Abstract
In this paper, we study the existence of multiple positive solutions for fourth-order impulsive differential equation with integral boundary conditions and deviating argument. The main tool is based on the Avery and Peterson fixed point theorem. Meanwhile, an example to demonstrate the main results is given.
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1 Introduction
As an important area of investigation, the theory and applications of the fourth-order ordinary differential equations are emerging. The study concerns mainly the description of the deformations of an elastic beam by means of a fourth-order differential equation boundary value problem (BVP for short). Owing to its various applications in physics, engineering, and material mechanics, a lot of attention has been received by fourth-order differential equation BVPs. For more information, see [1–8].
In the meantime, many very interesting and significant cases of BVPs are constituted by integral boundary conditions. They include two-, three-, multi-point, and nonlocal BVPs as special cases. Therefore, in recent years, increasing attention has been given to integral boundary conditions [9–15] of fourth-order BVPs. Especially, we intend to mention some recent results.
In [12], Ma studied the following fourth-order BVP:
where \(p, q \in L^{1}[0,1]\), h and f are continuous. From an application of the fixed point index in cones, the existence of at least one symmetric positive solution could be obtained.
In 2015, the authors [13] investigated the fourth-order differential equation with integral boundary conditions,
By using a novel technique and fixed point theories, they showed the existence and multiplicity of positive solutions.
Unlike [12] and [13], a class of fourth-order differential equations with advanced or delayed argument were considered by the author of [14],
subject to the boundary conditions
or
The existence of multiple positive solutions is obtained by using a fixed point theorem due to Avery and Peterson.
In addition, to deviating arguments, the authors of [15] studied a fourth-order impulsive BVP as follows:
The boundary conditions above are special Sturm-Liouville integral boundary conditions, since \(ay(0)-by'(0)=ay(1)+by'(1)=\int _{0}^{1}g(s)y(s)\,ds\). Several existence and multiplicity results were derived by using inequality techniques and fixed point theories. For most research papers on impulsive differential equation BVPs, see [16–19] and the references therein.
Motivated by the mentioned results, we investigate a fourth-order impulsive differential equation Sturm-Liouville integral BVP with deviating argument,
subject to
where \(\xi,\eta>0\). Compared with [13–15], in this paper, (1.2) contains the general Sturm-Liouville integral boundary conditions, where \(g_{1}(s)\), \(g_{2}(s)\) could be two different functions in \(L^{1}[0,1]\). In this case, we have to establish a more complicated expression of operator T and to find the proper lower and upper bounds of Green’s functions (Lemma 2.2 and Lemma 2.4). Further, by using the fixed point theorem due to Avery and Peterson, the existence and multiplicity of positive solutions are obtained.
In (1.2), \(t_{k}\) (\(k=1, 2, \ldots, m\)) are fixed points with \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{m}<t_{m+1}=1\), \(\triangle x'_{t_{k}}=x'({t_{k}}^{+})-x'({t_{k}}^{-})\), \(x'({t_{k}}^{+})\) and \(x'({t_{k}}^{-})\) represent the right-hand limit and the left-hand limit of \(x'(t_{k})\) at \(t=t_{k}\), respectively.
Throughout the paper, we always assume that \(J=[0,1]\), \(J_{0}=(0,1) \backslash\{t_{1}, t_{2}, \ldots, t_{m}\}\), \(\mathbb{R}^{+}=[0,+\infty)\), \(J_{k}=(t_{k}, t_{k+1}]\), \(k=0, 1, \ldots, m-1\). \(\alpha: J\rightarrow J\) is continuous and:
- (H1):
-
\(f\in C (J\times\mathbb{R}^{+}\times\mathbb{R}^{+}, \mathbb{R}^{+})\), with \(f(t,u,v)>0\) for \(t\in J\), \(u>0\), and \(v>0\);
- (H2):
-
h is a nonnegative continuous function defined on \((0,1)\); h is not identically zero on any subinterval on \(J_{0}\);
- (H3):
-
\(I_{k}:J\times\mathbb{R}^{+}\rightarrow\mathbb {R}^{+}\) is continuous with \(I_{k}(t,u)>0\) (\(k=1,2,\ldots,m\)) for all \(t\in J\) and \(u >0\);
- (H4):
-
\(g_{0}, g_{1}, g_{2} \in L^{1}[0,1]\) are nonnegative and \(\gamma=\int_{0}^{1}g_{0}(s)\,ds \in(0,1)\).
2 Expression and properties of Green’s function
For \(v(t)\in C(J)\), we consider the equation
with boundary conditions (1.2).
We shall reduce BVP (2.1) and (1.2) to two second-order problems. To this goal, first, by means of the transformation
we convert problem (2.1) and (1.2) into
and
Lemma 2.1
\(\lambda(t)\) is the solution of \(x''(t)=0\), \(x(0)=\xi\), \(x'(0)=1\). \(\mu(t)\) is the solution of \(y''(t)=0\), \(y(1)=\eta\), \(y'(1)=-1\). Then \(\lambda(t)\) is strictly increasing on J, \(\lambda(t)>0\) on \((0,1]\); \(\mu(t)\) is strictly decreasing on J, and \(\mu(t)>0\) on \([0,1)\). For any \(v(t)\in C(J)\), then the BVP (2.3) has a unique solution as follows:
where
and
where
Proof
Since λ and μ are two linearly independent solutions of the equation \(y''(t)=0\), we know that any solution of \(y''=v(t)\) can be represented by (2.5).
It is easy to check that the function defined by (2.5) is a solution of (2.3) if A and B are as in (2.9), respectively.
Now we show that the function defined by (2.5) is a solution of (2.3) only if A and B are as in (2.9), respectively.
Let \(y(t)=(Fv)(t)+A\lambda(t)+B\mu(t)\) be a solution of (2.3), then we have
and
Thus, by (2.8), we can obtain
Since
we have
Since
we have
From (2.11) and (2.12), we get
which implies that A and B satisfy (2.9), respectively. □
Assume that
- (H5):
-
\(\Delta<0\), \(\alpha[\mu]<\rho\), \(\beta[\lambda]<\rho\).
Lemma 2.2
Denote \(e_{1}(t)=G_{1}(t,t)\), \(\hat{e}_{1}(t)=\frac{1}{\rho}(1-t)(t+\xi)\), for \(t\in J\). Let \(\kappa[v]=\int_{0}^{1} e_{1}(s)v(s)\,ds\) and \(\hat{\kappa}[v]=\int_{0}^{1} \hat{e}_{1}(s)v(s)\,ds\), for \(v \in C(J, \mathbb{R}^{+})\). If (H5) is satisfied, then the following results are true:
-
(1)
\(\hat{e}_{1}(s)\hat{e}_{1}(t)\leq G_{1}(t,s)\leq e_{1}(s)\), for \(t,s \in J\);
-
(2)
\(0\leq\underline{A} \hat{\kappa}[v]\leq A(v)\leq\overline{A} \kappa[v]\), for \(v \in C(J, \mathbb{R}^{+})\);
-
(3)
\(0\leq\underline{B} \hat{\kappa}[v] \leq B(v)\leq\overline {B} \kappa[v]\), for \(v \in C(J, \mathbb{R}^{+})\),
where
Proof
Now we show that (1) is true. Obviously, \(G_{1}(t,s)\leq e_{1}(s) \) for \(t,s\in J\).
In fact, \(\hat{e}_{1}(s)\hat{e}_{1}(t)=\frac{1}{\rho^{2}}(1-s)(s+\xi)(1-t)(t+\xi)\), for \(s, t\in J\).
For \(0\leq s\leq t\leq1\), we notice that
It is easy to see that \(1-s\leq1\), \(1-t\leq\eta+1-t\), \(t+\xi\leq 1+\xi+\eta\), for \(s, t\in J\), \(\xi, \eta>0\), which implies
Hence, we have
Similarly, we can obtain
In the following we show (2) and (3) hold. In view of (H5), for \(v \in C(J, \mathbb{R}^{+})\), we have
In the same way, we have \(B(v)\leq\overline{B} \kappa[v]\), \(B(v)\geq\underline{B}\hat{\kappa}[v]\), for \(v \in C(J,\mathbb {R}^{+})\). □
Analogously to Lemma 2.1 in [12], we obtain the following result; we omit the proof.
Lemma 2.3
If (H4) holds, for any \(y \in C(J)\), the problem (2.4) has a unique solution x expressed in the form
where
From (2.14) and (2.15), we can prove that \(H(t,s)\) and \(G_{2}(t,s)\) have the following properties.
Lemma 2.4
Let \(H(t,s)\), \(G_{2}(t,s)\) be given as in Lemma 2.3. Assume that (H4) holds, then the following results are true:
-
(1)
\(e_{2}(s)e_{2}(t)\leq G_{2}(t,s) \leq e_{2}(s)\), for \(t, s \in J\);
-
(2)
\(\frac{\Gamma}{1-\gamma}e_{2}(s)\leq H(t,s) \leq\frac {1}{1-\gamma}e_{2}(s)\), for \(t, s \in J\),
where
Proof
It is easy to see that (1) holds. In the following, we prove that (2) is satisfied:
and
□
Lemma 2.5
Assume that (H1)-(H4) hold. Then problem (2.1) and (1.2) has a unique solution x given by
Lemma 2.6
Assume that (H3)-(H5) hold, for \(v\in C(J,\mathbb{R}^{+})\), the unique solution x of problem (2.1) and (1.2) satisfies \(x(t)\geq0\) on J.
Proof
By Lemma 2.4, we can obtain \(H(t,s)\geq0 \) for \(t, s \in J\). Hence, from Lemma 2.5, combining with Lemma 2.2 and (H3), we can obtain
This completes the proof. □
3 Background materials and definitions
Now, we present the prerequisite definitions in Banach spaces from the theory of cones.
Definition 3.1
Let E be a real Banach space. A nonempty convex closed set \(P\subset E \) is said to be a cone if
-
(i)
\(ku\in P \) for all \(u\in P\) and all \(k\geq0\), and
-
(ii)
\(u, -u\in P \) implies \(u=0\).
Definition 3.2
On a cone P of a real Banach space E, a map Λ is said to be a nonnegative continuous concave functional if \(\Lambda :P\rightarrow\mathbb{R}^{+} \) is continuous and
for all \(x, y \in P \) and \(t\in J\).
At the same time, on a cone P of a real Banach space E, a map φ is said to be a nonnegative continuous convex functional if φ: \(P\rightarrow\mathbb{R}^{+}\) is continuous and
for all \(x,y \in P \) and \(t\in J\).
Definition 3.3
If it is continuous and maps bounded sets into pre-compact sets, an operator is called completely continuous.
Let φ and Θ be nonnegative continuous convex functionals on P, Λ be a nonnegative continuous concave functional on P, and Ψ be a nonnegative continuous functional on P. Then for positive numbers a, b, c, and d, we define the following sets:
and
We will make use of the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1.1) and (1.2).
Theorem 3.1
(See [20])
Let P be a cone in a real Banach space E. Let φ and Θ be nonnegative continuous convex functionals on P, Λ be nonnegative continuous concave functional on P, and Ψ be a nonnegative continuous functional on P satisfying \(\Psi(kx)\leq k\Psi(x)\) for \(0 \leq{k} \leq1 \), such that for some positive numbers M and d,
for all \(x\in\overline{P(\varphi,d)}\). Suppose
is completely continuous and there exist positive numbers a, b, and c with \(a< b\) such that
- (S1):
-
\(\{x\in P(\varphi,\Theta,\Lambda,b,c,d):\Lambda(x)>b\} \neq \emptyset\) and \(\Lambda(Tx)>b\) for \(x\in P(\varphi,\Theta ,\Lambda,b,c,d)\);
- (S2):
-
\(\Lambda(Tx)>b\) for \(x\in P(\varphi,\Lambda,b,d)\) with \(\Theta(Tx)>c\);
- (S3):
-
\(0 \notin R(\varphi,\Psi,a,d)\) and \(\Psi(Tx)< a\) for \(x\in R(\varphi,\Psi,a,d)\) with \(\Psi(x)=a\).
Then T has at least three fixed points \(x_{1}, x_{2}, x_{3} \in\overline{P(\varphi,d)}\), such that
and
4 Existence result for the case of \(\alpha(t) \geq t\) on J
Function \(h(t)\) in (1.1) satisfies (H2). We introduce the notations
Let \(X=C(J,\mathbb{R})\) be our Banach space with the maximum norm \(\lVert{x}\lVert=\max_{t\in J} \lvert x \lvert\).
Set
where Γ is defined as in Lemma 2.4. We define the nonnegative continuous concave functional \(\Lambda=\Lambda_{1}\) on P by
where \(\delta\in(0,1)\) is such that \(0<\delta<1-\delta<1\). Set \(J_{\delta_{1}}=[\delta,1]\).
Note that \(\Lambda_{1}(x) \leq\lVert{x}\lVert\). Put \(\Psi(x)=\Theta (x)=\varphi(x)=\lVert{x}\lVert\).
Theorem 4.1
Let assumptions (H1)-(H5) hold and \(\alpha(t) \geq t \) on J. In addition, we assume that there exist positive constants a, b, c, d, ω, L with \(a< b\) such that
and
- (A1):
-
\(f(t, u, v)\leq\frac{d}{\omega}\), for \((t, u, v)\in J \times[0,d] \times[0,d]\), \(I_{k}(t,u)\leq\frac {d}{\omega}\), for \((t,u)\in J_{k} \times[0,d]\);
- (A2):
-
\(f(t, u, v)\geq\frac{b}{L}\), for \((t, u, v)\in J_{\delta_{1}} \times[b,\frac{b}{\Gamma}]\times[b,\frac{b}{\Gamma}]\), \(I_{k}(t,u)\geq\frac{b}{L}\), for \((t,u)\in J_{\delta_{1}}\cap J_{k} \times[b,\frac{b}{\Gamma}]\);
- (A3):
-
\(f(t, u, v)\leq\frac{a}{\omega}\), for \((t, u, v)\in J \times[0,a] \times[0,a]\), \(I_{k}(t,u)\leq\frac {a}{\omega}\), for \((t,u)\in J_{k} \times[0,a]\).
Then problem (1.1) and (1.2) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3} \) satisfying \(\|x_{i}\|\leq d \), \(i=1, 2, 3\), and
and
Proof
For any \(x\in C(J,\mathbb{R}^{+})\), we define operator T by
where \(hf_{x}(s)=h(s)f (s,x(s),x(\alpha(s)) )\). Indeed, \(T:X \rightarrow X\). Problem (1.1) and (1.2) has a solution x if and only if x solves the operator equation \(x=Tx\).
We need to prove the existence of at least three fixed points of T by verifying that operator T satisfies the Avery-Peterson fixed point theorem.
From the definition of T, we can obtain
In view of (H1), (H2), Lemma 2.1, and Lemma 2.2, we have
So Tx is concave on J. From (4.3) and (4.4), combining with Lemma 2.4 and (H3), we can obtain
Noting (2) in Lemma 2.4, it follows that
On the other hand, from the properties of \(H(t,s)\), we have
This proves that \(TP\subset P\).
Now we prove that the operator \(T: P\rightarrow P\) is completely continuous. Let \(x\in\overline{P_{r}}\), then \(\lVert x\lVert\leq r\). Note that h and f are continuous, so h is bounded on J and f is bounded on \(J\times[-r,r]\). It means that there exists a constant \(K>0\) such that \(\lVert Tx \lVert\leq K\). This proves that TP̅ is uniformly bounded. On the other hand, for \(t_{1}, t_{2} \in J \) there exists a constant \(L_{1} >0\) such that
This shows that TP̅ is equicontinuous on J, so T is completely continuous.
Let \(x\in\overline{P(\varphi,d)}\), so \(0\leq x(t)\leq d\), \(t\in J\), and \(\lVert x \lVert\leq d\). Note that also \(0\leq x (\alpha (t) )\leq d\), \(t\in J\) because \(0 \leq t \leq\alpha(t) \leq1\) on J. Hence
By (4.2), Lemma 2.2, Lemma 2.4, and (A1), we have
This shows that \(T :\overline{P(\varphi,d)}\rightarrow\overline {P(\varphi,d)}\).
To check condition (S1) we choose
Then
so
It proves that
Let \(b\leq x(t)\leq\frac{b}{\Gamma}\) for \(t\in[\delta,1]\), then \(\delta\leq t \leq\alpha(t) \leq1\) on \([\delta, 1]\). It yields \(b \leq x (\alpha(t) )\leq\frac{b}{\Gamma}\) on \([\delta,1]\). It gives
By (4.2), Lemma 2.2, Lemma 2.4, and (A2), we have
It proves condition (S1) holds.
Now we need to prove that condition (S2) is satisfied. Take
Then
So condition (S2) holds.
We finally show that condition (S3) also holds. Clearly, as \(\Psi(0)=0< a\), so \(0\notin R(\varphi,\Psi,a,d)\). Suppose that \(x \in R(\varphi,\Psi,a,d)\) with \(\Psi(x)=\lVert x \lVert=a\).
Similarly, by (4.2), Lemma 2.2, Lemma 2.4, and (A3), we have
It proves that condition (S3) is satisfied.
By Theorem 4.1, there exist at least three positive solutions \(x_{1} \), \(x_{2}\), \(x_{3}\) of problem (1.1) and (1.2) such that \(\lVert x_{i} \lVert\leq d \) for \(i= 1, 2, 3\),
and \(\lVert x_{3} \lVert < a\). This ends the proof. □
Example
We consider the following BVP:
with \(\alpha\in C(J,J)\), \(h(t)=Dt\), \(\alpha(t)\geq t\), \(\xi=\eta=\frac{1}{2}\), \(\rho=2\), and \(t_{1}=\frac{1}{12}\). It follows that \(\mu(t)=\frac{3}{2}-t\), \(\lambda(t)=t+\frac{1}{2}\), for \(t\in J\), and
Note that \(f\in C(\mathbb{R}^{+},\mathbb{R}^{+})\). As a function α we can take, for example, \(\alpha(t)=\sqrt{t}\).
In this case we have \(\gamma=\frac{1}{4}\), \(\Delta=-\frac{85}{36}\), \(\overline{A}=\frac{47}{170}\), \(\overline{B}=\frac{71}{170}\), \(\underline{A}=\frac{269}{6\text{,}800}\), \(\underline{B}=\frac{457}{6\text{,}800}\), \(l_{1}=\frac{11}{48}D\), \(\hat{l}_{1}=\frac{1}{12}D\), \(\Gamma=\frac{1}{24}\). Let \(a=1\), \(b=\frac{3}{2}\), \(c=36\), \(d\geq2\text{,}000\), \(D=2\text{,}784\). In this case we can take \(\omega=250\), \(L=1\). We see that all assumptions of Theorem 4.1 hold, so BVP (4.7) has at least three positive solutions.
5 Existence result for the case of \(\alpha(t) \leq t\) on J
The cone P is defined as in (4.1). We define the nonnegative continuous concave functional \(\Lambda=\Lambda_{2}\) on P by
where \(\delta\in(0,1)\) satisfying \(0<\delta<1-\delta<1\). Set \(J_{\delta_{2}}=[0,1-\delta]\). Similar to the proof of Theorem 4.1, we have the following result.
Theorem 5.1
Let assumptions (H1)-(H5) hold and \(\alpha(t) \leq t \) on J. In addition, we assume that there exist positive constants a, b, c, d, ω, L with \(a< b\) such that (4.2) holds and
- (A1):
-
\(f(t, u, v)\leq\frac{d}{\omega}\), for \((t, u, v)\in J \times[0,d] \times[0,d]\), \(I_{k}(t,u)\leq\frac {d}{\omega}\), for \((t,u)\in J_{k} \times[0,d]\);
- (A2):
-
\(f(t, u, v)\geq\frac{b}{L}\), for \((t, u, v)\in J_{\delta_{2}} \times[b,\frac{b}{\Gamma}]\times[b,\frac{b}{\Gamma}]\), \(I_{k}(t,u)\geq\frac{b}{L}\), for \((t,u)\in J_{\delta_{2}}\cap J_{k} \times[b,\frac{b}{\Gamma}]\);
- (A3):
-
\(f(t, u, v)\leq\frac{a}{\omega}\), for \((t, u, v)\in J \times[0,a] \times[0,a]\), \(I_{k}(t,u)\leq\frac {a}{\omega}\), for \((t,u)\in J_{k} \times[0,a]\).
Then problem (1.1) and (1.2) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3} \) satisfying \(\|x_{i}\|\leq d \), \(i=1, 2, 3\), and
and
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Acknowledgements
The work is supported by Chinese Universities Scientific Fund (Project No. 2016 LX002). The authors would like to thank the anonymous referees very much for helpful comments and suggestions, which lead to the improvement of presentation and quality of the work.
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Dou, J., Zhou, D. & Pang, H. Existence and multiplicity of positive solutions to a fourth-order impulsive integral boundary value problem with deviating argument. Bound Value Probl 2016, 166 (2016). https://doi.org/10.1186/s13661-016-0674-8
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DOI: https://doi.org/10.1186/s13661-016-0674-8