Abstract
In this paper we use the fixed point index to study the existence of positive solutions for a system of 2nth-order boundary value problems involving semipositone nonlinearities.
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1 Introduction
In this paper we investigate the existence of positive solutions for the following system of 2nth-order boundary value problems involving semipositone nonlinearities:
where \(n\in N\) with \(n\ge 1\), and \(f_{j}\in C ([0,1]\times R_{+} ^{4 n-2}, R )\) (\(R_{+}:=[0, \infty )\), \(R:=(-\infty ,+\infty )\), \(j=1,2\)) satisfy the semipositone condition:
- (H0)
there is a positive constant M such that
$$ f_{j}(t,z_{1},z_{2},\ldots,z_{4n-2}) \ge -M,\quad t\in [0,1], z_{i} \in R_{+}, i=1,2,\ldots,4n-2, j=1,2. $$
In recent years, coupled systems of boundary value problems have been investigated by many authors since such systems appear naturally in many real-world situations. Some recent results on the topic can be found in a series of papers [1–26] and the references therein. In [1], Yang used nonnegative matrix theory to study the existence of positive solutions for the system of generalized Lidstone problems,
where \(f_{1}, f_{2} \in C ([0,1] \times R_{+}^{m+n},R_{+} )\), and in [2] Xu and Yang used some concave functions to depict the coupling behaviors for the nonlinearities \(f_{i}\) (\(i=1,2\)), and they established the existence of positive solutions for (1.2). In [3], Wang and Yang used similar methods as in [1] to study the existence of positive solutions for the system of higher-order boundary value problems involving all derivatives of odd orders
where \(f, g \in C ([0,1] \times R_{+}^{m+n+2},R_{+} )\). Moreover, they used a condition of Berstein–Nagumo type to obtain a priori estimates for \(w^{(2m-1)}\) and \(z^{(2 n-1)}\). For related papers, we refer the reader to [27–33]. In [27] the authors used topological degree theory to study the existence of nontrivial solutions for the higher-order nonlinear fractional boundary value problem involving Riemann–Liouville fractional derivatives:
where \(D_{0+}^{\alpha }\), \(D_{0+}^{\beta }\), \(D_{0+}^{\beta _{i}}\) are the Riemann–Liouville fractional derivatives, and \(f\in C([0,1]\times R ^{n}, R)\).
Motivated by the above work, in this paper we investigate the positive solutions for the system of 2nth-order boundary value problems (1.1) involving semipositone nonlinearities. We first use the method of order reduction to transform (1.1) into an equivalent system of integro-integral equations, and then we establish a system of nonnegative operator equations. Using the fixed point index and nonnegative matrix theory, we study the existence of positive fixed points for the operator equations, and obtain positive solutions for (1.1).
2 Preliminaries
Let \(E=C[0,1]\), \(\|z\|=\max_{t\in [0,1]}|z(t)|\), \(P=\{t\in [0,1]: z(t) \ge 0, \forall t\in [0,1]\}\). Then \((E,\|\cdot \|)\) is a Banach space, and P a cone on E. Let
and
Note
and \(B_{i},B_{i}':E\to E\) are completely continuous linear operators, \(B_{i}\), \(B_{i}'\) are also positive operators, i.e., they will map P into P.
Lemma 2.1
([28])
Let \(\kappa _{\psi }=1-2/e\), and \(\psi (t)=te^{t}\), \(t\in [0,1]\). Then we have
Lemma 2.2
([28])
Let \(z\in P\). Then we have
Lemma 2.3
([34])
LetEbe a real Banach space andPa cone onE. Suppose that \(\varOmega \subset E\)is a bounded open set and that \(A:\overline{ \varOmega }\cap P\to P\)is a continuous compact operator. If there exists a \(\omega _{0}\in P\setminus \{0\}\)such that
then \(i(A,\varOmega \cap P,P)=0\), whereidenotes the fixed point index onP.
Lemma 2.4
([34])
LetEbe a real Banach space andPa cone onE. Suppose that \(\varOmega \subset E\)is a bounded open set with \(0\in \varOmega \)and that \(A:\overline{\varOmega }\cap P\to P\)is a continuous compact operator. If
then \(i(A,\varOmega \cap P,P)=1\).
Now, we consider the following auxiliary problem associated with (1.1):
where \(f\in C ([0,1]\times R_{+}^{2 n-1}, R )\) satisfies the condition:
- (H0)′:
there is a positive constant M such that
$$ f(t,z_{1},z_{2},\ldots,z_{2n-1})\ge -M, \quad t\in [0,1], z_{i}\in R _{+}, i=1,2,\ldots,2n-1. $$
Let \((-1)^{n-1} x^{(2 n-2)}(t)=z(t)\), \(t\in [0,1]\). Then we have
which can be expressed in the integral form
For convenience, let
As a result, we can also write (2.2) in the form
Let \(w(t)=M\int _{0}^{1}k_{1}(t,s)\,ds=M(t-t^{2}/2)\). We need to consider the following problem:
where
Note that (2.3) can be expressed in the integral form
Using (H0)′, we see that \(\widetilde{f}\in C ([0,1] \times R _{+}^{2 n-1}, R_{+} )\).
Lemma 2.5
-
(i)
If \(z^{*}\)is a positive solution of (2.1), then \(z^{*}+w\)is a positive solution of (2.3).
-
(ii)
If \(z^{**}\)is a positive solution of (2.3), and greater thanw, then \(z^{**}-w\)is a positive solution of (2.1).
Proof
Substituting \(z^{*}+w\) into (2.3), we have
Note that w satisfies the boundary value problem
By virtue of (2.4), we have
which is (2.1).
On the other hand, we substitute \(z^{**}-w\) into (2.1), and obtain
Note that, from the definitions of w and f̃, we have
which is (2.3). This completes the proof. □
From Lemma 2.5, if we wish to seek the positive solutions for (2.1), we only need to study the positive solutions for (2.3), which are greater than w. Consequently, we define an operator \(T:P\to E\) as follows:
Then T is a completely continuous operator, and if there exists a \(\overline{z}\in P\) with \(\overline{z}\ge w\) such that \(T\overline{z}= \overline{z}\), we see that \(\overline{z}-w\) is a positive solution of (2.1).
Let
Then \(P_{0}\) is also a cone on E, and we have the following lemma.
Lemma 2.6
\(T(P)\subset P_{0}\).
Note that, for \(t,s\in [0,1]\), \(tk_{1}(s,s)\le k_{1}(t,s)\le k_{1}(s,s)\) and \(k_{1}(s,s)=s\), so we can easily obtain this lemma (the details are omitted).
From Lemmas 2.5 and 2.6, we have \(\overline{z}\in P_{0}\) if z̅ is a fixed point of T. Consequently, if \(\| \overline{z}\|\ge M \) we have
Hence, we only need to seek T’s positive fixed point z̅ with \(\|\overline{z}\|\ge M\), and then \(\overline{z}-w\) is a positive solution of (2.1).
3 Main results
In (1.1), let \((-1)^{n-1} x^{(2 n-2)}=u\) and \((-1)^{n-1} y^{(2n-2)}=v\), then we obtain the following system of boundary value problems:
which has the integral form
For \(j=1,2\), let
Then we can define the operators \(T_{j}\ (j=1,2):P^{4n-2}\to P\) and \(T:P^{2} \to P^{2}\) as follows:
and
Then, if we find the positive fixed point \((u^{*},v^{*})\) of T with \(u^{*},v^{*}\ge w\), then \((u^{*}-w,v^{*}-w)\) is a positive solution for (3.1). Let
and we will obtain the positive solution for (1.1) (note from the discussion in Sect. 2, we need the norms of \(u^{*}\), \(v^{*}\) to be greater than M).
Now, we list our assumptions for \(F_{j}\) (\(j=1,2\)):
(H1) There exist \(a_{j1},b_{j1},c_{j1},d_{j1},l_{j}>0\) (\(j=1,2\)) such that
and, for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in R_{+}\), \(i=1,2,\ldots,2n-1\),
(H2) There exist \(Q_{j}\ (j=1,2):[0,1]\to R\) such that
and
for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in [0,M]\), \(i=1, 2,\ldots,2n-1\), \(j=1,2\).
(H3) There exist \(\widetilde{a}_{j1},\widetilde{b}_{j1},\widetilde{c} _{j1},\widetilde{d}_{j1},\widetilde{l}_{j}>0\) (\(j=1,2\)) such that
and, for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in R_{+}\), \(i=1,2,\ldots,2n-1\),
(H4) There exist \(\widetilde{Q}_{j}\ (j=1,2):[0,1]\to R\) and \(t_{1},t _{2}\in (0,1]\) such that
and
for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in [0,M]\), \(i=1, 2,\ldots,2n-1\), \(j=1,2\).
Let \(B_{\rho }=\{u\in P: \|u\|<\rho \}\) for \(\rho >0\) in the sequel. Then we easily have \(\partial B_{\rho }=\{u\in P: \|u\|=\rho \}\), \(\overline{B}_{\rho }=\{u\in P: \|u\|\le \rho \}\).
Theorem 3.1
Suppose that (H0)–(H2) hold. Then (1.1) has at least one positive solution.
Proof
We first prove that there exists \(R_{1}>M\) such that
where \(\phi _{i}\) (\(i=1,2\)) are given elements in the cone \(P_{0}\). We argue by contradiction. Suppose there exist \((u,v)\in \partial B_{R_{1}} \cap (P\times P)\) and \(\lambda _{0}\ge 0\) with
This, together with Lemma 2.6, implies that \(u,v\in P_{0}\). Moreover, from (H1) we have
Multiply by \(\psi (t)\) on both sides, integrate over \([0,1]\), and use Lemma 2.1, and we have
Using Lemma 2.2 we obtain
Let
Therefore, we have
and
Solving this matrix inequality, we obtain
Consequently, there exist \(\widetilde{\mathcal{N}}_{1}, \widetilde{\mathcal{N}}_{2}>0\) such that
Note that \(u,v\in P_{0}\), and we have
Therefore, we can choose \(R_{1}>\max \{ M,\frac{ \widetilde{\mathcal{N}}_{1}}{e-2}, \frac{\widetilde{\mathcal{N}}_{2}}{e-2} \} \) such that (3.4) is false, and thus (3.3) holds. From Lemma 2.3 we have
Next we prove that
If not, there exist \((u, v) \in \partial B_{{M}} \cap (P \times P)\) and \(\lambda _{1} \in [0,1]\) such that
This, combining with (H2), implies that
This is a contradiction, and thus (3.6) is true. From Lemma 2.4 we have
Therefore the operator T has at least one fixed point \((u^{*},v^{*})\) on \((B_{R_{1}} \setminus \overline{B}_{{M}} ) \cap (P \times P)\) with \(\|u^{*}\|\ge M\), \(\|v^{*}\|\ge M\), and note from (3.2) we see that (1.1) has at least one positive solution. This completes the proof. □
Theorem 3.2
Suppose that (H0), and (H3)–(H4) hold. Then (1.1) has at least one positive solution.
Proof
We first prove that
where \(\widetilde{\phi }_{i}\ (i=1,2)\in P\) are fixed elements. If this claim is false, there exist \((u,v)\in \partial B_{M}\cap (P\times P)\) and \(\lambda _{2}\ge 0\) such that
This, together with (H4), gives
This is a contradiction, and thus (3.8) holds. From Lemma 2.3 we have
Next we show that there is a large number \(R_{2}>M\) such that
We argue by contradiction, so we assume there exist \((u, v) \in \partial B_{R_{2}} \cap (P \times P)\) and \(\lambda _{3}\in [0,1]\) such that
Lemma 2.6 implies that \(u,v\in P_{0}\), and from (H3) we obtain
Multiply by \(\psi (t)\) on both sides, integrate over \([0,1]\), and use Lemma 2.1, we have
This, combining with Lemma 2.2, implies that
Consequently, we have
Solving this matrix inequality, we obtain
Therefore, there exist \(\widetilde{\mathcal{N}}_{3}, \widetilde{\mathcal{N}}_{4}>0\) such that
Note that \(u,v\in P_{0}\), and then we obtain
If we choose \(R_{2}>\max \{ M,\frac{\widetilde{\mathcal{N}} _{3}}{e-2},\frac{\widetilde{\mathcal{N}}_{4}}{e-2} \} \) then (3.10) holds. From Lemma 2.4 we have
Therefore the operator T has at least one fixed point \((u^{*},v^{*})\) on \((B_{R_{2}} \setminus \overline{B}_{{M}} ) \cap (P \times P)\) with \(\|u^{*}\|\ge M\), \(\|v^{*}\|\ge M\), and note from (3.2) we see that (1.1) has at least one positive solution. This completes the proof. □
Example 3.3
Let
Then
Consider
for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in R_{+}\), \(i=1,2,\ldots,2n-1\), and \(\delta _{1},\delta _{2}>1\).
For all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in [0,M]\), \(i=1, 2,\ldots,2n-1\), \(j=1,2\), we have
Consequently, if let \(Q_{1}(t)\equiv \frac{9}{5}M\), \(Q_{2}(t)\equiv \frac{19}{10}M\) for \(t\in [0,1]\), then (H2) holds.
On the other hand, for all \(t\in [0,1]\) we note that
and
Therefore, (H1) holds.
Example 3.4
Let \(t_{1}=\frac{1}{2}\), \(t_{2}=1\), and note that \(\int _{0}^{1} k_{1}(t,s)\,ds=t- \frac{1}{2}t^{2}\) for \(t\in [0,1]\), and if we consider the case \(\widetilde{Q}_{j}\equiv \mbox{constant}\), we have
To obtain the first inequality in (H4), we can take \(\widetilde{Q} _{1}=3M\), \(\widetilde{Q}_{2}=\frac{5}{2}M\).
Let
Then we have
Let
for all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in R_{+}\), \(i=1,2,\ldots,2n-1\).
For all \(t\in [0,1]\), \(z_{i},\widetilde{z}_{i}\in [0,M]\), \(i=1, 2,\ldots,2n-1\), \(j=1,2\), we have
and thus (H4) holds. On the other hand, for all \(t\in [0,1]\) we also have
and
Therefore, (H3) holds.
4 Conclusion
In this paper we use the fixed point index to study the existence of positive solutions for the system of 2nth-order boundary value problems (1.1) involving semipositone nonlinearities. Our nonlinearities not only depend on all derivatives of unknown functions, but they also grow superlinearly and sublinearly at infinity.
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Acknowledgements
The authors would like to thank the referees for their pertinent comments and valuable suggestions.
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This work is supported by the China Postdoctoral Science Foundation (Grant Nos. 2017M612230, 2019M652348), and the Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24).
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Hao, X., O’Regan, D. & Xu, J. Positive solutions for a system of 2nth-order boundary value problems involving semipositone nonlinearities. J Inequal Appl 2020, 20 (2020). https://doi.org/10.1186/s13660-020-2296-z
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DOI: https://doi.org/10.1186/s13660-020-2296-z