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.
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1 Introduction
We consider the existence of eigenvalues yielding positive solutions to the system of integral equations
where \(\lambda, \mu, \zeta\) are positive numbers and
-
(H1)
\(f, g, h\in C([0, 1]\times R^{+}\times R^{+}\times R^{+}, R^{+})\).
-
(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:
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:
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, [3–8] (ordinary differential equations), [9–15] (differential equations on time scales), and [16–20] (fractional differential equations). For \(n=2\), see [21–25] (differential equations on time scales), [26–36] (ordinary differential equations), and [37–54] (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
be a completely continuous operator such that
or
then A has a fixed point in \(K\cap(\Omega_{2}\setminus\overline {\Omega}_{1})\).
2 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
and the Banach space \(Y=X\times X\times X\) with the norm
We define the cone \(P\subset Y\) by
Define the operators \(T_{1}:Y\rightarrow X, T_{2}: Y\rightarrow X, T_{3}: Y\rightarrow X\), and \(T: Y\rightarrow Y\) by
and
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
Then for \((u, v, w)\in\Omega\), we have
Hence \(T(\Omega)\) is bounded.
By means of the Arzela-Ascoli theorem, we see that T is completely continuous. Furthermore, considering
Hence
Thus, we show that \(T: P\rightarrow P\) is a completely continuous operator.
Here we introduce the following extreme limits:
Denote the positive constants
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
□
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
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 \(u(t)+v(t)+w(t)\leq R_{1}\),
We define the set
Let \((u, v, w)\in P\cap\partial\Omega_{1}\),
Then for \((u, v, w)\in P\cap\partial\Omega_{1}\),
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 \(u(t)+v(t)+w(t)\geq\overline{R}_{2}\),
We consider \(R_{2}=\max\{2R_{1}, \overline{R}_{2}/\gamma\} \), and we define the set
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
Thus,
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
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
By condition (H1), there exists \(R_{3}>0\) such that for \(t\in[\xi, \eta], u(t), v(t), w(t)\geq0\) and \(u(t)+v(t)+w(t)\leq R_{3}\),
We define the set
Let \((u, v, w)\in P\cap \partial\Omega_{3}\),
Then for \((u, v, w)\in P\cap\partial\Omega_{3}\),
On the other side, we define the functions \(f^{\ast}, g^{\ast}, h^{\ast} [0, 1]\times R^{+}\longrightarrow R^{+}\),
Then
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
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 \(x\geq\overline {R}_{4}, t\in[0, 1]\),
Then
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
Thus
Then for \((u, v, w)\in P\cap\partial\Omega_{4}\),
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.
3 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
Define the positive constants
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,
Then
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
Define thepositive constants
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
Thus,
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
Define the positive constants
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
Thus,
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
there exist positive numbers \(m_{1}, m_{2}, m_{3}\) such that
Define the positive constants
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
Thus,
which is a contradiction. So the boundary value problem \((P_{\lambda, \mu, \zeta})\) has no positive solution. □
4 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.
4.1 Application to system of boundary value problems of ordinary differential equations
Consider the system of nonlinear second order differential equation (the problem (P1))
subject to the boundary conditions
where \(0<\eta_{1}, \eta_{2}, \eta_{3}<1\), and
By using the Green’s functions, we can formulate the problem (P1) as
where
and
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
and
Then:
-
(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)
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)
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.
4.2 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)),
subject to the boundary conditions
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
We can formulate the problem (P2) as
where
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
where
We consider the case
After an easy computation, we conclude
Choose \(a= \frac{1}{3}, b=\frac{1}{3}, c=\frac{1}{3}, d=\frac{1}{3}\). Then
Thus
-
(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)
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)
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.
4.3 Application to system of boundary value problems of fractional differential equations
Consider the system of nonlinear fractional differential equation (the problem (P3))
subject to the boundary conditions
where \(\alpha=3.5, \delta=1.5\), and
We can check that
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]
Thus
Choose \(a= \frac{1}{3}, b=\frac{1}{3}, c=\frac{1}{3}, d=\frac {1}{3}, \xi=\frac{1}{2}, \eta=1\). Then
Thus,
-
(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)
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)
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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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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Shen, C., Zhou, H. & Yang, L. Positive solution of a system of integral equations with applications to boundary value problems of differential equations. Adv Differ Equ 2016, 260 (2016). https://doi.org/10.1186/s13662-016-0953-9
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DOI: https://doi.org/10.1186/s13662-016-0953-9