Abstract
In this article, we establish some new existence results on positive solutions of a four-point integral boundary value problem for coupled nonlinear multi-term fractional differential equations. Our analysis rely on the well known fixed point theorems. Numerical examples are given to illustrate the main theorems.
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Introduction
Fractional differential systems have many applications in modeling of physical and chemical processes and in engineering [3, 14, 19], and have been of great interest recently. In its turn, mathematical aspects of studies on fractional differential systems were discussed by many authors, see the text books [5, 15] and papers [1, 6, 8, 13, 16, 17, 20, 21, 23–26]. A survey concerning the studies on solvability of two-point or four-point boundary value problems for fractional differential systems was given in [11].
In this paper, we discuss the existence of positive solutions of the following four-point integral type boundary value problem for the multi-term fractional differential system
where
-
(i)
\(1< \alpha ,\beta \le 2\), \(\alpha -1<m<\alpha \) and \(\beta -1<n<\beta \), \(D_{0^+}^*\) is the standard Riemann–Liouville differential derivative of order \(*>0\) with the starting point 0,
-
(ii)
\(0<\xi \le \eta <1\) and \(a,b,c,d\ge 0\),
-
(iii)
\(p,q:(0,1)\rightarrow \mathrm{I}\!\mathrm{R}\), p satisfies that there exist numbers \(k_1,l_1\) such that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\) and \(|p(t)|<t^{k_1}(1-t)^{l_1}\) for \(t\in (0,1)\), q satisfies that there exist numbers \(k_2,l_2\) such that \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\) and \(|q(t)|<t^{k_2}(1-t)^{l_2}\) for \(t\in (0,1)\), with \(p(t)\not \equiv 0\) and \(q(t)\not \equiv 0\) on (0, 1),
-
(iv)
\(f,g,\phi _i,\psi _i:(0,1)\times [0,+\infty )\times \mathrm{I}\!\mathrm{R}\rightarrow [0,+\infty )\), f is a strong \((n,\beta )\)-Carathéory function and g is a strong \((m,\alpha )\)-Carathéory function with \(f(t,0,0)\not \equiv 0\) and \(g(t,0,0)\not \equiv 0\) on (0, 1), \(\phi _1,\psi _1\) are \((n,\beta )\)-Carathéory functions and \(\phi _2,\psi _2\) are \((m,\alpha )\)-Carathéory functions.
A pair of functions (x, y) is called a solution of BVP (1) if \(x,y\in C^0(0,1]\) and x, y satisfy all equations in (1). We obtain the results on solutions of BVP(1) using Schauder’s fixed point theorem in Banach spaces. The salient features of this study are as follows:
-
(a)
the fractional differential equations in (1) are multi-term ones and their nonlinearities depend on the lower order fractional derivatives with order greater than \(\alpha -1\) and \(\beta -1\);
-
(b)
instead of the condition \(u(0)=0,v(0)=0\) we consider integral boundary conditions which are more suitable as \(D_{0^+}^\alpha x(t)=0\) with \(\alpha \in (1,2)\) implies \(x(t)=ct^{\alpha -1}\) and obviously x is not continuous at \(t=0\) while \(\lim \nolimits _{t\rightarrow 0}t^{2-\alpha }x(t)\) exists;
-
(c)
BVP(1) is a generalized form of known ones in references [4, 7, 9, 10, 21], the positive solutions of BVP(1) obtained are unbounded (discontinuous at \(t=0\)) which are different from those ones (continuous on [0,1]) in [1, 8, 23, 24];
-
(d)
this paper is a complement of [11] in which the existence of positive solutions of BVP(1) was studied under the assumptions \(m\in (0,\alpha -1]\) and \(n\in (0,\beta -1]\) while \(m\in (\alpha -1,\alpha ),n\in (\beta -1,\beta )\) are supposed in this paper.
The remainder of this paper is arranged as follows: in Sect. 2, we present preliminary results; in Sect. 3, the main results are presented; and two examples are given in Sect. 4 to illustrate the main results.
Preliminary results
For the convenience of readers, we present here the necessary definitions from fixed point theory and fractional calculus theory.
Definition 2.1
[2] Let X be a Banach space. An operator \(T:X\rightarrow X\) is completely continuous if it is continuous and maps bounded sets into pre-compact sets (or relatively compact sets).
Definition 2.2
[15] The left Riemann–Liouville fractional integral (left forward) of order \(\alpha >0\) of a function \(f:(0,\infty )\rightarrow \mathrm{I}\!\mathrm{R}\) is given by
provided that the right-hand side exists.
Definition 2.3
[15] The left Riemann–Liouville fractional derivative (left farward) of order \(\alpha >0\) of a continuous function \(f:(0,\infty )\rightarrow \mathrm{I}\!\mathrm{R}\) is given by
where \(n-1<\alpha < n\), provided that the right-hand side exists.
Definition 2.4
\(h:(0,1)\times \mathrm{I}\!\mathrm{R}\times \mathrm{I}\!\mathrm{R}\rightarrow \mathrm{I}\!\mathrm{R}\) is called a \((m,\alpha )-\)Carathédory function if it satisfies
-
(i)
\(t\rightarrow h\left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \) is measurable on (0, 1) for all \((x,y)\in \mathrm{I}\!\mathrm{R}^2\),
-
(ii)
\((x,y)\rightarrow h\left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \) is continuous for a.e. \(t\in (0,1)\),
-
(iii)
for each \(r > 0\), there exists nonnegative number \(M_r\) such that \(|u|,|v|\le r\) imply
$$\begin{aligned} \left| h \left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \right| \le M_r, \quad a.e. t \quad \in (0,1). \end{aligned}$$
Definition 2.5
\(h:(0,1)\times \mathrm{I}\!\mathrm{R}\times \mathrm{I}\!\mathrm{R}\rightarrow \mathrm{I}\!\mathrm{R}\) is called a \((m,\alpha )\)-Carathédory function if it satisfies
-
(i)
\(t\rightarrow h (t, t^{\alpha -2}x,t^{2+m-\alpha }y)\) is measurable on (0, 1) for all \((x,y)\in \mathrm{I}\!\mathrm{R}^2\),
-
(ii)
\((x,y)\rightarrow h (t, t^{\alpha -2}x,t^{2+m-\alpha }y)\) is continuous for a.e. \(t\in (0,1)\),
-
(iii)
for each \(r > 0\), there exists nonnegative function \(\phi _r\in L^1(0,1)\) such that \(|u|,|v|\le r\) imply
$$\begin{aligned} \left| h \left( t, t^{\alpha -2}x,t^{2+m-\alpha }y\right) \right| \le \phi _r(t), \quad a.e. t\quad \in (0,1).\end{aligned}$$
Lemma 2.1
[15] Let \(n-1\le \alpha < n,\) \(u\in C^0(0,\infty )\bigcap L^1(0,\infty ).\) Then
where \(C_{i}\in R\), \(i=1,2,\ldots n\)
Choose
with the norm
for \(x\in X.\) It is easy to show that X is a real Banach space.
Choose
with the norm
for \(y\in Y.\) It is easy to show that Y is a real Banach space.
Thus, \((X\times Y,||\cdot ||)\) is Banach space with the norm defined by
For a function \(x:(0,1]\rightarrow \mathrm{I}\!\mathrm{R},\) a number m and a function \(F:(0,1)\times \mathrm{I}\!\mathrm{R}^2\rightarrow \mathrm{I}\!\mathrm{R},\) denote \(F_{m,x}(t)=F(t,x(t),D_{0^+}^m x(t)).\)
Denote
Lemma 2.2
(Lemma 2.6 in [11]) Suppose that \(\Delta \not =0\) and
(B0) \(h\in C^0(0,1)\) and there exist \(k>-1\) and \(l\le 0\) such that \(2+l+k>0\) and \(|h(t)|\le t^k(1-t)^l\) for all \(t\in (0,1)\).
Then \(x\in X\) is a solution of problem
if and only if \(x\in X\) satisfies
Lemma 2.3
(Lemma 2.7 in [11]) Suppose that \(\nabla \not =0\) and (B0) holds. Then \(y\in Y\) is a solution of problem
if and only if \(y\in Y\) satisfies
Define the operator T on \(X\times Y,\) for \((x,y)\in X\times Y,\) by \(T(x,y)(t)=((T_1y)(t),(T_2x)(t))\) with
and
By Lemmas 2.2 and 2.3, we have that \((x,y)\in X\times Y\) is a solution of BVP(8) if and only if \((x,y)\in X\times Y\) is a fixed point of T.
Lemma 2.4
Suppose that (i)–(iv) defined in Sect. 1 hold, \(\Delta \not =0\) and \(\nabla \not =0.\) Then \(T:X\times Y\rightarrow X\times Y\) is completely continuous.
Proof
We will prove that both \(T_1\) and \(T_2\) are completely continuous. The proof of the completeness of \(T_1\) is divided into four steps and similarly we can prove that \(T_2\) is completely continuous.
Step 1 Suppose that \(\alpha -1<m<\alpha \). We prove that both \(T_1: Y\rightarrow X\) is well defined.
For \(y\in Y\), there exits \(r>0\) such that
Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that
for all \(t\in (0,1)\). Then
On the other hand, note \(D_{0^+}^mt^\mu =\frac{\Gamma (\mu +1)}{\Gamma (\mu +1-m)}t^{\mu -m}\), \(\Gamma (0)=\infty \) with \(\frac{1}{\Gamma (0)}=0\), we have
It is easy to show that \(T_1y\in X\). So \(T_1: Y\rightarrow X\) is well defined.
Step 2 Suppose that \(\alpha -1<m<\alpha \). Prove that \(T_1\) is continuous.
Let \(\{y_i\in Y\}\) be a sequence such that \(y_i\rightarrow y_0\) as \(i\rightarrow +\infty \) in Y. Then there exists \(r>0\) such that
holds for \(i=0,1,2,\ldots \).
Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that
for all \(t\in (0,1)\). By a direct computation, we get \((T_1y_i)(t)\) and \(D_{0^+}^m (T_1y_i)(t)\). One sees that
We can show using the dominant convergence theorem that \(T_1y_i\rightarrow T_1y_0\) as \(i\rightarrow +\infty \). Then \(T_1\) is continuous.
Now we prove that \(T_1\) maps bounded sets in Y into relatively compact sets in X. Let \(\Omega \subset Y\) be a bounded subset. Then there exists \(r>0\) such that
holds for all \(y\in \Omega. \) Then (iii) and (iv) imply that there exists a number \(M_r>0\) and \(\phi _0,\psi _0\in L^1(0,1)\) such that
for all \(t\in (0,1)\).
Step 3 Suppose that \(\alpha -1<m<\alpha \). Prove that \(\{T_1y:y\in \Omega \}\) is a bounded set in X.
Similar to Step 1 and Step 2, we can show that
So \(T_1\) maps bounded sets into bounded sets in X.
Step 4 Suppose that \(\alpha -1<m<\alpha \). Prove that \(\{T_1y:y\in \Omega \}\) is a relatively compact set in X.
We prove first that both \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^m(T_1y)(t):y\in \Omega \}\) are equi-continuous on (0, 1]. By the definition of \(T_1\), it suffices to show that both
are equi-continuous on (0, 1] (we can prove that the other parts of \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^m(T_1y)(t):y\in \Omega \}\) are equi-continuous on (0, 1] similar to [1]). Then, we prove that both \(\{t^{2-\alpha }(T_1y)(t):y\in \Omega \}\) and \(\{t^{2+m-\alpha }D_{0^+}^n(T_1y)(t):y\in \Omega \}\) are equi-convergent as \(t\rightarrow 0\). By the definition of \(T_1\), it suffices to show that both
are equi-convergent as \(t\rightarrow 0\).
First, let \(t_1,t_2\in [e,f]\subset (0,1]\) with \(t_1<t_2\), \(0<e<f\le 1\), and \(y\in \Omega \). Then we have
Second, let \(t_1,t_2\in [e,f]\subset (0,1]\) with \(0<e\le t_1<t_2\le f\le 1\) and \(y\in \Omega \). Then we have
Third, we have
Fourth, we have
Therefore, \(T_1\Omega \) is relatively compact.
From above discussion, \(T_1\) is completely continuous. The proof is completed. \(\square \)
Define
and
Now, we rewrite
Lemma 2.5
(Lemma 2.9 in [11]) Suppose that \(a,b,c,d\ge 0,\) and
Then
Main results
In this section, we prove existence result on solutions of BVP(1). Let \(\mu _i,\upsilon _i,\omega _i,\lambda _i(i=1,2)\) and \(\Delta ,\nabla \) be defined by (10). For \(\Phi \in L^1(0,1)\), denote \(||\Phi ||_1=\int _0^1|\Phi (s)|\mathrm{{d}}s\). The following assumption will be used in the main theorem.
A function \(\Phi :[0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) is called a bi-increasing function if both \(u\rightarrow \Phi (u,v)\) and \(v\rightarrow \Phi (u,v)\) are increasing. We now list the following assumption:
(B1) there exist \(\overline{\phi }_i,\overline{\psi }_i\in L^1(0,1)(i=1,2)\) and bi-increasing functions \(\Phi ,\Psi ,\Phi _i,\Psi _i(i=1,2)\) such that
For ease expression, denote
and
Theorem 3.1
Suppose that (12) holds, (i)–(iv) defined in Sect. 1 and (B1) hold. Then BVP(1) has at least one positive solution if
has a solution \((r_1,r_2)\) satisfying \(r_1>0,r_2>0\).
Proof
From Lemmas 2.2 and 2.3, we know that (x, y) is a solution of BVP(1) if and only if (x, y) is a fixed point of T. From Lemma 2.4, \(T:X\times Y\rightarrow X\times Y\) is completely continuous. By Lemma 2.5 and (i)–(iv), (x, y) is a positive solution if (x, y) is a solution of BVP(1).
To get a fixed point of T, we apply the Schauder’s fixed point theorem. We should define a closed convex bounded subset \(\Omega \) of E such that \(T(\Omega )\subseteq \Omega \). It is easy to see that \( \Omega =\{(x,y)\in E:||x||\le r_1,||y||\le r_2\}\) is a closed convex bounded subset \(\Omega \) of E.
For \((x,y)\in \Omega \), we get \( ||x||\le r_1,\;\;||y||\le r_2. \) Furthermore, we have
By the definition of T, we have
and similarly we get
We get
Similarly, we get
Since (13) has positive solution \(r_1>0,r_2>0\), we choose \(\Omega =\{(x,y)\in E:||x||\le r_1,||y||\le r_2\}\). Then we get \(T(\Omega )\subset \Omega \). Hence, the Schauder’s fixed point theorem implies that T has a fixed point \((x,y)\in \Omega \). So (x, y) is a positive solution of BVP(1).
The proof of Theorem 3.1 is completed. \(\square \)
Theorem 3.2
Suppose
(B2) there exists \(\overline{\phi }_i,\overline{\psi }_i\in L^1(0,1)(i=1,2)\) and nonnegative constants \(M_\Phi ,M_\Psi ,M_{\Phi _i},M_{\Psi _i}(i=1,2)\) such that
Then BVP(1) has at least one positive solution.
Proof
Let \(M_i,N_i,Q_i(i=1,2)\) be defined in Theorem 3.1. Choose \(\Phi (u,v)=M_\Phi ,\) \(\Psi (u,v)=M_\Psi ,\) \(\Phi _i(u,v)=M_{\Phi _i}\) and \(\Psi _i(u,v)=M_{\Psi _i}(i=1,2)\). We see that (13) has positive solution
The results follows from Theorem 3.1 directly. \(\square \)
Numerical examples
In this section, we present two examples for the illustration of our main result (Theorems 3.1 and 3.2).
Example 4.1
We consider the following boundary value problem
Then
-
(i)
BVP(13) has at least one positive solution if there exists a constant \(H>0\) such that
$$\begin{array}{l} |f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ |g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R}. \end{array}$$ -
(ii)
BVP(13) has at least one positive solution if
$$\begin{array}{l} |f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le c_1+b_1|u|^{\epsilon _1}+a_1|v|^{\delta _1},\;c_1,b_1,a_1\ge 0,\;\epsilon _1,\delta _1>0,\\ \\ |g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le c_2+b_2|u|^{\sigma _1}+a_2|v|^{\gamma _1},\;c_2,b_2,a_2\ge 0,\;\sigma _1,\gamma _1>0 \end{array} $$
and one of the followings holds:
-
(a)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}<1\);
-
(b)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}=1\) with \((38.1089b_1)^{1/\sigma _1}34.0678b_2<1\) or \(38.1089b_1(34.0678b_2)^{1/\tau _1}<1\)
-
(c)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}>1\) for sufficiently small \(b_1,a_1,b_2,a_2\).
Proof
Corresponding to BVP(1), we have \(\alpha =\frac{19}{10},\beta =\frac{39}{20}\), \(m=\frac{19}{20}\) and \(n=\frac{39}{40}\), \(\xi =\frac{1}{2},\) \(\eta =\frac{3}{4}\), \(a=b=c=d=\frac{1}{2}\) and \(\phi _i(t,u,v)=\psi _i(t,u,v)\equiv 0(i=1,2)\) and \(p(t)=t^{-\frac{1}{10}}(1-t)^{-\frac{17}{20}},\) \(q(t)=t^{-\frac{1}{10}}(1-t)^{-\frac{13}{20}}.\)
It is easy to see that (i)–(iv) hold with \(k_1=-\frac{1}{10}=k_2\), and \(l_1=-\frac{17}{20}\), \(l_2=-\frac{13}{20}\). One sees that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\), \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\). Hence, (i)-(iv) defined in Sect. 1 hold.
By direct calculation using Matlab7, we find that
and
-
(i)
By
$$\begin{array}{ll} &{}|f(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ &{}|g(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)|\le H,\;\;t\in (0,1),u,v\in \mathrm{I}\!\mathrm{R},\\ \\ &{}\phi _1(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)=\psi _1(t,t^{-\frac{1}{20}}u,t^{-\frac{41}{40}}v)=\phi _2(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)\\ &{}=\psi _2(t,t^{-\frac{1}{10}}u,t^{-\frac{21}{20}}v)=0, \end{array} $$It follows from Theorem 3.2 that BVP(13) has at least one positive solution.
-
(ii)
One sees that (B1) holds with
$$\begin{aligned}\Phi (u,v)&=c_1+b_1|u|^{\epsilon _1}+a_1v^{\delta _1},\\ \Psi (u,v)&=c_2+b_2u^{\sigma _1}+a_2v^{\gamma _1},\\ \Phi _i(u,v)&=\Psi _i(u,v)=0\; (i=1,2)\end{aligned}$$. Furthermore, we have by direct computation (use Mathlab7.0) that
$$\begin{aligned} Q_1= & {} \left[ 1+\frac{\upsilon _1+\mu _1}{\Delta }+ \frac{b\upsilon _1+b\mu _1}{\Delta }\eta ^{\alpha +k_1+l_1} +\frac{a\lambda _1+a\omega _1}{\Delta }\xi ^{\alpha +k_1+l_1}\right] \frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{\mathbf{B}(\alpha -m+l_1,k_1+1)}{\Gamma (\alpha -m)}+\frac{\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+ \frac{b\upsilon _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+b\mu _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\eta ^{\alpha +k_1+l_1}{} \mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\\&+\frac{a\lambda _1\frac{\Gamma (\alpha )}{\Gamma (\alpha -m)}+a\omega _1\frac{\Gamma (\alpha -1)}{|\Gamma (\alpha -m-1)|}}{\Delta }\frac{\xi ^{\alpha +k_1+l_1}\mathbf{B}(\alpha +l_1,k_1+1)}{\Gamma (\alpha )}\simeq 67.8769\le 68 , \end{aligned}$$and
$$\begin{aligned} Q_2= & {} \left[ 1+\frac{\upsilon _2+\mu _2}{\nabla }+ \frac{c\upsilon _2+d\mu _2}{\nabla }\eta ^{\beta +k_2+l_2} +\frac{c\lambda _2+c\omega _2}{\nabla }\xi ^{\beta +k_2+l_2}\right] \frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{\mathbf{B}(\beta -n+l_2,k_2+1)}{\Gamma (\beta -n)}+\frac{\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+ \frac{d\upsilon _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+d\mu _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\eta ^{\beta +k_2+l_2}{} \mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\\&+\frac{c\lambda _2\frac{\Gamma (\beta )}{\Gamma (\beta -n)}+c\omega _2\frac{\Gamma (\beta -1)}{|\Gamma (\beta -n-1)|}}{\nabla }\frac{\xi ^{\beta +k_2+l_2}\mathbf{B}(\beta +l_2,k_2+1)}{\Gamma (\beta )}\simeq 56.4653\le 57 . \end{aligned}$$One sees that inequality system (13) has positive solutions if
$$\begin{array}{l} 68[c_1+b_1r_2^{\epsilon _1}+a_1r_2^{\delta _1}]\le r_1,\\ \\ 57[c_2+b_2r_1^{\sigma _1}+a_2r_1^{\gamma _1}]\le r_2 \end{array} $$(14)has positive solutions. One sees that if
$$\begin{array}{l} 68[c_1+(b_1+a_1)r_2^{\max \{\epsilon _1,\delta _1\}}]\le r_1,\\ \\ 57[c_2+(b_2+a_2)r_1^{\max \{\sigma _1,\gamma _1\}}]\le r_2 \end{array} \qquad \qquad \qquad \qquad (14)' $$has positive solution \((r_1,r_2)\) with \(r_1>1,r_2>1\), then (14) has positive solution \((\max \{1,r_1\},\max \{1,r_2\}\).
-
(ii)-(a)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}<1\). It is easy to see that (14) has positive a positive solution \((r_1,r_2)\) with \(r_1>0,r_2>0\). It follows from Theorem 3.1 that BVP(13) has at least one solution if one of the followings holds:
-
(ii)-(b)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}=1\). One sees that (14)\('\) becomes
$$\begin{array}{l} 68[c_1+(b_1+a_1)r_2]\le r_1,\;\; 57[c_2+(b_2+a_2)r_1]\le r_2. \end{array}$$It is easy to see that the latest inequality system holds for sufficiently large \(r_1',r_2'>0\) if \(68\times 57(a_1+b_1)(a_2+b_2)<1\). Hence (15) has positive solution \((\max \{1,r_1'\},\max \{1,r_2'\}\). Then BVP(1) has positive solution by Theorem 3.1.
-
(ii)-(c)
\(\max \{\epsilon _1,\delta _1\}\max \{\sigma _1,\gamma _1\}>1\). By
$$\begin{aligned} \lim \limits _{(a_1,b_1,c_1)\rightarrow (0,0,0)}Q_1[c_1+b_1|u|^{\epsilon _1}+a_1v^{\delta _1}]=\lim \limits _{(a_2,b_2,c_2)\rightarrow (0,0,0)}Q_2[c_2+b_2u^{\sigma _1}+a_2v^{\gamma _1}]=0, \end{aligned}$$we know that (15) has positive solution \((r_1,r_2)\) with \(r_i>0\). Then Theorem 3.1 implies that BVP(1) has at least one positive solution if \(a_1,b_1,c_1,a_2,b_2,c_2\) are sufficiently small. The proof is completed.
\(\square \)
Example 4.2
We consider the following boundary value problem
where
Then BVP(15) has at least one positive solution for sufficiently small \(a_i,b_i (i=1,2)\).
Proof
Corresponding to BVP(1), we have \(\alpha =\frac{19}{10},\beta =\frac{39}{20}\), \(m=\frac{19}{20}\) and \(n=\frac{39}{40}\), \(a=b=c=d=\frac{1}{2}\) and \(\phi _1(t,u,v)=A,\psi _1(t,u,v)=B,\phi _2(t,u,v)=C,\psi _2(t,u,v)=D\) and \(p(t)=t^{-\frac{1}{2}}(1-t)^{-\frac{1}{5}},\) \(q(t)=t^{-\frac{1}{2}}(1-t)^{\frac{1}{10}}.\)
It is easy to see that (i)–(iv) hold with \(k_1=-\frac{1}{10}=k_2\), and \(l_1=-\frac{1}{5}\), \(l_2=-\frac{1}{10}\). One sees that \(k_1>-1\), \(\alpha -m+l_1>0\), \(2+k_1+l_1>0\), \(k_2>-1\), \(\beta -n+l_2>0\), \(2+k_2+l_2>0\). One sees \(m>\alpha -1\), \(n>\beta -1\).
Then similar to Example 4.1, we know that BVP(15) has at least one positive solution by Theorem 3.2. \(\square \)
Conclusions
In this paper, we establish sufficient conditions for the existence of positive solutions of four-point integral type boundary value problems for singular fractional differential systems. We allow the nonlinearities p(t)f(t, x, y) and q(t)g(t, x, y) in fractional differential equations to be singular at \(t=0\). Both f and g may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. This paper contributes within the domain of fractional differential equations. The methods can be applied to solve other kinds of four-point integral type boundary value problems for singular fractional differential systems.
In [12, 22], authors studied the existence of positive solutions of two-point boundary value problems for fractional order elastic beam equations. One can discuss the following boundary value problem for nonlinear singular coupled fractional order elastic beam equations of the form
where \(3<\alpha ,\beta \le 4\), \(D_{0^+}^{*}\) (\(D^*\) for short) is the Riemann–Liouville fractional derivative of order \(*\), and \(f,g:(0,1)\times [0,\infty )\times \mathrm{I}\!\mathrm{R}^2\rightarrow [0,\infty )\) is continuous. f, g depend on the lower order fractional derivatives \(u',v'\) and \(u'',v''\) and may be singular at \(t=0\) and \(t=1\), f, g are non-Carathéodory functions.
References
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)
Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered banach spaces. Comput. Math. Appl. 42, 313–322 (2001)
Basset, A.B.: On the descent of a sphere in a vicous liquid. Q. J. Pure Appl. Math. 41, 369–381 (1910)
Bai, C.Z., Fang, J.X.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150(3), 611–621 (2004)
Caponetto, R., Dongona, G., Fortuna, L.: Fractional Differential Systems, Modeling and control applications, World Scientific Series on Nonlinear Science, Ser. A, vol. 72. World Scientific Publishing Co. Pte. Ltd. (2010)
Duan, J., Temuer, C.: Solution for system of linear fractional differential equations with constant coefficients. J. Math. 29, 599–603 (2009)
Gaber, M., Brikaa, M.G.: Existence results for a coupled system of nonlinear fractional differential equation with four-point boundary conditions. ISRN Math. Anal. Article ID 468346, pp 14 (2011)
Goodrich, C.S.: Existence of a positive solution to systems of differential equations of fractional order. Comput. Math. Appl. 62, 1251–1268 (2011)
Liu, Y.: Existence and non-existence of positive solutions of BVPs for fractional order elastic beam equations with a non-Caratheodory nonlinearity. Appl. Math. Model. 38(2), 620–640 (2014)
Liu, Y.: Existence of positive solutions of fractional order elastic beam equation with a non-Carathodory nonlinearity. Math. Methods Appl. Sci. 39(6), 1311–1324 (2015)
Liu, Y.: New existence results for positive solutions of boundary value problems for coupled systems of multi-term fractional differential equations. Hacet. Univ. Bull. Nat. Sci. Eng. 2(45), 391–416 (2016)
Liang, S., Zhang, J.: Positive solutions for boundary value problems of nonlinear frac- tional differential equations. Nonlinear Anal. 71, 5545–5550 (2009)
Liu, L., Zhang, X., Wu, Y.: On existence of positive solutions of a two-point boundary value problem for a nonlinear singular semipositone system. Appl. Math. Comput. 192, 223–232 (2007)
Mainardi, F.: Fraction Calculus: Some basic problems in continuum and statistical machanics. In: Carpinteri, A., Mainardi, F. (eds.) Fratals and Fractional Calculus in Continuum Machanics, pp. 291–348. Springer, Vien (1997)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993)
Mamchuev, M.O.: Boundary value problem for a system of fractional partial differential equations. Partial Differ. Equ. 44, 1737–1749 (2008)
Rehman, M., Khan, R.: A note on boundaryvalueproblems for a coupled system of fractionaldifferential equations. Comput. Math. Appl. 61, 2630–2637 (2011)
Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22(1), 64–69 (2009)
Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)
Trujillo, J.J., Rivero, M., Bonilla, B.: On a Riemann–Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231, 255–265 (1999)
Wang, J., Xiang, H., Liu, Z.: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. Article ID 186928, pp 12. (2010). doi:10.1155/2010/186928
Xu, X., Jiang, D., Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 4676–4688 (2009)
Yuan, C.: Multiple positive solutions for (n-1, 1)-type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations, E. J. Qual. Theory of Differ. Equ. 13, 1–12 (2011)
Yang, A., Ge, W.: Positive solutions for boundary value problems of N-dimension nonlinear fractional differential systems, Boundary Value Problems, article ID 437453. (2008). doi:10.1155/2008/437453
Yuan, C., Jiang, D., O’Regan, D., Agarwal, R.P.: Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order three-point non-linear differential and difference equations. Appl. Anal. 87, 921–932 (2008)
Yuan, C., Jiang, D., O’regan, D., Agarwal, R.P.: Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Eur. J. Qual. Theory Diff. Equ. 13, 1–17 (2012)
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The author would like to thank the referees and the editors for their careful reading and some useful comments on improving the presentation of this paper.
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Supported by the Natural Science Foundation of Guangdong province (No. S2011010001900) and Natural science research project for colleges and universities of Guangdong Province (No: 2014KTSCX126).
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Yang, X., Liu, Y. New existence results on positive solutions of four-point integral type BVPs for coupled multi-term fractional differential equations. Math Sci 10, 227–240 (2016). https://doi.org/10.1007/s40096-016-0197-6
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DOI: https://doi.org/10.1007/s40096-016-0197-6
Keywords
- Four-point integral boundary value problem
- Multi-term fractional differential system
- Non-Carathéodory function
- Fixed-point theorem