 1], x\geq0. \end{aligned}$$

Then

$$\begin{aligned}& f(t, u, v, w)\leq f^{\ast}(t, x),\quad t\in[0, 1], u, v, w\geq0, u+v+w\leq x, \\& g(t, u, v, w)\leq g^{\ast}(t, x),\quad t\in[0, 1], u, v, w\geq0, u+v+w\leq x, \\& h(t, u, v, w)\leq h^{\ast}(t, x),\quad t\in[0, 1], u, v, w\geq0, u+v+w\leq x. \end{aligned}$$

The functions \(f^{\ast}(t, \cdot), g^{\ast}(t, \cdot), h^{\ast }(t, \cdot)\) are nondecreasing for each \(t\in[0, 1]\) and satisfy the conditions

$$\begin{aligned}& \limsup_{x\rightarrow\infty}\max_{t\in[0, 1]}\frac{f^{\ast}(t, x)}{x} \leq f^{s}_{\infty},\qquad \limsup_{x\rightarrow\infty}\max _{t\in[0, 1]}\frac{g^{\ast}(t, x)}{x}\leq g^{s}_{\infty},\\& \limsup_{x\rightarrow\infty}\max_{t\in [0, 1]}\frac{h^{\ast}(t, x)}{x}\leq h^{s}_{\infty}, \end{aligned}$$

which can be proved similar to Lemma 2.8 in [56]. Thus, for \(\varepsilon>0\), there exists \(\overline{R}_{4}>0\) such that for all \(x\geq\overline {R}_{4}, t\in[0, 1]\),

$$\begin{aligned}& \frac{f^{\ast}(t, x)}{x}\leq\limsup_{x\rightarrow\infty}\max_{t\in [0, 1]} \frac{f^{\ast}(t, x)}{x}+\varepsilon\leq f^{s}_{\infty }+\varepsilon, \\& \frac{g^{\ast}(t, x)}{x}\leq\limsup_{x\rightarrow\infty}\max_{t\in [0, 1]} \frac{g^{\ast}(t, x)}{x}+\varepsilon\leq g^{s}_{\infty }+\varepsilon, \\& \frac{h^{\ast}(t, x)}{x}\leq\limsup_{x\rightarrow\infty }\max_{t\in[0, 1]} \frac{h^{\ast}(t, x)}{x}+\varepsilon\leq h^{s}_{\infty}+\varepsilon. \end{aligned}$$

Then

$$f^{\ast}(t, x)\leq\bigl(f^{s}_{\infty}+\varepsilon \bigr)x,\qquad g^{\ast}(t, x)\leq \bigl(g^{s}_{\infty}+ \varepsilon\bigr)x,\qquad h^{\ast}(t, x)\leq\bigl(h^{s}_{\infty }+ \varepsilon\bigr)x. $$

Let \(R_{4}=\max\{2R_{3}, \overline{R}_{4}\}\) and \(\Omega _{4}=\{(u, v, w)\in Y, \Vert (u, v, w)\Vert _{Y}< R_{4}\}\). Let \((u, v, w)\in P\cap\partial\Omega_{4}\), then

$$\begin{aligned}& f(t, u, v, w)\leq f^{\ast}\bigl(t, \bigl\Vert (u, v, w)\bigr\Vert _{Y}\bigr),\quad t\in[0, 1], \\& g(t, u, v, w)\leq g^{\ast}\bigl(t, \bigl\Vert (u, v, w)\bigr\Vert _{Y}\bigr),\quad t\in[0, 1], \\& h(t, u, v, w)\leq h^{\ast}\bigl(t, \bigl\Vert (u, v, w)\bigr\Vert _{Y}\bigr),\quad t\in[0, 1]. \end{aligned}$$

Thus

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\lambda \int^{1}_{0}k_{1}(t, s)f^{\ast} \bigl(t, \bigl\Vert (u, v, w)\bigr\Vert _{Y}\bigr)\,ds \\ &\leq\lambda \int^{1}_{0}k_{1}(t, s) \bigl(f^{s}_{\infty }+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\leq\lambda A\bigl(f^{s}_{\infty}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq c\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\mu \int^{1}_{0}k_{2}(t, s)g^{\ast} \bigl(t, \bigl\Vert (u, v, w)\bigr\Vert _{Y}\bigr)\,ds \\ &\leq\mu \int^{1}_{0}k_{2}(t, s) \bigl(g^{s}_{\infty }+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\leq\mu B\bigl(g^{s}_{\infty}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq d\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\zeta \int^{1}_{0}k_{3}(t, s)h^{\ast} \bigl(t, \bigl\Vert \bigl(u, v, w(s)\bigr)\bigr\Vert _{Y}\bigr)\,ds \\ &\leq\zeta \int^{1}_{0}k_{3}(t, s) \bigl(h^{s}_{\infty }+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\leq\zeta C\bigl(h^{s}_{\infty}+\varepsilon\bigr)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\leq(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Then for \((u, v, w)\in P\cap\partial\Omega_{4}\),

$$\begin{aligned} \bigl\Vert T(u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &\leq c\bigl\Vert (u, v, w)\bigr\Vert _{Y}+d\bigl\Vert (u, v, w)\bigr\Vert _{Y}+(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

By using Lemma 1.1, T has a fixed point \((u, v, w)\in P\cap(\Omega _{4}\setminus\overline{\Omega}_{3})\). □

We can also consider the case that the above limits reach 0 or ∞.

Theorem 2.17

Assume that (H1)-(H2) hold. If \(f^{s}_{\infty }=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, g^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{3}< L_{4}\), and \(L_{5}< L_{6}\), then for \(\lambda\in(L_{1}, \infty), \mu\in(L_{3}, L_{4}), \zeta\in (L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.18

Assume that (H1)-(H2) hold. If \(g^{s}_{\infty }=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, f^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{1}< L_{2}\), and \(L_{5}< L_{6}\), then for \(\lambda\in(L_{1}, L_{2}), \mu\in(L_{3}, \infty), \zeta\in (L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.19

Assume that (H1)-(H2) hold. If \(h^{s}_{\infty }=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, g^{s}_{\infty}, f^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{1}< L_{2}\), and \(L_{3}< L_{4}\), then for \(\lambda\in(L_{2}, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in (L_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.20

Assume that (H1)-(H2) hold. If \(f^{s}_{\infty }=g^{s}_{\infty}=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{5}< L_{6}\), then for \(\lambda\in (L_{1}, \infty), \mu\in(L_{3}, \infty), \zeta\in(L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.21

Assume that (H1)-(H2) hold. If \(f^{s}_{\infty }=h^{s}_{\infty}=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, g^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{3}< L_{4}\), then for \(\lambda\in (L_{1}, \infty), \mu\in(L_{3}, L_{4}), \zeta\in(L_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.22

Assume that (H1)-(H2) hold. If \(h^{s}_{\infty }=g^{s}_{\infty}=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}, f^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{1}< L_{2}\), then for \(\lambda\in (L_{1}, L_{2}), \mu\in(L_{3}, \infty), \zeta\in(L_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.23

Assume that (H1)-(H2) hold. If \(f^{s}_{\infty }=g^{s}_{\infty}=h^{s}_{\infty}=0, f^{i}_{0}, g^{i}_{0}, h^{i}_{0}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in[0, 1], d\in(0, 1)\), then for \(\lambda\in(L_{1}, \infty), \mu\in(L_{3}, \infty), \zeta\in(L_{5}, \infty)\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.24

Assume that (H1)-(H2) hold. If \(f^{i}_{0}=\infty , g^{i}_{0}, h^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{3}< L_{4}\), and \(L_{5}< L_{6}\), then for \(\lambda\in(0, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in (L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.25

Assume that (H1)-(H2) hold. If \(g^{i}_{0}=\infty , f^{i}_{0}, h^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{1}< L_{2}\), and \(L_{5}< L_{6}\), then for \(\lambda\in(L_{1}, L_{2}), \mu\in(0, L_{4}), \zeta\in (L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.26

Assume that (H1)-(H2) hold. If \(h^{i}_{0}=\infty , g^{i}_{0}, f^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1), L_{3}< L_{4}\), and \(L_{1}< L_{2}\), then for \(\lambda\in(L_{1}, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in (0, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.27

Assume that (H1)-(H2) hold. If \(f^{i}_{0}=g^{i}_{0}=\infty, h^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty }, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{5}< L_{6}\), then for \(\lambda\in (0, L_{2}), \mu\in(0, L_{4}), \zeta\in(L_{5}, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.28

Assume that (H1)-(H2) hold. If \(f^{i}_{0}=h^{i}_{0}=\infty, g^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty }, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{3}< L_{4}\), then for \(\lambda\in (0, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in(0, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.29

Assume that (H1)-(H2) hold. If \(h^{i}_{0}=g^{i}_{0}=\infty, f^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty }, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), and \(L_{1}< L_{2}\), then for \(\lambda\in (L_{1}, L_{2}), \mu\in(0, L_{4}), \zeta\in(0, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

Theorem 2.30

Assume that (H1)-(H2) hold. If \(f^{i}_{0}=g^{i}_{0}=h^{i}_{0}=\infty, f^{s}_{\infty}, g^{s}_{\infty }, h^{s}_{\infty}\in(0, \infty), a\in[0, 1], b\in(0, 1), c\in [0, 1], d\in(0, 1)\), then for \(\lambda\in(0, L_{2}), \mu\in(0, L_{4}), \zeta\in(0, L_{6})\), the problem \((P_{\lambda, \mu, \zeta})\) has at least one positive solution \((u(t), v(t), w(t)), t\in[0, 1]\).

The proof is similar to Theorem 2.16, we omit it here.

Nonexistence results of positive solutions

In this section we shall consider sufficient conditions on \(\lambda, \mu,\zeta, f, g\), and h such that the problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Theorem 3.1

Assume that (H1)-(H2) hold. If \(f^{s}_{0}, f^{s}_{\infty}, g^{s}_{0}, g^{s}_{\infty}, h^{s}_{0}, h^{s}_{\infty}<\infty\), then there exists a positive constant \(\lambda _{0}, \mu_{0}, \zeta_{0}\) such that for every \(\lambda\in(0, \lambda_{0}), \mu\in(0, \mu_{0}), \zeta\in(0, \zeta_{0})\), the problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Proof

From the condition \(f^{s}_{0}, f^{s}_{\infty}, g^{s}_{0}, g^{s}_{\infty}, h^{s}_{0}, h^{s}_{\infty}<\infty\), there exist \(M_{1}>0, M_{2}>0, M_{3}>0\) such that

$$\begin{aligned}& f(t, u, v, w)\leq M_{1}(u+v+w),\qquad g(t, u, v, w)\leq M_{2}(u+v+w),\\& h(t, u, v, w)\leq M_{3}(u+v+w),\quad t \in[0, 1], u, v, w\geq0. \end{aligned}$$

Define the positive constants

$$\lambda_{0}= \frac{a}{AM_{1}},\qquad \mu_{0}= \frac{b}{BM_{2}},\qquad \zeta _{0}= \frac{1-a-b}{CM_{3}}. $$

Let \(\lambda\in(0, \lambda_{0}), \mu\in(0, \mu_{0}), \zeta\in(0, \zeta_{0})\), suppose that the problem \((P_{\lambda, \mu, \zeta})\) has a positive solution \((u(t), v(t)), t\in[0, 1]\). Thus,

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\lambda \int^{1}_{0}k_{1}(t, s)M_{1} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\leq\lambda M_{1} \int^{1}_{0}k_{1}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\lambda AM_{1}\bigl(\Vert u\Vert +\Vert v\Vert +\Vert w \Vert \bigr) \\ &< \lambda_{0} AM_{1}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &< a\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\mu \int^{1}_{0}k_{2}(t, s)M_{2} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\leq\mu M_{2} \int^{1}_{0}k_{2}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\mu BM_{2}\bigl(\Vert u\Vert +\Vert v\Vert +\Vert w \Vert \bigr) \\ &< \mu_{0} BM_{2}\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ &< b\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\leq\zeta \int^{1}_{0}k_{3}(t, s)M_{3} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\leq\zeta M_{3} \int^{1}_{0}k_{3}(t, s) \bigl(\Vert u \Vert +\Vert v\Vert +\Vert w\Vert \bigr)\,ds \\ &\leq\zeta CM_{3}\bigl(\Vert u\Vert +\Vert v\Vert +\Vert w \Vert \bigr) \\ &< \zeta_{0} CM_{3}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &< (1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert (u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &< a\bigl\Vert (u, v, w)\bigr\Vert _{Y}+b\bigl\Vert (u, v, w) \bigr\Vert _{Y}+(1-a-b)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \end{aligned}$$

which is a contradiction. So the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution. □

Theorem 3.2

Assume that (H1)-(H2) hold. If \(f^{i}_{0}, f^{i}_{\infty}>0\), then there exists a positive constant \(\tilde{\lambda}_{0}\) such that for every \(\lambda\in(\tilde {\lambda }_{0}, \infty), \mu>0, \zeta>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Proof

From the definitions of \(f^{i}_{0}, f^{i}_{\infty}\), and the condition \(f^{i}_{0}, f^{i}_{\infty}>0\), there exist positive numbers \(m_{1}\) such that

$$f(t, u, v, w)\geq m_{1}(u+v+w), \quad t\in[\xi, \eta], u, v, w\geq0. $$

Define thepositive constants

$$\tilde{\lambda}_{0}= \frac{1}{\gamma\tilde{A}m_{1}}. $$

Let \(\lambda\in(\tilde{\lambda}_{0}, \infty), \mu>0, \zeta>0\), we suppose that the problem \((P_{\lambda, \mu, \zeta})\) has a positive solution \((u(t), v(t), w(t)), t\in[0, 1]\). Then for \(t\in[0, 1]\), we have

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\lambda\gamma \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\lambda\gamma\tilde{A}m_{1}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &> \bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Thus,

$$\bigl\Vert (u, v, w)\bigr\Vert _{Y}\geq\bigl\Vert T_{1}(u, v, w)\bigr\Vert >\bigl\Vert (u, v, w)\bigr\Vert _{Y}, $$

which is a contradiction. So the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution. □

Theorem 3.3

Assume that (H1)-(H2) hold. If \(g^{i}_{0}, g^{i}_{\infty}>0\), then there exists a positive constant \(\tilde{\mu}_{0}\) such that for every \(\mu\in(\tilde{\mu}_{0}, \infty), \lambda>0, \zeta>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Theorem 3.4

Assume that (H1)-(H2) hold. If \(h^{i}_{0}, h^{i}_{\infty}>0\), then there exists a positive constant \(\tilde{\zeta}_{0}\) such that for every \(\zeta\in(\tilde{\zeta }_{0}, \infty), \lambda>0, \mu>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Theorem 3.5

Assume that (H1)-(H2) hold. If \(f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty}>0\), then there exist positive constants \(\tilde{\lambda}_{0}, \tilde{\mu}_{0}, a\in[0, 1]\), such that for every \(\lambda\in(\tilde{\lambda}_{0}, \infty), \mu\in(\tilde{\mu}_{0}, \infty), \zeta>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Proof

From the definitions of \(f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty} \), and the condition \(f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty}>0\), there exist positive numbers \(m_{1}, m_{2}\) such that

$$\begin{aligned}& f(t, u, v, w)\geq m_{1}(u+v+w),\quad t\in[\xi, \eta], u, v, w\geq0. \\& g(t, u, v, w)\geq m_{2}(u+v+w),\quad t\in[\xi, \eta], u, v, w\geq0. \end{aligned}$$

Define the positive constants

$$\tilde{\lambda}_{0}= \frac{a}{\gamma\tilde{A}m_{1}},\qquad \tilde{\mu }_{0}= \frac{1-a}{\gamma\tilde{B}m_{2}}. $$

Let \(\lambda\in(\tilde{\lambda}_{0}, \infty), \mu\in (\tilde{\mu}_{0}, \infty), \zeta>0\), we suppose that the problem \((P_{\lambda, \mu, \zeta})\) has a positive solution \((u(t), v(t), w(t)), t\in[0, 1]\). Then for \(t\in[0, 1]\), we have

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\lambda\gamma \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\lambda\gamma\tilde{A}m_{1}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq a\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)m_{2} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\mu\gamma \int^{\eta}_{\xi}k_{2}(t, s)m_{2} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\mu\gamma\tilde{B}m_{2}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq(1-a)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Thus,

$$\bigl\Vert (u, v, w)\bigr\Vert _{Y}>a\bigl\Vert (u, v, w)\bigr\Vert _{Y}+(1-a)\bigl\Vert (u, v, w)\bigr\Vert _{Y}= \bigl\Vert (u, v, w)\bigr\Vert _{Y}, $$

which is a contradiction. So the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution. □

Theorem 3.6

Assume that (H1)-(H2) hold. If \(f^{i}_{0}, f^{i}_{\infty}, h^{i}_{0}, h^{i}_{\infty}>0\), then there exist positive constants \(\tilde{\lambda}_{0}, \tilde{\zeta}_{0}\) such that for every \(\lambda\in(\tilde{\lambda}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty), \mu>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Theorem 3.7

Assume that (H1)-(H2) hold. If \(g^{i}_{0}, g^{i}_{\infty}, h^{i}_{0}, h^{i}_{\infty}>0\), then there exist positive constants \(\tilde{\mu}_{0}, \tilde{\zeta}_{0}\) such that for every \(\mu\in(\tilde{\mu}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty), \lambda>0\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Theorem 3.8

Assume that (H1)-(H2) hold. If \(f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty}, h^{i}_{0}, h^{i}_{\infty}>0\), then there exist positive constants \(\tilde {\lambda }_{0}, \tilde{\mu}_{0}, \tilde{\zeta}_{0}\) such that for every \(\lambda\in(\tilde{\lambda}_{0}, \infty), \mu\in(\tilde {\mu }_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty)\), the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution.

Proof

From the definitions of \(f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty}, h^{i}_{0}, h^{i}_{\infty}\), and the condition

$$f^{i}_{0}, f^{i}_{\infty}, g^{i}_{0}, g^{i}_{\infty}, h^{i}_{0}, h^{i}_{\infty}>0, $$

there exist positive numbers \(m_{1}, m_{2}, m_{3}\) such that

$$\begin{aligned}& f(t, u, v, w)\geq m_{1}(u+v+w),\quad t\in[0, 1], u, v, w\geq0, \\& g(t, u, v, w)\geq m_{2}(u+v+w),\quad t\in[0, 1], u, v, w\geq0, \\& h(t, u, v, w)\geq m_{3}(u+v+w),\quad t\in[0, 1], u, v, w\geq0. \end{aligned}$$

Define the positive constants

$$\tilde{\lambda}_{0}= \frac{c}{\gamma\tilde{A}m_{1}},\qquad \tilde{\mu }_{0}= \frac{d}{\gamma\tilde{B}m_{2}},\qquad \tilde{\zeta}_{0}= \frac {1-c-d}{\gamma\tilde{C}m_{3}}. $$

Let \(\lambda\in(\tilde{\lambda}_{0}, \infty), \mu\in (\tilde{\mu}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty)\), we suppose that the problem \((P_{\lambda, \mu, \zeta})\) has a positive solution \((u(t), v(t), w(t)), t\in[0, 1]\). Then for \(t\in[0, 1]\), we have

$$\begin{aligned} T_{1}(u, v, w) (t)&=\lambda \int^{1}_{0}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)f\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\lambda \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\lambda\gamma \int^{\eta}_{\xi}k_{1}(t, s)m_{1} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\lambda\gamma\tilde{A}m_{1}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq c\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{2}(u, v, w) (t)&=\mu \int^{1}_{0}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)g\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\mu \int^{\eta}_{\xi}k_{2}(t, s)m_{2} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\mu\gamma \int^{\eta}_{\xi}k_{2}(t, s)m_{2} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\mu\gamma\tilde{B}m_{2}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq d\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \\ T_{3}(u, v, w) (t)&=\zeta \int^{1}_{0}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s)h\bigl(s, u(s), v(s), w(s)\bigr)\,ds \\ &\geq\zeta \int^{\eta}_{\xi}k_{3}(t, s)m_{3} \bigl(u(s)+v(s)+w(s)\bigr)\,ds \\ &\geq\zeta\gamma \int^{\eta}_{\xi}k_{3}(t, s)m_{3} \bigl\Vert (u, v, w)\bigr\Vert _{Y}\,ds \\ &\geq\zeta\gamma\tilde{C}m_{3}\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &\geq(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y}. \end{aligned}$$

Thus,

$$\begin{aligned} \bigl\Vert (u, v, w)\bigr\Vert _{Y}&=\bigl\Vert T_{1}(u, v, w)\bigr\Vert +\bigl\Vert T_{2}(u, v, w)\bigr\Vert +\bigl\Vert T_{3}(u, v, w)\bigr\Vert \\ &>c\bigl\Vert (u, v, w)\bigr\Vert _{Y}+d\bigl\Vert (u, v, w) \bigr\Vert _{Y}+(1-c-d)\bigl\Vert (u, v, w)\bigr\Vert _{Y} \\ &=\bigl\Vert (u, v, w)\bigr\Vert _{Y}, \end{aligned}$$

which is a contradiction. So the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution. □

Examples

In this section we show how our results may be applied to consider the existence and nonexistence of positive solutions for a system of boundary value problems for differential equations of integral or fractional order. The study of these problems was mainly initiated by Il’in and Moiseev [57, 58]. Since then positive solutions of boundary value problems have been extensively studied by many researchers in recent years, not only because of their mathematical interest but also because of their wide use in a variety of applications.

Application to system of boundary value problems of ordinary differential equations

Consider the system of nonlinear second order differential equation (the problem (P1))

$$\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u^{\prime\prime}(t)+\lambda f(t, u(t), v(t), w(t))=0,&{t\in[0, 1],}\\ v^{\prime\prime}(t)+\mu g(t, u(t), v(t), w(t))=0,&{ t\in[0, 1],}\\ w^{\prime\prime}(t)+\zeta h(t, u(t), v(t), w(t))=0,&{ t\in[0, 1],} \end{array}\displaystyle \right . $$

subject to the boundary conditions

$$\begin{aligned}& u(0)=0, \qquad u(1)=\beta_{1} u(\eta_{1}), \\& v'(0)=0,\qquad v(1)=\beta_{2} v(\eta_{2}), \\& w'(0)=0, \qquad w(1)=\beta_{3} w(\eta_{3}), \end{aligned}$$

where \(0<\eta_{1}, \eta_{2}, \eta_{3}<1\), and

$$\begin{aligned}& f(t, u, v, w)=\frac{\sqrt [3]{t}(70(u+v+w)+1)(3+\sin(v))(u+v+w)}{u+v+w+1}, \\& g(t, u, v, w)=\frac{\sqrt{t+1}(40(u+v+w)+1)(4+\cos(w))(u+v+w)}{u+v+w+1}, \\& h(t, u, v, w)=\frac{\sqrt{t+1}(50(u+v+w)+1)(3+\cos(u))(u+v+w)}{u+v+w+1}. \end{aligned}$$

By using the Green’s functions, we can formulate the problem (P1) as

$$\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(t)=\lambda \int^{1}_{0}k_{1}(t, s)f(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,}\\ v(t)=\mu \int^{1}_{0}k_{2}(t, s)g(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,}\\ w(t)=\zeta \int^{1}_{0}k_{3}(t, s)h(s, u(s), v(s), w(s))\,ds,&{0\leq t\leq1,} \end{array}\displaystyle \right . $$

where

$$\begin{aligned}& k_{1}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{t(1-s)}{1-\beta_{1}\eta_{1}}-\frac{\beta_{1}(\eta _{1}-s)t}{1-\beta_{1}\eta_{1}}-(t-s),& 0\leq s\leq t\leq1, s\leq\eta _{1},\\ \frac{t(1-s)}{1-\beta_{1}\eta_{1}}-\frac{\beta_{1}(\eta _{1}-s)t}{1-\beta_{1}\eta_{1}},& 0\leq t\leq s\leq\eta_{1},\\ \frac{t(1-s)}{1-\beta_{1}\eta_{1}},& 0\leq t\leq s\leq1, s\geq \eta_{1},\\ \frac{t(1-s)}{1-\beta_{1}\eta_{1}}-(t-s), &\eta_{1}\leq s\leq t\leq1, \end{array}\displaystyle \right . \\& k_{2}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1-s}{1-\beta_{2}}-\frac{\beta_{2}(\eta_{2}-s)t}{1-\beta _{2}}-(t-s),& 0\leq s\leq t\leq1, s\leq\eta_{2},\\ \frac{1-s}{1-\beta_{2}}-\frac{\beta_{2}(\eta_{2}-s)t}{1-\beta _{2}},& 0\leq t\leq s\leq\eta_{2},\\ \frac{t(1-s)}{1-\beta_{2}},& 0\leq t\leq s\leq1, s\geq\eta_{2},\\ \frac{1-s}{1-\beta_{2}}-(t-s),& \eta_{1}\leq s\leq t\leq1, \end{array}\displaystyle \right . \end{aligned}$$

and

$$k_{3}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1-s}{1-\beta_{3}}-\frac{\beta_{3}(\eta_{3}-s)t}{1-\beta _{3}}-(t-s),& 0\leq s\leq t\leq1, s\leq\eta_{3},\\ \frac{1-s}{1-\beta_{3}}-\frac{\beta_{3}(\eta_{3}-s)t}{1-\beta _{3}}, &0\leq t\leq s\leq\eta_{3},\\ \frac{t(1-s)}{1-\beta_{3}},& 0\leq t\leq s\leq1, s\geq\eta_{3},\\ \frac{1-s}{1-\beta_{3}}-(t-s),& \eta_{3}\leq s\leq t\leq1. \end{array}\displaystyle \right . $$

We consider the case \(\eta_{1}= \frac{1}{2}, \eta _{2}=\eta_{3}= \frac{1}{3}, \beta_{1}= \frac{1}{3}, \beta_{2}=\beta_{3}= \frac{1}{2}\). After an easy computation, we conclude

$$\begin{aligned}& A=\frac{1}{5}, \qquad\tilde{A}=\frac{1}{30},\qquad B= \frac{13}{9}, \qquad\tilde{B}=\frac{26}{45},\qquad C= \frac{13}{9}, \qquad\tilde {C}=\frac{26}{45},\qquad \gamma= \frac{1}{6}, \\& f^{s}_{0}=3,\qquad f^{i}_{0}=3 \sqrt[3]{\frac{1}{2}},\qquad f^{s}_{\infty}=280,\qquad f^{i}_{\infty}=140\sqrt[3]{\frac{1}{2}}, \\& g^{s}_{0}=4\sqrt{2},\qquad g^{i}_{0}=4, \qquad g^{s}_{\infty }=200\sqrt{2},\qquad g^{i}_{\infty}=120, \\& h^{s}_{0}=3\sqrt{2},\qquad h^{i}_{0}=3, \qquad h^{s}_{\infty }=200\sqrt{2},\qquad h^{i}_{\infty}=100, \\& M_{1}=280,\qquad M_{2}=200\sqrt{2},\qquad M_{3}=200\sqrt{2}, \\& m_{1}=3\sqrt[3]{\frac{1}{2}},\qquad m_{2}=4,\qquad m_{3}=3,\qquad a=b=c=d=\frac{1}{3}, \end{aligned}$$

and

$$\begin{aligned}& K_{1}=\frac{b}{\gamma\tilde{A}f^{i}_{\infty}}\thickapprox0.5400,\qquad K_{2}= \frac{a}{Af^{s}_{0}}\thickapprox0.5556,\qquad K_{3}=\frac{1-b}{\gamma\tilde{B}g^{i}_{\infty}} \thickapprox0.0288, \\& K_{4}=\frac{1-a}{Bg^{s}_{0}}\thickapprox0.0408,\qquad K_{5}= \frac{1-a-b}{\gamma\tilde{C}h^{i}_{\infty}}\thickapprox 0.0346,\qquad K_{6}=\frac{1-c-d}{Ch^{s}_{0}} \thickapprox0.0544, \\& \lambda_{0}= \frac{a}{AM_{1}}\thickapprox0.0060,\qquad \mu _{0}= \frac{b}{BM_{2}}\thickapprox0.0008, \qquad \zeta_{0}= \frac {1-a-b}{CM_{3}}\thickapprox 0.0008, \\& \tilde{\lambda}_{0}= \frac{c}{\gamma\tilde {A}m_{1}}\thickapprox25.1894,\qquad \tilde{\mu}_{0}= \frac{d}{\gamma\tilde{B}m_{2}}\thickapprox 0.8654,\qquad \tilde{ \zeta}_{0}= \frac{1-c-d}{\gamma\tilde{C}m_{3}}\thickapprox1.1538. \end{aligned}$$

Then:

  1. (1)

    from Theorem 2.1, for \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, K_{4}), \zeta\in(K_{5}, K_{6})\), the problem (P1) has a positive solution;

  2. (2)

    from Theorem 3.1, for \(\lambda\in(0, \lambda_{0}), \mu\in(0, \mu_{0}), \zeta\in(0, \zeta_{0})\), the problem (P1) has no positive solution;

  3. (3)

    from Theorem 3.8, for \(\lambda\in(\tilde{\lambda}_{0}, \infty ), \mu\in(\tilde{\mu}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty)\) the problem (P1) has no positive solution.

Application to system of boundary value problems of differential equations on time scales

Consider the system of boundary value problems of nonlinear differential equation on time scale \(\mathbb{T}\) (the problem (P2)),

$$\begin{aligned}& u^{\triangle\nabla}(t)+\lambda f\bigl(t, u(t), v(t), w(t)\bigr)=0,\quad t\in[0, T] \subset\mathbb{T}, \\& v^{\triangle\nabla}(t)+\mu g\bigl(t, u(t), v(t), w(t)\bigr)=0,\quad t\in[0, T] \subset\mathbb{T}, \\& w^{\triangle\nabla}(t)+\zeta h\bigl(t, u(t), v(t), w(t)\bigr)=0,\quad t\in[0, T] \subset\mathbb{T}, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned}& u(0)=\beta_{1} u(\eta_{1}),\qquad u(T)=\alpha_{1} u(\eta_{1}), \\& v(0)=\beta_{2} v(\eta_{2}), \qquad v(T)=\alpha_{2} v(\eta_{2}), \\& w(0)=\beta_{3} w(\eta_{3}),\qquad w(T)= \alpha_{3} w(\eta_{3}), \end{aligned}$$

where \(\mathcal{T}\) is a time scale and \(0<\eta<T, 0<\alpha< \frac {T}{\eta}, 0<\beta<\frac{T-\alpha\eta}{T-\eta}\), and

$$\begin{aligned}& f(t, u, v, w)=\frac{\sqrt{t+3}(u+v+w+1)(3+\sin(v))(u+v+w)}{4{,}000(u+v+w)+1}, \\& g(t, u, v, w)=\frac{\sqrt{t+1}(u+v+w+1)(4+\cos(w))(u+v+w)}{6{,}000(u+v+w)+1}, \\& h(t, u, v, w)=\frac{\sqrt{t+1}(u+v+w+1)(3+\cos(u))(u+v+w)}{6{,}000(u+v+w)+1}. \end{aligned}$$

We can formulate the problem (P2) as

$$\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(t)=\lambda \int^{1}_{0}k_{1}(t, s)f(s, u(s), v(s), w(s))\nabla s,&{t\in[0, T],}\\ v(t)=\mu \int^{1}_{0}k_{2}(t, s)g(s, u(s), v(s), w(s))\nabla s,&{t\in[0, T],}\\ w(t)=\zeta \int^{1}_{0}k_{3}(t, s)h(s, u(s), v(s), w(s))\nabla s,&{t\in[0, T]}, \end{array}\displaystyle \right . $$

where

$$\begin{aligned}& k_{1}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{((1-\beta_{1})t+\beta_{1}\eta_{1})(T-s)}{(T-\alpha_{1}\eta _{1})-\beta_{1}(T-\eta_{1})}-\frac{((\beta_{1}-\alpha_{1})t-\beta_{1} T)(\eta_{1}-s)}{(T-\alpha_{1}\eta_{1})-\beta_{1}(T-\eta _{1})}-(t-s), & 0\leq s\leq t\leq T, s\leq\eta_{1},\\ \frac{((1-\beta_{1})t+\beta_{1}\eta_{1})(T-s)}{(T-\alpha_{1}\eta _{1})-\beta_{1}(T-\eta_{1})}-\frac{((\beta_{1}-\alpha_{1})t-\beta_{1} T)(\eta_{1}-s)}{(T-\alpha_{1}\eta_{1})-\beta_{1}(T-\eta_{1})},& 0\leq t\leq s\leq\eta_{1},\\ \frac{((1-\beta_{1})t+\beta_{1}\eta_{1})(T-s)}{(T-\alpha_{1}\eta _{1})-\beta_{1}(T-\eta_{1})},& 0\leq t\leq s\leq T, s\geq\eta_{1},\\ \frac{((1-\beta_{1})t+\beta_{1}\eta_{1})(T-s)}{(T-\alpha_{1}\eta _{1})-\beta_{1}(T-\eta_{1})}-(t-s),& \eta_{1}\leq s\leq t\leq T, \end{array}\displaystyle \right . \\& k_{2}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{((1-\beta_{2})t+\beta_{2}\eta_{2})(T-s)}{(T-\alpha_{2}\eta _{2})-\beta_{2}(T-\eta_{2})}-\frac{((\beta_{2}-\alpha_{2})t-\beta_{2} T)(\eta_{2}-s)}{(T-\alpha_{2}\eta_{2})-\beta_{2}(T-\eta _{2})}-(t-s), & 0\leq s\leq t\leq T, s\leq\eta_{2},\\ \frac{((1-\beta_{2})t+\beta_{2}\eta_{2})(T-s)}{(T-\alpha_{2}\eta _{2})-\beta_{2}(T-\eta_{2})}-\frac{((\beta_{2}-\alpha_{2})t-\beta_{2} T)(\eta_{2}-s)}{(T-\alpha_{2}\eta_{2})-\beta_{2}(T-\eta_{2})},& 0\leq t\leq s\leq\eta_{2},\\ \frac{((1-\beta_{2})t+\beta_{2}\eta_{2})(T-s)}{(T-\alpha_{2}\eta _{2})-\beta_{2}(T-\eta_{2})},& 0\leq t\leq s\leq T, s\geq\eta_{2},\\ \frac{((1-\beta_{2})t+\beta_{2}\eta_{2})(T-s)}{(T-\alpha_{2}\eta _{2})-\beta_{2}(T-\eta_{2})}-(t-s), &\eta_{1}\leq s\leq t\leq T, \end{array}\displaystyle \right . \\& k_{3}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{((1-\beta_{3})t+\beta_{3}\eta_{3})(T-s)}{(T-\alpha_{3}\eta _{3})-\beta_{3}(T-\eta_{3})}-\frac{((\beta_{3}-\alpha_{3})t-\beta_{3} T)(\eta_{3}-s)}{(T-\alpha_{3}\eta_{3})-\beta_{3}(T-\eta _{3})}-(t-s), &0\leq s\leq t\leq T, s\leq\eta_{3},\\ \frac{((1-\beta_{3})t+\beta_{3}\eta_{3})(T-s)}{(T-\alpha_{3}\eta _{3})-\beta_{3}(T-\eta_{3})}-\frac{((\beta_{3}-\alpha_{3})t-\beta_{3} T)(\eta_{3}-s)}{(T-\alpha_{3}\eta_{3})-\beta_{3}(T-\eta_{3})},& 0\leq t\leq s\leq\eta_{3},\\ \frac{((1-\beta_{3})t+\beta_{3}\eta_{3})(T-s)}{(T-\alpha_{3}\eta _{3})-\beta_{3}(T-\eta_{3})}, & 0\leq t\leq s\leq T, s\geq\eta_{3},\\ \frac{((1-\beta_{3})t+\beta_{3}\eta_{3})(T-s)}{(T-\alpha_{3}\eta _{3})-\beta_{3}(T-\eta_{3})}-(t-s),& \eta_{1}\leq s\leq t\leq T. \end{array}\displaystyle \right . \end{aligned}$$

Lemma 4.1

Let \(0<\alpha_{i}< \frac{\eta_{i}}{T}, 0<\beta _{i}<\frac{T-\alpha_{i}\eta_{i}}{T-\eta_{i}}, (u(t), v(t), w(t))\) be a solution of the problem (P2), then

$$\min_{t\in[0, T]}u(t)\geq\gamma_{1}\max _{t\in[0, T]}u(t),\quad \min_{t\in[0, T]}v(t)\geq \gamma_{2}\max_{t\in[0, T]}v(t),\quad \min _{t\in [0, T]}w(t)\geq\gamma_{3}\max_{t\in[0, T]}w(t), $$

where

$$\gamma_{i}=\min \biggl\{ \frac{\alpha_{i}(T-\eta_{i})}{T-\alpha _{i}\eta _{i}}, \frac{\alpha_{i}\eta_{i}}{T}, \frac{\beta_{i}(T-\eta _{i})}{T}, \frac{\beta_{i}T}{T} \biggr\} ,\quad i=1,2,3. $$

We consider the case

$$T=1,\qquad \eta_{1}=\eta_{2}=\eta_{3}= \frac{1}{16},\qquad \beta_{1}=\beta _{2}= \beta_{3}=\frac{1}{3},\qquad \alpha_{1}= \alpha_{2}=\alpha_{3}=8. $$

After an easy computation, we conclude

$$\begin{aligned}& A=B=C=\frac{48}{27},\qquad \gamma=\frac{1}{48},\qquad \tilde {A}= \tilde{B}=\tilde{C}=\frac{65}{1{,}296}, \\& f^{s}_{0}=6,\qquad f^{i}_{0}=3 \sqrt{3},\qquad f^{s}_{\infty}=\frac {1}{500}, \qquad f^{i}_{\infty}=\frac{\sqrt{3}}{2{,}000}, \\& g^{s}_{0}=4\sqrt{2},\qquad g^{i}_{0}=4, \qquad g^{s}_{\infty}=\frac{\sqrt {2}}{1{,}200},\qquad g^{i}_{\infty}=\frac{1}{2{,}000}, \\& h^{s}_{0}=3\sqrt{2},\qquad h^{i}_{0}=3, \qquad h^{s}_{\infty}=\frac {\sqrt{2}}{1{,}500},\qquad h^{i}_{\infty}=\frac{1}{3{,}000}, \\& M_{1}=6,\qquad M_{2}=4\sqrt{2},\qquad M_{3}=3 \sqrt{2}, \\& m_{1}=\frac{\sqrt{3}}{2{,}000},\qquad m_{2}=\frac{1}{2{,}000}, \qquad m_{3}=\frac{1}{3{,}000}. \end{aligned}$$

Choose \(a= \frac{1}{3}, b=\frac{1}{3}, c=\frac{1}{3}, d=\frac{1}{3}\). Then

$$\begin{aligned}& L_{1}=\frac{b}{\gamma\tilde{A}f^{i}_{0}}\thickapprox61.3945,\qquad L_{2}= \frac{C}{Af^{s}_{\infty}}\thickapprox93.7500,\qquad L_{3}=\frac{a}{\gamma\tilde{B}g^{i}_{0}} \thickapprox79.7538, \\& L_{4}=\frac{d}{Bg^{s}_{\infty}}\thickapprox159.0990,\qquad L_{5}= \frac{1-a-b}{\gamma\tilde{C}h^{i}_{0}}\thickapprox106.3385,\\& L_{6}=\frac{1-c-d}{Ch^{s}_{\infty}} \thickapprox198.8738, \\& \lambda_{0}= \frac{a}{AM_{1}}\thickapprox0.0313,\qquad \mu _{0}= \frac{b}{BM_{2}}\thickapprox0.0331,\qquad \zeta_{0}= \frac {1-a-b}{CM_{3}}\thickapprox 0.0442, \\& \tilde{\lambda}_{0}= \frac{c}{\gamma\tilde {A}m_{1}}\thickapprox368{,}370,\qquad \tilde{\mu}_{0}= \frac{d}{\gamma\tilde{B}m_{2}}\thickapprox 638{,}370,\qquad \tilde{ \zeta}_{0}= \frac{1-c-d}{\gamma\tilde{C}m_{3}}\thickapprox957{,}050. \end{aligned}$$

Thus

  1. (1)

    from Theorem  2.16, for \(\lambda\in(L_{1}, L_{2}), \mu\in(L_{3}, L_{4}), \zeta\in(L_{5}, L_{6})\), the problem (P2) has a positive solution;

  2. (2)

    from Theorem  3.1, for \(\lambda\in(0, \lambda_{0}), \mu\in(0, \mu_{0}), \zeta\in(0, \zeta_{0})\), the problem (P2) has no positive solution;

  3. (3)

    from Theorem  3.8, for \(\lambda\in(\tilde{\lambda}_{0}, \infty ), \mu\in(\tilde{\mu}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty)\) the problem (P2) has no positive solution.

Application to system of boundary value problems of fractional differential equations

Consider the system of nonlinear fractional differential equation (the problem (P3))

$$\begin{aligned}& D^{\alpha}_{0+}u(t)+\lambda f\bigl(t, u(t), v(t), w(t)\bigr)=0, \quad t\in(0, 1), \\& D^{\alpha}_{0+}v(t)+\mu g\bigl(t, u(t), v(t), w(t)\bigr)=0,\quad t\in (0, 1), \\& D^{\alpha}_{0+}w(t)+\zeta h\bigl(t, u(t), v(t), w(t)\bigr)=0, \quad t\in(0, 1), \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned}& u(0)=u'(0)=u'(1)=0,\qquad \bigl[D^{\delta}_{0+}u(t) \bigr]_{t=1}=0, \\& v(0)=v'(0)=v'(1)=0,\qquad \bigl[D^{\delta}_{0+}v(t) \bigr]_{t=1}=0, \\& w(0)=w'(0)=w'(1)=0, \qquad \bigl[D^{\delta}_{0+}w(t) \bigr]_{t=1}=0, \end{aligned}$$

where \(\alpha=3.5, \delta=1.5\), and

$$\begin{aligned}& f(t, u, v, w)=(u+v+w) \frac{100(u+v+w)+1}{u+v+w+1}\bigl(3+\sin(u+v+w)\bigr), \\& g(t, u, v, w)=(u+v+w) \frac{80(u+v+w)+1}{u+v+w+1}\bigl(4+\sin(u+v+w)\bigr), \\& h(t, u, v, w)=(u+v+w) \frac{125(u+v+w)+1}{u+v+w+1}\bigl(3+\cos(u+v+w)\bigr). \end{aligned}$$

We can check that

$$k_{1}(t, s)=k_{2}(t,s)=k_{3}(t, s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{t^{\alpha-1}(1-s)^{\alpha-\delta-1}-(t-s)^{\alpha -1}}{\Gamma (\alpha)} ,&{0\leq s\leq t\leq1,}\\ \frac{t^{\alpha-1}(1-s)^{\alpha-\delta-1}}{\Gamma(\alpha)} ,&{0\leq t\leq s\leq1}. \end{array}\displaystyle \right . $$

It is easy to verify that \(\max_{t\in[0,1]} k_{i}(t, s)=k_{i}(1, s), i=1, 2, 3\), for each \(s\in[0, 1]\) and there is a positive constant \(\gamma\in(0, 1)\) such that [58]

$$\min_{t\in[\frac{1}{2},1]}k_{i}(t, s)\geq\gamma k_{i}(1, s),\quad 0< s< 1, i=1, 2, 3, \gamma=\min \biggl\{ \frac{(1/2)^{\alpha-\delta -1}}{2^{\delta}-1}, \biggl( \frac{1}{2} \biggr)^{\alpha-1} \biggr\} . $$

Thus

$$\begin{aligned}& A=B=C= \int^{1}_{0}k_{i}(1, s)\,ds= \frac{3}{4\Gamma (4.5)}\thickapprox0.0645,\qquad \gamma= \biggl(\frac{1}{2} \biggr)^{2.5},\\& \tilde{A}=\tilde{B}=\tilde{C}=0.0114, \\& f^{s}_{0}=3,\qquad f^{i}_{0}=3,\qquad f^{s}_{\infty}=400,\qquad f^{i}_{\infty}=200, \\& g^{s}_{0}=4,\qquad g^{i}_{0}=4,\qquad g^{s}_{\infty}=400,\qquad g^{i}_{\infty}=240, \\& h^{s}_{0}=3,\qquad h^{i}_{0}=3,\qquad h^{s}_{\infty}=500,\qquad h^{i}_{\infty}=250, \\& M_{1}=400,\qquad M_{2}=400,\qquad M_{3}=500, \qquad m_{1}=3,\qquad m_{2}=4,\qquad m_{3}=3. \end{aligned}$$

Choose \(a= \frac{1}{3}, b=\frac{1}{3}, c=\frac{1}{3}, d=\frac {1}{3}, \xi=\frac{1}{2}, \eta=1\). Then

$$\begin{aligned}& K_{1}=\frac{b}{\gamma\tilde{A}f^{i}_{\infty}}\thickapprox0.8269,\qquad K_{2}= \frac{a}{Af^{s}_{0}}\thickapprox1.7227,\qquad K_{3}=\frac{1-b}{\gamma\tilde{B}g^{i}_{\infty}} \thickapprox0.6891, \\& K_{4}=\frac{1-a}{Bg^{s}_{0}}\thickapprox1.2920,\qquad K_{5}= \frac{1-a-b}{\gamma\tilde{C}h^{i}_{\infty}}\thickapprox 0.6615,\qquad K_{6}=\frac{1-c-d}{Ch^{s}_{0}} \thickapprox1.7227, \\& \lambda_{0}= \frac{a}{AM_{1}}\thickapprox0.0129,\qquad \mu _{0}= \frac{b}{BM_{2}}\thickapprox0.0129,\qquad \zeta_{0}= \frac {1-a-b}{CM_{3}}\thickapprox 0.0103, \\& \tilde{\lambda}_{0}= \frac{c}{\gamma\tilde {A}m_{1}}\thickapprox55.1249,\qquad \tilde{\mu}_{0}= \frac{d}{\gamma\tilde{B}m_{2}}\thickapprox 41.3437, \qquad\tilde{ \zeta}_{0}= \frac{1-c-d}{\gamma\tilde{C}m_{3}}\thickapprox55.1249. \end{aligned}$$

Thus,

  1. (1)

    From Theorem 2.1, for \(\lambda\in(K_{1}, K_{2}), \mu\in(K_{3}, K_{4}), \zeta\in(K_{5}, K_{6})\), the problem (P3) has a positive solution;

  2. (2)

    from Theorem 3.1, for \(\lambda\in(0, \lambda_{0}), \mu\in(0, \mu_{0}), \zeta\in(0, \zeta_{0})\), the problem (P3) has no positive solution;

  3. (3)

    from Theorem 3.8, for \(\lambda\in(\tilde{\lambda}_{0}, \infty ), \mu\in(\tilde{\mu}_{0}, \infty), \zeta\in(\tilde{\zeta}_{0}, \infty)\) the problem (P3) has no positive solution.

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Acknowledgements

The work is sponsored by the NSFC (11201109), Anhui Provincial Natural Science Foundation (1408085QA07), the Higher School Natural Science Project of Anhui Province (KJ2014A200) and the outstanding talents plan of Anhui High school.

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Correspondence to Liu Yang.

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The authors declare that they have no competing interests.

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The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

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MSC

  • 45C05
  • 34B18

Keywords

  • positive solution
  • boundary value problem
  • integral equation
  • fixed point
  • cone