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Monotone iterative technique and positive solutions to a thirdorder differential equation with advanced arguments and Stieltjes integral boundary conditions
Advances in Difference Equationsvolume 2018, Article number: 218 (2018)
Abstract
We treat the existence of monotonic iteration positive solutions to a thirdorder boundary value problem with advanced arguments and Stieltjes integral boundary conditions. In our work, the main tool is a monotone iterative technique. Meanwhile, at the end of this paper, an example is presented to show that this method can be well used to get the main results.
Introduction
Thirdorder equations play a significant role in various aspects of applied mathematics and physics. Take draining or coating fluidflow problems for example, surface tension forces are important in these problems and thirdorder ordinary differential equations serve well to describe them. Besides, thirdorder equations also work out well in the deflection of a curved beam having a constant or varying crosssection, a threelayer beam, electromagnetic waves, or gravity driven flows, etc. To learn more about the applications of the thirdorder boundary value problems, readers can refer to [1] and related themes.
Recently, many authors have widely studied the existence of multiple solutions to some boundary value problems; to see the details, we refer the readers to [1–5] and the references therein.
Since science and technology are developing at an unprecedented speed, a lot of boundary value problems with integral boundary conditions are applied in different industries and fields, for instance, thermal conduction, chemical engineering, semiconductor, underground water flow, hydrodynamic, thermoelasticity, etc.; these can be found in [6–8] and related topics. The point is that boundary value problems with integral boundary conditions are made up of a very interesting and significant class of problems, since they include two, three, multipoint, and nonlocal boundary value problems as special cases.
Although the existence of multiple solutions to some boundary value problems with integral boundary conditions has been studied widely by many authors nowadays, we find that most authors study the secondorder and fourthorder differential equations involving integral boundary conditions; to give example, the readers can see [9–13] and the references therein. As far as we are concerned, there are few papers dealing with thirdorder differential equations with Stieltjes integral boundary conditions in the existing literature.
We noted that Graef and Webb in [14] dealt with the existence of multiple positive solutions to the boundary value problem
where \(p>\frac{1}{2}\), and \(\lambda[v]=\int^{1}_{0}v(t)\, d\Lambda(t)\) is a typical linear function on \(C[0,1]\) along with a Riemann–Stieltjes integral and Λ is a function of bounded variation which suits well. Given that λ can cover sums and integrals as well, it is a more ordinary setup.
We also noted that in [15] Jankowski proved the existence of at least three nonnegative solutions to the following nonlocal boundary value problem by using a fixed point theorem:
where λ is a linear function under the circumstance that \(\lambda[x]=\int^{1}_{0}x(t)\, d\Lambda(t)\), which bears Stieltjes integral with a befitting function Λ of bounded variation.
We cannot help pointing out that only \(x(1)\) is related to a Stieltjes integral in the above boundary condition in (1.2), and this is the specific source where our thoughts came from. That is, whether there will be some interesting findings when \(x(0)\) is also related to a Stieltjes integral.
Under the motivation of the above works we talked about, we finally decided to research the successive iteration and positive solutions to the following thirdorder boundary value problem with advanced arguments and Stieltjes integral boundary conditions in this paper:
where \(0<\eta<1\), \(0\leq\gamma^{2}\leq\beta<\gamma<1\), \(\alpha (t)\in C([0,1]\rightarrow[0,1])\), and \(\alpha(t)\geq t\) for \(t\in[0,1]\), Λ is an appropriate bounded variation function and \(\lambda [u]=\int^{1}_{0}u(t)\, d\Lambda(t)\) bearing a Riemann–Stieltjes integral function. It is worthy to mention that \(\lambda[u]\) is not always positive for all positive u here.
When it comes to our work here, we will show that not only can we prove the existence of positive solutions to problem (1.3), but also a few of successive iterative schemes with either a known constant function starting point or a simple linear function one will be set up to approach the solutions. Last but not least, a perfect example is shown at the end of our paper to represent the applicability of the above mentioned methods and results. We must point out that acquiring the knowledge of how to find the solutions is perhaps the most significant skill when we turn to numerical analysis and application.
Preliminaries
At the beginning, we will show some important and necessary definitions here by using the theories of cones in Banach spaces.
Definition 2.1
Let E be a real Banach space. A nonempty closed set \(P\subset E\) is said to be a cone provided the following hypotheses are satisfied:

(i)
\(au+bv\in P\) for all \(u,v\in P\) and all \(a\geq0\), \(b\geq0\), and

(ii)
\(u,u\in P \) implies \(u=0\).
Definition 2.2
The map α is identified as being concave on \([0,1]\) if
for all \(u,v\in[0,1]\) and \(t\in[0,1]\).
Definition 2.3
An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
We are concerned with the Banach space \(E=C[0,1]\) facilitated with the norm \(\u\=\max_{0\leq t\leq1}u(t)\). And we denote the cone \(P\subset E\) by
where
Lemma 2.1
For \(y\in L[0,1]\), then the boundary value problem
has a unique solution
where
Proof
Facilitated by (2.1), we have
And then we can obtain
Under the boundary condition in (2.1), we can get that
Thus
Therefore, we get the solution to the problem as follows:
So, the proof is completed. □
Now we define an operator \(T: P\rightarrow E\) by
According to Lemma 2.1, boundary value problem (1.3) has a solution \(u=u(t)\) if and only if u is a fixed point of T.
During the following proof, we will assume that (H_{1})–(H_{4}) are satisfied.
 (H_{1})::

\(f(t,x)\in C([0,1]\times[0,+\infty)\rightarrow[0,+\infty ))\).
 (H_{2})::

\(q(t)\) is a nonnegative continuous function on \([0,1]\), \(q(t)\not\equiv0\) on any subinterval of \((0,1)\).
 (H_{3})::

\(\int^{1}_{0}d\Lambda(t)\geq0\), \(\int^{1}_{0}t\, d\Lambda (t)\geq0\), \(\int^{1}_{0}F(t,s)\, d\Lambda(t)\geq0\), \(0< s<1\).
 (H_{4})::

\(1\beta(\gamma\beta)\eta>0\).
Lemma 2.2
Since (H_{1})–(H_{4}) hold, then \(T:P\rightarrow P\) defined by (2.3) is completely continuous.
Proof
Through (2.3), we have
It is easy to notice that \((Tu)(t)\) is concave on \([0,1]\).
Besides, by (2.3)
and
Then it follows that Tu is nonnegative on \([0,1]\).
Then, according to (H_{3}), we can get
On the other hand, we must show that \(\min_{\eta\leq t\leq1} (Tu)(t)\geq\delta\Tu\\).
Because of the concavity of Tu, we can obtain that there exists \(\sigma\in[0,1]\) such that \(\Tu\=(Tu)(\sigma)\).
When \(\eta\geq\sigma\), we have \(\min_{\eta\leq t\leq 1} (Tu)(t)=(Tu)(1)\) and
then
Uniting the boundary condition in (1.3), we can have that
When \(\eta< \sigma\), we have \(\min_{\eta\leq t\leq1} (Tu)(t)=\min\{(Tu)(\eta),(Tu)(1)\}\) and
then
Combining the boundary condition in (1.3), we can obtain that
and
Hence, we get
Given (2.4) and (2.5), we can get that \(\min_{\eta \leq t\leq1} (Tu)(t)\geq\delta\Tu\\).
It is easy to see that T is continuous. Then, let \(\Omega\subset P\) be a bounded set, the proof that TΩ is bounded and equicontinuous is easy and obvious. Then the Arzela–Ascoli theorem makes sure that TΩ is relatively compact, which means T is compact. Then we obtain that T is completely continuous.
So, based on what has been discussed above, we can arrive at the conclusion that \(T:P\rightarrow P\) is completely continuous. □
Main results
For the convenience of next work, we denote
Theorem 3.1
Assume that (H_{1})–(H_{4}) hold and there exists \(0< b< a\) such that
 (H_{5})::

\(f(t,x_{1})\leq f(t,x_{2})\) for any \(0\leq t\leq1\), \(0\leq x_{1}\leq x_{2}\leq a\);
 (H_{6})::

\(\sup_{0\leq t\leq1} f(t,a)\leq\frac{a}{A}\), \(\inf_{\eta\leq t\leq1} f(t,\delta b)\geq\frac {b}{B}\);
 (H_{7})::

\(f(t,0)\not\equiv0\) for \(0\leq t\leq1\).
where \((Tu)(t)\) is defined by (2.3).
The successive iterative schemes in the theorem are \(w_{0}(t)=a\), \(w_{n+1}=Tw_{n}=T^{n}w_{0}\), \(n=0,1,2,\ldots\) , which starts off with the constant function, and \(v_{0}(t)=\frac{b}{\gamma} (\beta+(\gamma\beta)t )\), \(v_{n+1}=Tv_{n}=T^{n}v_{0}\), \(n=0,1,2,\ldots\) , which starts off with a known simple linear function.
Proof
We denote \(P[b,a]=\{u\in P \mid b\leq\u\\leq a\}\).
In the following content of proof, we will firstly prove that \(T:P[b,a]\rightarrow P[b,a]\).
If \(u\in P[b,a]\), then
Therefore, under assumptions (H_{5}) and (H_{6}), we get
For any \(u\in P[b,a]\), by Lemma 2.2, we know that \(Tu\in P\) and
and
Thus, we get \(b\leq\Tu\\leq a\). So, we get that \(T:P[b,a]\rightarrow P[b,a]\).
Let \(w_{0}(t)=a\), \(0\leq t\leq1\), then \(w_{0}(t)\in P[b,a]\). Let \(w_{1}=Tw_{0}\), then \(w_{1}\in P[b,a]\). We denote \(w_{n+1}=Tw_{n}\), \(n=0,1,2,\ldots\) . Then we have \(w_{n}\subseteq P[b,a]\), \(n=1,2,\ldots\) . Since T is completely continuous, we assert that \(\{w_{n}\}_{n=1}^{\infty}\) is a sequentially compact set.
Then we will search the convergence property of the iterative scheme since
After calculation, the iterative scheme is clear, then
Thus, we can get that there exists \(w^{\ast}\in P[b,a]\) such that \(w_{n}\rightarrow w^{\ast}\). Combining with the continuity of T and \(w_{n+1}=Tw_{n}\), we obtain that \(Tw^{\ast}=w^{\ast}\).
On the other hand, another way to approach this is to start off with a linear function. Let \(v_{0}(t)=\frac{b}{\gamma} (\beta+(\gamma\beta)t )\), \(0\leq t\leq1\), then \(v_{0}(t)\in P[b,a]\). Let \(v_{1}=Tv_{0}\), then \(v_{1}\in P[b,a]\). We denote \(v_{n+1}=Tv_{n}\), \(n=0,1,2,\ldots\) . Then we have \(v_{n}\subseteq P[b,a]\), \(n=1,2,\ldots\) . Since T is completely continuous, we assert that \(\{v_{n}\}_{n=1}^{\infty}\) is a sequentially compact set.
In a similar way, since \(v_{1}=Tv_{0}\in P[b,a]\) and
Through a similar calculation argument, we can easily get that
Hence there exists \(v^{\ast}\in P[b,a]\) such that \(v_{n}\rightarrow v^{\ast}\). Combining with the continuity of T and \(v_{n+1}=Tv_{n}\), we get \(Tv^{\ast}=v^{\ast}\).
Assumption (H_{7}) indicates that \(f(t,0)\not\equiv0\), \(0\leq t\leq 1\), then the zero function is not the solution of (1.3). Thus we have \(v^{\ast}>0\) for \(0< t<1\).
It is well known that each fixed point of T in P is a solution of (1.3). Hence, we assert that the boundary value problem (1.3) has at least two positive concave solutions \(w^{\ast}\) and \(v^{\ast}\).
The proof is completed. □
Remark 3.1
If \(\lim_{n\rightarrow\infty}w_{n}\neq\lim_{n\rightarrow\infty}v_{n}\), then \(w^{\ast}\) and \(v^{\ast}\) are two positive concave solutions of problem (1.3). And if \(\lim_{n\rightarrow\infty}w_{n}=\lim_{n\rightarrow\infty}v_{n}\), then \(w^{\ast}=v^{\ast}\) is a positive concave solution of problem (1.3).
The following corollary can be obtained easily.
Corollary 3.1
Assume that (H_{1})–(H_{5}) and (H_{7}) hold, and there exists \(a>0\) such that
 (H_{8})::

\(\varlimsup_{l\rightarrow0} \inf_{\eta \leq t\leq1} \frac{f(t,l)}{l}>\frac{1}{\delta B}\), \(\varliminf_{l\rightarrow+\infty} \sup_{0\leq t\leq1} \frac{f(t,l)}{l}<\frac{1}{A}\)
(particularly, \(\varlimsup_{l\rightarrow0} \inf_{\eta\leq t\leq 1} \frac{f(t,l)}{l}=+\infty\), \(\varliminf_{l\rightarrow +\infty} \sup_{0\leq t\leq1} \frac{f(t,l)}{l}=0\).)
Example
In the following part, we will discuss an example and simulations. Then we will get a perfect result by using the method above.
Example 4.1
Let \(\eta=\frac{1}{2}\) and \(q(t)=1\), we consider the following boundary value problem:
where
From calculation we can get that
The verification of conditions (H_{1})–(H_{5}), (H_{7}) is very easy to complete, then we set \(b=1\), \(a=21\), then it also satisfies
So all the hypotheses of Theorem 3.1 are fulfilled, and we can obtain that the boundary value problem (4.1) has at least two positive concave solutions \(w^{\ast}\) and \(v^{\ast}\) such that
where \((Tu)(t)\) is defined by (2.3).
For \(n=0,1,2,\ldots\) , the two iterative schemes are as follows:
which starts off with a constant function, and
which starts off with a known simple linear function.
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Acknowledgements
The author would like to thank the referees for careful reading and several constructive comments and making some useful corrections that have improved the presentation of the paper.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11126245), China Scholarship Council (CSC[2014]3072), and Discipline Construction Fund of Central University of Finance and Economics.
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Correspondence to Bo Sun.
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MSC
 34K10
 34B15
Keywords
 Stieltjes integral boundary conditions
 Monotone Iterative
 Completely continuous