Abstract
In this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditions
where \((-1)^{n}f>0\) is continuous, and \(k_{i}(t)\in L^{1}[0,1]\) (\(i=0,1,\ldots,n-1\)) are nonnegative. The associated Green’s function for the higher order differential equations with integral boundary conditions is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem.
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1 Introduction
The multi-point boundary value problems (BVPs) for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. The study of nonlocal BVPs for second order ordinary differential equations has been widely investigated in [1–3]. Since then, nonlinear high order nonlocal BVPs have been studied by many authors. We refer the reader to [4–13] and references therein. Recently, Guo et al. [14] used Leggett-Williams fixed point theorem to obtain the existence of at least three positive solutions for the 2nth order m-point BVP
where \(k_{ij}>0\) (\(i=0,1,\ldots,n-1\); \(j=1,2,\ldots,m-2\)), \(0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1\), and \((-1)^{n}f: [0,1]\times\mathbb{R}^{n}\rightarrow[0, +\infty)\) is continuous.
BVPs with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various physical, biological and chemical processes, such as heat conduction, chemical engineering, thermo-elasticity, underground water flow, population dynamics, and plasma physics. Such problems include two-, three-, multi-point and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention; see [15–19] and references therein. In particular, we would mention the result of [19], Zhang and Ge investigated the existence and nonexistence of positive solutions of the following fourth-order BVP with integral boundary conditions
where ω may be singular at \(t=0\) and (or) \(t=1\), \(f\in C([0,1]\times[0,+\infty)\times(-\infty,0],[0,+\infty))\), and \(g, h\in L^{1}[0,1]\) are nonnegative.
Motivated by [14, 19], in this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditions
where \((-1)^{n}f>0\) is continuous, and \(k_{i}(t)\in L^{1}[0,1]\) (\(i=0,1,\ldots,n-1\)) are nonnegative.
For more precise conditions on f, let \({(-1)}^{j}[a,b]=[a,b]\) if j is even and \({(-1)}^{j}[a,b]=[-b,-a]\) if j is odd. Let
We shall require that
We shall suppose the following conditions are satisfied:
- (H1):
-
\(k_{i}(t)\in L^{1}[0,1]\) are nonnegative, and \(K_{i}\in [0,1)\), where
$$ K_{i}= \int_{0}^{1}(1-s)k_{i}(s) \,\mathrm{d}s, \quad 0\leq i\leq n-1; $$ - (H2):
-
\((-1)^{n}f: [0,1]\times \prod_{j=0}^{n-1}(-1)^{j}[0, +\infty)\rightarrow[0, +\infty)\) is continuous.
2 Preliminary results
Definition 2.1
Let E be a Banach space over \(\mathbb{R}\). A nonempty convex closed set \(K\subset E\) is said to be a cone provided that
-
(i)
\(au\in K\) for all \(u\in K\) and all \(a\geq0\);
-
(ii)
\(u,-u\in K\) implies \(u=0\).
Definition 2.2
The map α is said to be a nonnegative continuous concave functional on K provided that \(\alpha:K\rightarrow[0,\infty) \) is continuous and
for all \(x,y\in K \) and \(0\leq t \leq1\). Similarly, we say the map γ is a nonnegative continuous convex functional on K provided that \(\gamma:K\rightarrow[0,\infty) \) is continuous and
for all \(x,y\in K \) and \(0\leq t \leq1\).
Definition 2.3
Let \(0< a< b\) be given and let α be a nonnegative continuous concave functional on K. Define the convex sets \(P_{r}\) and \(P(\alpha,a,b)\) by
Theorem 2.4
(Leggett-Williams fixed point theorem [20])
Let \(A:\overline{P}_{c}\rightarrow\overline{P}_{c}\) be a completely continuous operator and let α be a nonnegative continuous concave functional on K such that \(\alpha(x)\leq\|x\|\) for all \(x\in\overline{P}_{c}\). Suppose there exist \(0< a< b< d\leq c\) such that
- (C1):
-
\(\{x\in P(\alpha,b,d)|\alpha(x)>b\}\neq\emptyset\), and \(\alpha(Ax)>b\) for \(x\in P(\alpha,b,d)\),
- (C2):
-
\(\|Ax\|< a\) for \(\|x\|\leq a\), and
- (C3):
-
\(\alpha(Ax)>b\) for \(x\in P(\alpha,b,c)\), with \(\|Ax\| >d\).
Then A has at least three fixed points \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
Remark 2.5
If we have \(d=c\), then condition (C1) of Theorem 2.4 implies condition (C3) of Theorem 2.4.
3 Preliminary lemmas
Lemma 3.1
Suppose (H1) holds. Then \(g_{i}(t,s)\leq0\) (\(0\leq i\leq n-1\)), where \(g_{i}(t,s)\) is the Green’s function for the problem
Proof
It is easy to see that \(g_{i}(t,s)\leq0\) (\(0\leq i\leq n-1\)) by using Lemma 2.1 of [19]. □
Let \(G_{1}(t,s)=g_{n-2}(t,s)\), then for \(2\leq j\leq n-1\), we recursively define
Lemma 3.2
Suppose (H1) holds. If \(y\in C[0,1]\), then the BVP
has a unique solution for each \(1\leq l\leq n-1\), where \(G_{l}(t, s)\) is the associated Green’s function for the BVP (3.1).
Proof
We will prove the result by using mathematical induction.
When \(l=1\), which implies that \(i=0\), then the BVP (3.1) reduces to
By using Lemma 3.1, it is easy to see that the BVP (3.2) has a unique solution
Therefore, the result holds for \(l=1\).
We assume that the result holds for \(l-1\). Now, we deal with the case for l. Let \(x''(t)=u(t)\), then the BVP (3.1) is equivalent to the following BVPs:
and
By applying Lemma 3.1, the BVP (3.3) has a unique solution
Replacing l by \(l-1\) and x by u in (3.1), by applying the inductive hypothesis, the BVP (3.4) has also a unique solution
Substituting (3.6) into (3.5), we see that the BVP (3.1) has a unique solution
Therefore, the result holds for l. Lemma 3.2 is now completed. □
For each \(1\leq l\leq n-1\), we define \(A_{l}:C[0,1]\rightarrow C[0,1]\) by
With the use of Lemma 3.2, for each \(1\leq l\leq n-1\), we have
Therefore (1.1) has a solution if and only if the BVP
has a solution. If x is a solution of (1.1), then \(u= x^{(2(n-1))}\) is a solution of (3.7). Conversely, if u is a solution of (3.7), then \(x=A_{n-1}u\) is a solution of (1.1). In addition if \((-1)^{n-1}u(t)\geq0\) (≠0) on \([0,1]\), then \(x=A_{n-1}u\) is a positive solution of (1.1).
For \(0\leq i\leq n-1\), let
Obviously, \(0< m_{i}<M_{i}\). Let E denote the Banach space \(C[0,1]\) with the maximum norm
and define the cone \(K \subset E\) by
Finally, we define the nonnegative continuous concave functional α on K by
for each \(u\in K\) and it is easy to see that \(\alpha(u)\leq\|u\|\).
4 Main results
Theorem 4.1
Suppose conditions (H1), (H2) hold. In addition assume there exist nonnegative numbers a, b, and c such that \(0< a < b\leq\min \{\tau^{2}, \frac{m_{n-1}}{M_{n-1}} \}c\) and \(f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\) satisfies the following growth conditions:
- (H3):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\leq\frac{c}{M_{n-1}}\), for \((t, u_{n-1}, u_{n-2},\ldots, u_{1}, u_{0})\in[0, 1]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [0, \prod_{i=2}^{j+1}M_{n-i}c ]\times(-1)^{n-1}[0, c]\);
- (H4):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})<\frac{a}{M_{n-1}}\), for \((t, u_{n-1}, u_{n-2},\ldots, u_{1}, u_{0})\in[0, 1]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [0, \prod_{i=2}^{j+1}M_{n-i}a ]\times(-1)^{n-1}[0, a]\);
- (H5):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\geq \frac{b}{m_{n-1}}\), for \((t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\in[\tau, 1-\tau]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [\prod_{i=2}^{j+1}m_{n-i}b, \prod_{i=2}^{j+1}M_{n-i}\frac{b}{\tau^{2}} ]\times (-1)^{n-1}[b, \frac{b}{\tau^{2}}]\).
Then the BVP (1.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\), which satisfy
Proof
Define the completely continuous operator A by
We will first verify that \(A:K\rightarrow K\). Let \(u\in K\), then \((-1)^{n-1}Au(t)\geq0\), \(((-1)^{n-1}Au)''(t)=(-1)^{n-1}f (t,A_{n-1}u(t),\ldots ,A_{1}u(t),u(t) )\leq 0\), \(0\leq t\leq1\), and by Proposition 2.2 of [19], we have
On the other hand, by Proposition 2.3 of [19], we obtain
Consequently, \(A:K\rightarrow K\).
It is a standard argument to show that the operator A is completely continuous. Equicontinuity and uniform boundedness follow readily from the properties of \(G_{l}\), \(1\leq l\leq n-1\).
If \(u\in\overline{P}_{c}\), then \(\|u\|\leq c\). For each \(1\leq j\leq n-1\), note that inductively (using (3.8)) we have
From the condition (H3) and (3.8), we obtain
So, \(A:\overline{P}_{c}\rightarrow\overline{P}_{c}\).
In a completely analogous argument, the condition (H4) implies the condition (C2) of Theorem 2.4 is satisfied.
We now show that condition (C1) is satisfied. Note that, for \(0\leq t\leq1\),
Thus,
Also, if \(u\in P(\alpha,b,\frac{b}{\tau^{2}})\), then \(b\leq (-1)^{n-1}u(t)\leq\frac{b}{\tau^{2}}\) for \(t\in[\tau, 1-\tau]\), implies for each \(1\leq j\leq n-1\), \(t\in[\tau, 1-\tau]\), inductively,
With the use of condition (H5) and (3.8), we get
Therefore, condition (C1) is satisfied.
Finally, we show that condition (C3) is also satisfied. That is, we show that if \(u\in P(\alpha,b,c)\) and \(\|Au\|>d=\frac{b}{\tau^{2}}\), then \(\alpha(Au)>b\). This follows since \(A:K\rightarrow K\). In particular, since \((-1)^{n-1}(Au)\) is concave and \(\min_{t\in[\tau, 1-\tau]}(-1)^{n-1}(Au)(t)\geq \tau^{2}\|Au\|\).
That is,
Therefore, condition (C3) is also satisfied. By Theorem 2.4, there exist three solutions \(u_{1}, u_{2}, u_{3}\in K\) for the BVP (3.7). Moreover, let
then \(x_{1}\), \(x_{2}\), \(x_{3}\) are three positive solutions for the BVP (1.1) and satisfy
□
Consider the following 2nth order differential equations with integral boundary conditions:
where \((-1)^{n}f\in C([0,1]\times \prod_{j=0}^{n-1}(-1)^{j}[0, +\infty)\rightarrow[0, +\infty))\) and \(k^{\ast}_{i}(t)\in L^{1}[0,1]\) (\(i=0,1,\ldots,n-1\)) are nonnegative.
Now we deal with problem (4.1). The method is just similar to what we have done for the problem (1.1), so we omit the proof of main results in this section.
For convenience, we list the following assumptions:
- (\(\mathrm{H}^{\ast}_{1}\)):
-
\(k^{\ast}_{i}(t)\in L^{1}[0,1]\) are nonnegative, and \(K^{*}_{i}\in[0,1)\), where
$$ K^{*}_{i}= \int_{0}^{1}sk^{\ast}_{i}(s) \, \mathrm{d}s, \quad 0\leq i\leq n-1. $$
By analogous methods, we have the following results.
Lemma 4.2
Suppose (\(\mathrm{H}^{\ast}_{1}\)) holds. Then \(g^{\ast }_{i}(t,s)\leq0\) (\(0\leq i\leq n-1\)), where \(g^{\ast}_{i}(t,s)\) is the Green’s function for the problem
For \(0\leq i\leq n-1\), let
Theorem 4.3
Suppose condition (\(\mathrm{H}^{\ast}_{1}\)) holds. In addition assume there exist nonnegative numbers a, b, and c such that \(0< a < b\leq\min \{\tau^{2}, \frac{m^{\ast}_{n-1}}{M^{\ast}_{n-1}} \}c\) and \(f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\) satisfies the following growth conditions:
- (\(\mathrm{H}^{\ast}_{2}\)):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\leq\frac{c}{M^{\ast}_{n-1}}\), for \((t, u_{n-1}, u_{n-2},\ldots, u_{1}, u_{0})\in[0, 1]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [0, \prod_{i=2}^{j+1}M^{\ast}_{n-i}c ]\times(-1)^{n-1}[0, c]\);
- (\(\mathrm{H}^{\ast}_{3}\)):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})<\frac{a}{M^{\ast}_{n-1}}\), for \((t, u_{n-1}, u_{n-2},\ldots, u_{1}, u_{0})\in[0, 1]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [0, \prod_{i=2}^{j+1}M^{\ast}_{n-i}a ]\times(-1)^{n-1}[0, a]\);
- (\(\mathrm{H}^{\ast}_{4}\)):
-
\((-1)^{n}f(t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\geq\frac{b}{m^{\ast}_{n-1}}\), for \((t,u_{n-1},u_{n-2},\ldots,u_{1}, u_{0})\in[\tau, 1-\tau]\times \prod_{j=n-1}^{1}(-1)^{n-1-j} [\prod_{i=2}^{j+1}m^{\ast}_{n-i}b, \prod_{i=2}^{j+1}M^{\ast}_{n-i}\frac{b}{\tau^{2}} ]\times (-1)^{n-1}[b, \frac{b}{\tau^{2}}]\).
Then the BVP (4.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\), which satisfy
5 Example
Example 5.1
As an example of problem (1.1), consider the following sixth order BVP:
where
We notice that \(n=3\), \(k_{i}(s)=s^{i}\) (\(i=0, 1, 2\)) and \(K_{0}=\frac{1}{2}\), \(K_{1}=\frac{1}{6}\), \(K_{2}=\frac{1}{12}\).
If we take \(\tau=\frac{1}{4}\), by calculation we obtain
In addition, if we take \(a=1\), \(b=2\), \(c=256\), then
and \(f(t,u_{2},u_{1},u_{0})\) satisfies the growth conditions (H3)-(H5).
Therefore all the conditions of Theorem 4.1 are satisfied. Hence, the problem (5.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\), which satisfy
Example 5.2
As another example of problem (4.1), consider the following sixth order BVP:
where
We notice that \(n=3\), \(k^{\ast}_{i}(s)=s^{i}\) (\(i=0, 1, 2\)) and \(K^{\ast}_{0}=\frac{1}{2}\), \(K^{\ast}_{1}=\frac{1}{3}\), \(K^{\ast}_{2}=\frac{1}{4}\).
If we take \(\tau=\frac{1}{4}\), by calculation we obtain
In addition, if we take \(a=1\), \(b=2\), \(c=256\), then
and \(f(t,u_{2},u_{1},u_{0})\) satisfies the growth conditions (\(\mathrm{H}_{2}^{\ast}\))-(\(\mathrm{H}_{4}^{\ast}\)).
Therefore all the conditions of Theorem 4.3 are satisfied. Hence, the problem (5.2) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\), which satisfy
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Acknowledgements
The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very helpful suggestions. The project was supported by the Natural Science Foundation of China (11371120), the Natural Science Foundation of Hebei Province (A2013208147, A2015208051).
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Ji, Y., Guo, Y. & Yao, Y. Positive solutions for higher order differential equations with integral boundary conditions. Bound Value Probl 2015, 214 (2015). https://doi.org/10.1186/s13661-015-0485-3
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DOI: https://doi.org/10.1186/s13661-015-0485-3