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Existence of positive solutions for fractional differential equation with integral boundary conditions on the half-line
Boundary Value Problems volume 2016, Article number: 104 (2016)
Abstract
This paper considers the existence of positive solutions for fractional-order nonlinear differential equation with integral boundary conditions on the half-infinite interval. By using the fixed point theorem in a cone, sufficient conditions for the existence of at least one or at least two positive solutions of a boundary value problem are established. These theorems also reveal the properties of solutions on the half-line.
1 Introduction
Boundary value problems are often studies in the areas of applied mathematics and physics. With the development of technology, applications of boundary value problems on the infinite interval attract increasing attention; see [1–4] and the references therein. Recently, fractional differential equations have also aroused great interest; see [5–8]. At the same time, the existence of positive solutions for nonlinear fractional differential equation boundary value problems have been widely studied by many authors; see [9–17] and the references therein.
In [3], the authors, using fixed point theorems in a cone, established the existence of one positive solution and three positive solutions for the following second-order nonlinear boundary value problems with integral boundary conditions on an infinite interval:
where \(f \in C((0,+\infty)\times[0,+\infty)\times \mathbb{R},[0,+\infty))\), f may be singular at \(t=0\), \(g_{1},g_{2}:[0,+\infty)\rightarrow[0,+\infty)\) and \(\psi:[0,+\infty)\rightarrow(0,+\infty) \) are continuous, \(\int _{0}^{+\infty} \psi(s)\,\mathrm{d}s<+\infty\), \(p\in C[0,+\infty)\cap C^{1}(0,+\infty)\) with \(p(t)>0\) on \((0,+\infty)\) and \(\int_{0}^{+\infty}\frac {\,\mathrm{d}s}{p(s)}<+\infty\), \(a_{1}+a_{2}>0\), and \(b_{i}>0\) for \(i=1,2\). Iterative schemes for approximating the solutions of a nonlinear fractional boundary value problem on the half-line were presented in [15]. The authors, based on the monotone iterative technique, obtained the existence of positive solutions of the following fractional boundary value problem:
where \(1<{\alpha}<2\), and \(D_{0+}^{\alpha} \) is the standard Riemann-Liouville fractional derivative. For an overview of the literature on differential equations boundary value problems, see [6–8] and the references therein.
Motivated by all the works mentioned, we study the following fractional boundary value problem on the half-line:
where \({}^{\mathrm{C}} D^{\alpha}\) is the Caputo fractional derivative of order \(\alpha\in(0,1)\), \(p\in C^{1}([0,{+\infty}),(0,{+\infty}) )\), \(f:(0,{+\infty})\times[0,{+\infty})\rightarrow[0,{+\infty}) \) is a continuous function and may be singular at \(t=0\); \(a_{i}>0\), \(b_{i}>0 \), \(g_{i}\in C([0,{+\infty}),[0,{+\infty}))\), and \(\psi_{i}\in L^{1}([0,{+\infty}))\) is nonnegative for \(i=1,2\).
We assume that the following conditions are satisfied:
-
(H0)
\(\lim_{t\to{+\infty}}\int_{0}^{t}\frac{(t-s)^{\alpha -1}}{p(s)}\,\mathrm{d}s<{+\infty}\), \(\frac{b_{2}}{a_{2}}>M\), where
$$ M= \frac{1}{\Gamma(\alpha)} \sup_{t\in[0,+\infty)} \int_{0}^{t}\frac {(t-s)^{\alpha-1}}{p(s)}\,\mathrm{d}s. $$(1.2) -
(H1)
There exist functions \(h\in C([0,{+\infty}),[0,{+\infty}))\) and \(v\in C((0,{+\infty}),(0,{+\infty}))\) such that
$$f(t,u)\leq v(t)h(u),\quad t\in(0,{+\infty}); \qquad \int_{0}^{+\infty}p(s)v(s)\,\mathrm{d}s< {+\infty}. $$
2 Preliminaries
In this section, we present some useful definitions and the related theorems.
Definition 2.1
Let \(\alpha>0\). For a function \(u:(0,+\infty)\rightarrow\mathbb{R}\), the Riemann-Liouville fractional integral operator of order α of u is defined by
provided that the integral exists.
Definition 2.2
The Caputo derivative of order α for a function \(u:(0,+\infty )\rightarrow\mathbb{R}\) is given by
provided that the right side is pointwise defined on \((0, +\infty)\), where \(n =[\alpha]+1\) and \(n-1<\alpha<n\).
If \(\alpha=n\), then \({}^{\mathrm{C}} D^{\alpha}_{0+} u(t)=u^{(n)}(t)\).
Lemma 2.1
(See [7])
Let \(\alpha>0\). Then the differential equation
has solutions
where n is the smallest integer greater than or equal to α.
Lemma 2.2
If (H0) holds and \(y\in C((0,{+\infty}),[0,{+\infty}))\) with \(\int _{0}^{+\infty}p(s)y(s)\,\mathrm{d}s<{+\infty}\), then the fractional boundary value problem
has a unique solution
where
and
Proof
It is well known that the fractional differential equation in (2.1) is equivalent to the integral equation
Hence,
where \(c_{0}\in\mathbb{R} \) is any constant. It follows from (2.2) and (2.3) that
and
By the boundary conditions in (2.1) we have
and
Substituting them into (2.3), we get
By (1.2), \(M<\frac{b_{2}}{a_{2}}\), and \(\int_{0}^{+\infty }p(s)y(s)\,\mathrm{d}s<{+\infty}\), we have
Hence,
Therefore, the unique solution of the fractional boundary value problem (2.1) is
 □
For convenience, we denote
Lemma 2.3
If (H0) holds, then \(G(t,s)\), \(F_{1}(t)\), and \(F_{2}(t)\) defined in Lemma 2.2 satisfy
-
(1)
\(G(t,s)\) is a continuous function and \(G(t,s)>0\) for \((t,s)\in [0,{+\infty})\times[0,{+\infty})\);
-
(2)
\(F_{1}(t)\), \(F_{2}(t)\) are continuous functions, and \(F_{1}(t), F_{2}(t)\geq0\) for \(t\in[0,{+\infty})\);
-
(3)
$$\begin{aligned}& G_{m}\leq G(t,s)\leq G_{M} \quad \textit{for }(t,s) \in[0,{+\infty})\times[0,{+\infty}), \\& F_{im}\leq F_{i}(t)\leq F_{iM} \quad \textit{for }t\in[0,{+\infty}), i=1,2; \end{aligned}$$
-
(4)
there exist constants \(0< l_{1}< l_{2}<{+\infty}\) such that
$$\begin{aligned}& G(t,s)\geq\gamma_{0} G_{M}\quad \textit{for }(t,s) \in[l_{1},l_{2}]\times[0,+\infty), \\& F_{i}(t)\geq\gamma_{0} F_{iM} \quad \textit{for }t\in[l_{1},l_{2}]\textit{ and }i=1,2; \end{aligned}$$ -
(5)
for any \(s\in[0,+\infty)\), \(\lim_{t\to{+\infty}}G(t,s)=\overline {G}<{+\infty}\), \(\lim_{t\to{+\infty}}F_{1}(t)=\overline{F}_{1}<{+\infty}\), \(\lim_{t\to{+\infty}}F_{2}(t)=\overline{F}_{2}<{+\infty}\).
Proof
(1) For \(0\leq t\leq s\), it is easy to see that \(G(t,s)>0\).
For \(0\leq s< t\), by (H0) and (1.2) we have
Hence, \(G(t,s)>0\) for \((t,s)\in[0,{+\infty})\times[0,{+\infty})\).
It is easy to see that \(G(t,s)\) is a continuous function.
(2) It follows from (1.2) and (H0) that (2) holds.
(3) By (H0), for \(t,s\in[0,+\infty)\), we have
and
It is easy to see that \(F_{im}\leq F_{i}(t)\leq F_{iM}\) for \(t\in [0,{+\infty})\), \(i=1,2\).
(4) By (3) there exist constants \(0< l_{1}< l_{2}<{+\infty}\) such that
Similarly, we have
(5) By (H0) and \(0<\alpha<1\), for any \(s\in[0,+\infty)\), we can show that
It is obvious that \(\lim_{t\to{+\infty}}F_{1}(t)=\overline{F}_{1}<{+\infty}\) and \(\lim_{t\to{+\infty}}F_{2}(t)=\overline{F}_{2}<{+\infty}\). □
Let
be a Banach space with the norm \(\|u\|=\sup_{t\in[0,+\infty)}|u(t)|\), and
be a cone in E.
For \(r>0\), we denote
and
It follows from (H1) that \(S_{r}\), \(S_{r}'\), and \(S_{r}''<+\infty\).
We define the operator \(T: P\rightarrow E\) by
We can easily get the following Lemma 2.4 from Lemma 2.2.
Lemma 2.4
If \(u\in P\), then the boundary value problem (1.1) is equivalent to the integral equation
Lemma 2.5
Let E be defined by (2.4), and \(\Omega\subset E\). Then Ω is relatively compact in E if the following conditions hold:
-
(a)
Ω is uniformly bounded in E;
-
(b)
the functions belonging to M are equicontinuous on any compact interval of \([0,+\infty)\);
-
(c)
the functions from Ω are equiconvergent, that is, for any given \(\varepsilon>0\), there exists \(T(\varepsilon)>0\) such that \(|f(t)-f(+\infty)|<\varepsilon\) for any \(t>T(\varepsilon)\) and \(f\in\Omega\).
Lemma 2.6
If (H0) and (H1) hold, then \(T:P\rightarrow P\) is completely continuous.
Proof
We divide the proof into three steps.
Step 1: We show that \(T:P\rightarrow P\) is well defined.
For \(u\in P\), there exists a constant \(r_{0}>0\) such that \(\|u\|\leq r_{0}\). By (H1) and Lemma 2.3, for \(t,s\in[0,+\infty)\), we have
Since \(G(t,s)\), \(F_{1}(t)\), \(F_{2}(t)\) are continuous with respect to t, by using the Lebesgue dominated convergence theorem, for \(t_{0}\in[0,+\infty )\), we have
So, \(Tu\in C[0,+\infty)\), and we get
It is obvious that \(Tu(t)\geq0\), \(t\in[0,+\infty)\), by Lemma 2.3. Moreover,
By Lemma 2.3 (4) we have
Hence, \(T:P\rightarrow P\) is well defined.
Step 2: We can verify that \(T:P\rightarrow P\) is continuous.
Let \(u_{n},u\in P\) and \(\|u_{n}-u\|\rightarrow0\) as \(n\rightarrow+\infty\). Then there exists a constant \(r_{1}>0\) such that \(\|u_{n}\|,\|u\|\leq r_{1}\). We have
and, for \(s\in[0,+\infty)\),
Then, by the Lebesgue dominated convergence theorem we have
Therefore, \(T:P\rightarrow P\) is a continuous operator.
Step 3: We can show that \(T:P\rightarrow P\) is relatively compact.
Let Ω be a bounded subset of P. Then there exists a constant \(r_{2}>0\) such that \(\|u\|\leq r_{2}\) for each \(u\in\Omega\).
By Lemma 2.3 and (H1) we have
So, \(T(\Omega)\) is uniformly bounded.
For any \(\overline{T}\in(0,+\infty)\), since \(G(t,s)\), \(F_{1}(t)\), and \(F_{2}(t)\) are continuous, we have that G is uniformly continuous on \([0,\overline{T}]\times[0,\overline{T}]\) and \(F_{1}\) and \(F_{1}\) are uniformly continuous on \([0,\overline{T}]\). This implies that, for any \(\varepsilon>0\), there exists \(\delta>0\) such that, when \(t_{1}, t_{2}\in [0,\overline{T}]\), whenever \(|t_{2}-t_{1}|<\delta\) and \(s\in[0,\overline {T}]\), we have
Therefore, for \(t_{1}, t_{2}\in[0,\overline{T}]\), whenever \(|t_{2}-t_{1}|<\delta\) and \(u\in\Omega\), we can show that
Hence, \(T(\Omega)\) is locally equicontinuous on \([0,+\infty)\).
Since
By Lemma 2.3 we conclude that
Hence, \(T(\Omega)\) is equiconvergent at infinity.
By Lemma 2.5 we obtain that \(T:P\rightarrow P\) is completely continuous. □
Lemma 2.7
(See [19])
Let E be a Banach space, \(P\subseteq E\) be a cone, and \(\Omega_{1}\), \(\Omega_{2}\) be two bounded open subsets of E with \(\theta\in\Omega _{1}\subset\overline{\Omega}_{1}\subset\Omega_{2}\). Suppose that \(T: P\cap(\overline{\Omega}_{2}\setminus\Omega _{1})\rightarrow P \) is a completely continuous operator such that either
-
(i)
\(\|Tx\|\leq\|x\|\), \(x\in P\cap\partial\Omega_{1}\) and \(\| Tx\|\geq\|x\|\), \(x\in P\cap\partial\Omega_{2}\), or
-
(ii)
\(\|Tx\|\geq\|x\|\), \(x\in P\cap\partial\Omega_{1}\), and \(\| Tx\|\leq\|x\|\), \(x\in P\cap\partial\Omega_{2}\),
holds. Then the operator T has at least one fixed point in \(P\cap (\overline{\Omega}_{2}\setminus\Omega_{1})\).
3 The existence of positive solutions
For convenience, we give the following notation:
where \(\varphi=0 \) or +∞, and \(i=1,2\). We denote
Theorem 3.1
Suppose that (H0) and (H1) hold. If
then the boundary value problem (1.1) has at least one positive solution.
Proof
Since \(A(h^{0}+g_{1}^{0}+g_{2}^{0})<1\), then there exists a constant \(r_{1}>0\) such that, for \(u\leq r_{1}\), we have
where \(\varepsilon_{1}\) satisfies \(A(h^{0}+g_{1}^{0}+g_{2}^{0}+\varepsilon_{1})\leq1\).
Therefore, for any \(t\in[0,+\infty)\), \(u\in\partial K_{r_{1}}\), we can get
On the other hand, since \(B(f_{+\infty}+g_{1,+\infty}+g_{2,+\infty })>1\), there exist constants \(\overline{r}_{2}>0\) and \(M_{i}\), \(i=0,1,2\), with \(f_{+\infty}>M_{0}>0\), \(g_{1,+\infty}>M_{1}>0\), \(g_{2,+\infty}>M_{2}>0\) such that, for \(t\in[l_{1},l_{2}]\),
where \(\varepsilon_{2}\) satisfies \(B(M_{0}+M_{1}+M_{2}-\varepsilon_{2})\geq1\).
Let \(r_{2}=\max \{r_{1},\frac{\overline{r}_{2}}{\gamma_{0}} \}\). In view of the definition of P,
According to Lemma 2.3, for \(u\in\partial K_{r_{2}}\), we have
Therefore, by (i) of Lemma 2.7 and Lemma 2.3, the boundary value problem (1.1) has at least one positive solution \(u\in\overline{K}_{r_{2}}\setminus K_{r_{1}}\). □
Remark 3.1
It follows from the proof of Theorem 3.1 that the boundary value problem (1.1) has at least one positive solution \(u\in P\) if one of the conditions \(f_{+\infty }=+\infty\), \(g_{1,{+\infty}}=+\infty\), and \(g_{2,{+\infty}}=+\infty\) holds.
Theorem 3.2
Suppose that (H0) and (H1) hold. If
then the boundary value problem (1.1) has at least one positive solution.
Proof
It follows from \(B(f_{0}+g_{1,0}+g_{2,0})>1\) that there exist constants \(r_{3}>0\), \(M'_{i}\), \(i=0,1,2\), with \(f_{0}>M'_{0}>0\), \(g_{1,0}>M'_{1}>0\), \(g_{2,0}>M'_{2}>0\) such that, for \(t\in[l_{1},l_{2}]\) and \(0< u\leq r_{3}\), we have
where \(i=1,2\), and \(\varepsilon_{3} \) satisfies \(B(M'_{0}+M'_{1}+M'_{2}-\varepsilon_{3})\geq1\).
Thus, for any \(t\in[0,+\infty)\) and \(u\in\partial K_{r_{3}}\), we have \(\inf_{t\in[l_{1},l_{2}]}u(t)\geq\gamma_{0}\|u\|\) and
It follows from (3.4) that
On the other hand, since \(A(h^{+\infty}+g_{1}^{+\infty}+g_{2}^{+\infty })<1\), there exists a constant \(\overline{r}_{4}>0\) such that, for \(u\geq \overline{r}_{4}\), we have
where \(\varepsilon_{4}\) with \(A(h^{+\infty}+g_{1}^{+\infty}+g_{2}^{+\infty }+\varepsilon_{4})<1\).
Let
For any \(t\in[0,+\infty)\), \(u\in\partial K_{r_{4}}\), we denote
We have
Hence, by using (ii) of Lemma 2.7 and Lemma 2.3, the boundary value problem (1.1) has at least one positive solution \(u\in\overline{K}_{r_{4}}\setminus K_{r_{3}}\). □
Remark 3.2
It follows from the proof of Theorem 3.2 that the boundary value problem (1.1) has one positive solution \(u\in P\) if at least one of the conditions \(f_{0}=+\infty\), \(g_{1,0}=+\infty\), and \(g_{2,0}=+\infty\) holds.
Theorem 3.3
Suppose that (H0) and (H1) hold. If
-
(1)
\(B(f_{0}+g_{1,0}+g_{2,0})>1\), \(B(f_{+\infty}+g_{1,+\infty }+g_{2,+\infty})>1\) and
-
(2)
there exists a constant \(c>0\) such that \(\max\{S_{c},S'_{c},S''_{c}\} :=S^{*}< A^{-1}c\),
then the boundary value problem (1.1) has at least two positive solutions.
Proof
Since \(B(f_{0}+g_{1,0}+g_{2,0})>1\), similarly to the proof of Theorem 3.2, there exists a constant \(0< r< c\) with
Since \(B(f_{+\infty}+g_{1,+\infty}+g_{2,+\infty})>1\), there also exists a constant \(R>c\) such that
On the other hand, by condition (2), for any \(u\in\partial K_{c}\),
Namely,
According to Lemma 2.7, the boundary value problem (1.1) has at least two positive solutions \(u_{1}\), \(u_{2}\) with \(0<\|u_{1}\| <c<\|u_{2}\|\). □
Remark 3.3
It follows from the proof of Theorem 3.3 that the boundary value problem (1.1) has at least two positive solutions \(u\in P\) if one of the conditions \(f_{0}=+\infty \), \(g_{1,{0}}=+\infty\), \(g_{2,{0}}=+\infty\), \(f_{+\infty}=+\infty\), \(g_{1,{+\infty}}=+\infty\), and \(g_{2,{+\infty}}=+\infty\) holds.
Theorem 3.4
Suppose that (H0) and (H1) hold. If
-
(1)
\(A(h^{0}+g_{1}^{0}+g_{2}^{0})<1\), \(A(h^{+\infty}+g_{1}^{+\infty}+g_{2}^{+\infty })<1\), and
-
(2)
there exists a constant \(C>0\) such that, for any \(t\in[l_{1},l_{2}]\) and \(u\in[\gamma_{0}C,C]\), we have
$$\min\bigl\{ f(t,u),g_{1}(u),g_{2}(u)\bigr\} > \gamma_{0}B^{-1}C, $$
then the boundary value problem (1.1) has at least two positive solutions.
Proof
The proof is similar to that of Theorem 3.3.
It is easy to get the two positive solutions \(u_{3}\), \(u_{4}\) with \(0<\|u_{3}\| <C<\|u_{4}\|\). □
4 Illustration
Example
We consider the following boundary value problem:
where \(f(t,u)=\frac{\mathrm{e}^{-3t}(1+t)(u+\mathrm{e}^{-u})}{10\sqrt {t}}\), \(a_{1}=1\), \(a_{2}=1\), \(b_{1}=1\), \(b_{2}=2\), \(p(t)=\mathrm{e}^{t}\), \(\psi _{1}(s)=\psi_{2}(s)=\frac{1}{1+s^{2}}\), and
It is obvious that \(f:(0,{+\infty})\times[0,{+\infty})\rightarrow [0,{+\infty}) \) is a continuous function and singular at \(t=0\).
Let
Then we have \(M\approx0.610503\), \(A=2.19551\), \(h^{+\infty}=\frac {1}{10}\), \(g_{1}^{+\infty}=g_{2}^{+\infty}=0\), and \(A(h^{+\infty }+g_{1}^{+\infty}+g_{2}^{+\infty})=0.219551<1\).
On the other hand, let \(l_{1}=\frac{1}{2}\), \(l_{2}=1\). So we have \(B\approx 0.03\), \(f_{0}=+\infty\), \(g_{i,0}=\frac{1}{10}\), \(i=1,2\). Namely, \(B(f_{0}+g_{1,0}+g_{2,0})=+\infty>1\).
By using Theorem 3.2 the boundary value problem (4.1) has at least one positive solution.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11171220) and the Hujiang Foundation of China (B14005).
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Jia, M., Zhang, H. & Chen, Q. Existence of positive solutions for fractional differential equation with integral boundary conditions on the half-line. Bound Value Probl 2016, 104 (2016). https://doi.org/10.1186/s13661-016-0614-7
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DOI: https://doi.org/10.1186/s13661-016-0614-7
MSC
- 34B18
- 34B10
- 26A33
Keywords
- Caputo derivative
- integral boundary conditions
- half-line
- positive solutions
- fixed point theorems