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Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems

Boundary Value Problems20162016:43

https://doi.org/10.1186/s13661-016-0553-3

  • Received: 6 November 2015
  • Accepted: 4 February 2016
  • Published:

Abstract

In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form
$$ \left \{ \textstyle\begin{array}{l} D^{\alpha} x(t)=\phi(t,x(t),y(t)), \quad t\in I=[0,1], \\ D^{\beta} y(t)=\psi(t,x(t),y(t)),\quad t\in I=[0,1], \\ x(0)=g(x),\qquad x(1)=\delta x(\eta),\quad 0< \eta< 1, \\ y(0)=h(y),\qquad y(1)=\gamma y(\xi),\quad 0< \xi< 1, \end{array}\displaystyle \right . $$
where \(\alpha, \beta\in(1,2]\), D denotes the Caputo fractional derivative, \(0<\delta, \gamma<1\) are parameters such that \(\delta\eta^{\alpha}<1\), \(\gamma\xi^{\beta}<1\), \(h, g\in C(I,\mathbb{R})\) are boundary functions and \(\phi,\psi:I\times\mathbb{R} \times \mathbb{R} \rightarrow\mathbb{R}\) are continuous. We use the technique of topological degree theory to obtain sufficient conditions for existence and uniqueness of positive solutions to the system. Finally, we provide an example to illustrate our main results.

Keywords

  • coupled system
  • boundary value problems
  • fractional differential equations
  • existence and uniqueness of solutions
  • topological degree theory

MSC

  • 34A08
  • 35R11

1 Introduction

Recently much attention has been paid to investigate sufficient conditions for existence of positive solutions to boundary value problems for fractional order differential equations, we refer to [113] and the references therein. This is because of many applications of fractional differential equations in various field of science and technology as in [1420]. Existence of solutions to boundary value problems for coupled systems of fractional order differential equations has also attracted some attentions, we refer to [13, 2024]. In these papers, classical fixed point theorems such as Banach contraction principle and Schauder fixed point theorem are used for existence of solutions. The use of these results require stronger conditions on the nonlinear functions involved which restricts the applicability to limited classes of problems and very specialized systems of boundary value problems. In order to enlarge the class of boundary value problems and to impose less restricted conditions, one needs to search for other sophisticated tools of functional analysis. One such tool is topological degree theory. Few results can be found in the literature which use degree theory arguments for the existence of solutions to boundary value problems (BVPs) [2532]. However, to the best of our knowledge, the existence and uniqueness of solutions to coupled systems of multi-point boundary value problems for fractional order differential equations with topological degree theory approach have not been studied previously. Wang et al. [28] studied the existence and uniqueness of solutions via topological degree method to a class of nonlocal Cauchy problems of the form
$$ \left \{ \textstyle\begin{array}{l} D^{q} u(t)=f(t,u(t)),\quad t\in I=[0,T], \\ u(0)+g(u)=u_{0}, \end{array}\displaystyle \right . $$
where \(D^{q}\) is the Caputo fractional derivative of order \(q\in (0,1]\), \(u_{0}\in\mathbb{R}\), and \(f:I\times\mathbb{R}\rightarrow \mathbb{R}\) is continuous. The result was extended to the case of a boundary value problem by Chen et al. [26] who studied sufficient conditions for existence results for the following two point boundary value problem:
$$ \left \{ \textstyle\begin{array}{l} D_{0+}^{\alpha} \phi_{p} (D_{0+}^{\beta}u(t))=f(t,u(t),D_{0+}^{\beta }u(t)), \\ D_{0+}^{\beta}u(0)=D_{0+}^{\beta}u(1)=0, \end{array}\displaystyle \right . $$
where \(D_{0+}^{\alpha}\) and \(D_{0+}^{\beta}\) are Caputo fractional derivatives, \(0<\alpha, \beta\leq1\), \(1<\alpha+\beta\leq2\). Wang et al. [27] studied the following two point boundary value problem for fractional differential equations with different boundary conditions:
$$ \left \{ \textstyle\begin{array}{l} D_{0+}^{\alpha} \phi_{p} (D_{0+}^{\beta}u(t))=f(t,u(t),D_{0+}^{\beta }u(t)), \\ u(0)=0, \qquad D_{0+}^{\beta}u(0)=D_{0+}^{\beta}u(1), \end{array}\displaystyle \right . $$
where \(D_{0+}^{\alpha}\) and \(D_{0+}^{\beta}\) are Caputo fractional derivatives, \(0<\alpha, \beta\leq1\), \(1<\alpha+\beta\leq2\).
Motivated by the work cited above, in this paper, we use a coincidence degree theory approach for condensing maps to obtain sufficient conditions for the existence and uniqueness of solutions to more general coupled systems of nonlinear multi-point boundary value problems. The boundary conditions are also nonlinear. The system is of the form
$$ \left \{ \textstyle\begin{array}{l} D^{\alpha} x(t)=\phi(t,x(t),y(t)), \quad t\in I=[0,1], \\ D^{\beta} y(t)=\psi(t,x(t),y(t)), \quad t\in I=[0,1], \\ x(0)=g(x), \qquad x(1)=\delta x(\eta), \quad 0< \eta< 1, \\ y(0)=h(y), \qquad y(1)=\gamma y(\xi), \quad 0< \xi< 1, \end{array}\displaystyle \right . $$
(1)
where \(\alpha,\beta\in(1,2]\), D is used for standard Caputo fractional derivative and \(0<\delta,\gamma<1\) such that \(\delta\eta^{\alpha}<1\), \(\gamma\xi^{\beta}<1\), \(h,g\in C(I,\mathbb{R})\) are boundary functions and \(\phi,\psi:I\times\mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}\) are continuous.

2 Preliminaries

In this section we give some fundamental definitions and results from fractional calculus and topological degree theory. For further detailed study, we refer to [19, 24, 26, 27, 33, 34].

Definition 2.1

The fractional integral of order \(\rho\in\mathbb{R}_{+}\) of a function \(u\in L^{1}([a,b], \mathbb{R})\) is defined as
$$ I_{0+}^{\rho}u(t)=\frac{1}{\Gamma(\rho)} \int_{a}^{t}(t-s)^{\rho-1} u(s)\, ds. $$

Definition 2.2

The Caputo fractional order derivative of a function u on the interval \([a, b]\) is defined by
$$ D_{0+}^{\rho}u(t)=\frac{1}{\Gamma(m-\rho)} \int_{a}^{t}(t-s)^{m-\rho -1}u^{(m)}(s)\, ds, $$
where \(m=[\rho]+1\) and \([\rho]\) represents the integer part of ρ.

Lemma 2.1

The following result holds for fractional differential equations:
$$ I^{\rho}D^{\rho}u(t)=u(t)+d_{0}+d_{1}t+d_{2}t^{2}+ \cdots+d_{m-1}t^{m-1} $$
for arbitrary \(d_{i}\in R\), \(i=0,1,2,\ldots,m-1\).

The spaces \(X=C([0,1], \mathbb{R})\), \(Y=C([0,1], \mathbb{R})\) of all continuous functions from \([0,1]\rightarrow\mathbb{R}\) are Banach spaces under the topological norms \(\lVert x\rVert=\sup\{\lvert x(t)\rvert:t\in[0,1] \}\) and \(\lVert y\rVert=\sup\{\lvert y(t)\rvert:t\in[0,1] \}\), respectively. The product space \(X\times Y\) is a Banach space under the norm \(\lVert(x,y)\rVert=\lVert x\rVert+\lVert y\rVert\). It is also a Banach space under the norm \(\lvert(x,y)\rvert= \max\{\lVert x\rVert,\lVert y\rVert\}\).

Let \(\mathbb{S}\) be a family of all bounded sets of \(\mathfrak {P}(X\times Y)\), where \(X\times Y\) is a Banach space. Then we recall the following notions [35].

Definition 2.3

The Kuratowski measure of non-compactness \(\mu: \mathbb {S}\rightarrow\mathbb{R}_{+}\) is defined as
$$ \mu(S)= \inf \{d>0:S \text{ admits a finite cover by sets of diameter}\leq d\}, $$
where \(S\in\mathbb{S}\).

Proposition 2.1

The Kuratowski measure μ satisfies the following properties:
  1. (i)

    \(\mu(S)=0\) iff S is relatively compact.

     
  2. (ii)

    μ is a seminorm, i.e., \(\mu(\lambda S)=\lvert\lambda\rvert\mu(S)\), \(\lambda\in\mathbb{R}\) and \(\mu(S_{1}+S_{2})\leq\mu(S_{1})+\mu(S_{2})\).

     
  3. (iii)

    \(S_{1}\subset S_{2}\) implies \(\mu(S_{1})\leq\mu(S_{2})\); \(\mu(S_{1}\cup S_{2})=\max\{\mu (S_{1}),\mu(S_{2})\}\).

     
  4. (iv)

    \(\mu(\operatorname{conv} S)=\mu(S)\).

     
  5. (v)

    \(\mu(\bar{S})=\mu(S)\).

     

Definition 2.4

Let \(F:\Omega\rightarrow X\) be continuous bounded map, where \(\Omega\subset X\). Then F is μ-Lipschitz if there exists \(K\geq0\) such that
$$ \mu\bigl(F(S)\bigr)\leq K \mu(S),\quad \forall S\subset\Omega \text{ bounded}. $$
Further, F will be strict μ-contraction if \(K<1\).

Definition 2.5

A function F is μ-condensing if
$$ \mu\bigl(F(S)\bigr)< \mu(S),\quad \forall S\subset\Omega \text{ bounded, with } \mu(S)>0. $$
In other words, \(\mu(F(S))\geq\mu(S)\) implies \(\mu(S)=0\).

Here, we denote the class of all strict μ-contractions \(F:\Omega\rightarrow X\) by \(\vartheta C_{\mu}(\Omega)\) and denote the class of all μ-condensing maps \(F:\Omega\rightarrow X\) by \(C_{\mu}(\Omega)\).

Remark 1

\(\vartheta C_{\mu}(\Omega)\subset C_{\mu}(\Omega)\) and every \(F\in C_{\mu}(\Omega)\) is μ-Lipschitz with constant \(K=1\).

Moreover, we recall that \(F:\Omega\rightarrow X\) is Lipschitz if there exists \(K>0\) such that
$$ \bigl\lVert F(x)-F(y) \bigr\rVert \leq K\lvert x-y \rvert,\quad \forall x, y \in \Omega, $$
and that if \(K<1\), then F is a strict contraction. For the following results, we refer to [34].

Proposition 2.2

If \(F,G:\Omega\rightarrow X \) are μ-Lipschitz with constants K and \(K'\), respectively, then \(F+G:\Omega\rightarrow X\) is μ-Lipschitz with constant \(K+K'\).

Proposition 2.3

If \(F:\Omega\rightarrow X\) is compact, then F is μ-Lipschitz with constant \(K=0\).

Proposition 2.4

If \(F:\Omega\rightarrow X\) is Lipschitz with constant K, then F is μ-Lipschitz with the same constant K.

The following theorem due to Isaia [34] plays an important role for our main result.

Theorem 2.1

Let \(F:X\rightarrow X\) be μ-condensing and
$$ \Psi=\bigl\{ x\in X: \exists \lambda\in[0,1] \textit{ such that } x=\lambda Fx \bigr\} . $$
If Ψ is a bounded set in X, so there exists \(r>0\) such that \(\Psi\subset S_{r}(0)\), then the degree
$$ D\bigl(I-\lambda F, S_{r}(0),0\bigr)=1,\quad \forall \lambda\in[0,1]. $$
Consequently, F has at least one fixed point and the set of the fixed points of F lies in \(S_{r}(0)\).
Now, we list the following hypotheses.
(C1): 
There exist constants \(K_{g}, K_{h} \in[0,1)\) such that
$$\begin{aligned}& \bigl\lvert g(x_{2})-g(x_{1}) \bigr\rvert \leq K_{g}\lvert x_{2}-x_{1} \rvert, \\& \bigl\lvert h(y_{2})-h(y_{1}) \bigr\rvert \leq K_{h} \lvert y_{2}-y_{1} \rvert\quad \text{for }x_{1},x_{2},y_{1},y_{2}, \in\mathbb{R}. \end{aligned}$$
(C2): 
There exist constants \(C_{g}, C_{h}, M_{g},M_{h} >0\) such that, for \(x,y \in \mathbb{R}\),
$$ \bigl\lvert g(x) \bigr\rvert \leq C_{g}\lvert x \rvert+M_{g}, \qquad \bigl\lvert h(y) \bigr\rvert \leq C_{h} \lvert y \rvert+M_{h}. $$
(C3): 
There exist constants \(c_{i}\), \(d_{i}\) (\(i=1,2\)) and \(M_{\phi}\), \(M_{\psi}\) such that, for \(x,y \in\mathbb{R}\),
$$\begin{aligned}& \bigl\lvert \phi(t,x,y) \bigr\rvert \leq c_{1} \lvert x \rvert+c_{2}\lvert y\rvert +M_{\phi}, \\& \bigl\lvert \psi(t,x,y) \bigr\rvert \leq d_{1} \lvert x \rvert+d_{2}\lvert y\rvert+M_{\psi}. \end{aligned}$$
(C4): 
There exist constants \(L_{\phi}\), \(L_{\psi}\) such that, for \(x_{1},x_{2},y_{1},y_{2} \in \mathbb{R}\),
$$\begin{aligned}& \bigl\lvert \phi(t,x_{2},y_{2})-\phi(t,x_{1},y_{1}) \bigr\rvert \leq L_{\phi}\bigl( \lvert x_{2}-x_{1} \rvert+ \lvert y_{2}-y_{1} \rvert\bigr), \\& \bigl\lvert \psi(t,x_{2},y_{2})-\psi(t,x_{1},y_{1}) \bigr\rvert \leq L_{\psi}\bigl( \lvert x_{2}-x_{1} \rvert+ \lvert y_{2}-y_{1} \rvert\bigr). \end{aligned}$$

3 Main results

In this section, we discuss the existence and uniqueness of solutions to the BVP (1).

Lemma 3.1

If \(h:I\rightarrow\mathbb{R}\) is α times integrable function, then solutions of the BVP
$$ \begin{aligned} & D^{\alpha} x(t)=h(t),\quad t\in I=[0,1], \\ &x(0)=g(x),\qquad x(1)=\delta x(\eta), \quad 0< \eta< 1, \end{aligned} $$
(2)
are a solution of the following Fredholm integral equation:
$$ x(t)= \biggl(1-\frac{t(1-\delta)}{1-\delta\eta} \biggr)g(x)+ \int _{0}^{1}G_{\alpha}(t,s) h(s)\, ds,\quad t \in[0,1], $$
(3)
where \(G_{\alpha}(t,s)\) is defined by
$$ G_{\alpha}(t,s)= \frac{1}{\Gamma(\alpha)}\left \{ \textstyle\begin{array}{l@{\quad}l} (t-s)^{\alpha-1}+\frac{t\delta(\eta-s)^{\alpha-1}}{1-\delta\eta}- \frac{t(1-s)^{\alpha-1}}{1-\delta\eta},& 0\leq s\leq t\leq\eta \leq 1, \\ (t-s)^{\alpha-1}-\frac{t(1-s)^{\alpha-1}}{1-\delta\eta }, &0\leq \eta\leq s\leq t\leq1, \\ \frac{t\delta(\eta-s)^{\alpha-1}}{1-\delta\eta}-\frac {t(1-s)^{\alpha-1}}{1-\delta\eta},& 0\leq t\leq s\leq\eta\leq1, \\ -\frac{t(1-s)^{\alpha-1}}{1-\delta\eta},& 0\leq\eta\leq t\leq s\leq 1. \end{array}\displaystyle \right . $$
(4)

Proof

Applying \(I^{\alpha}\) on \(D^{\alpha}x(t)=h(t)\) and using Lemma 2.1, we have
$$ x(t)=I^{\alpha}h(t)+c_{0}+c_{1}t $$
(5)
for some \(c_{0},c_{1}\in R\). The conditions \(x(0)=g(x)\) and \(x(1)=\delta x(\eta)\) imply \(c_{0}=g(x)\) and \(c_{1}=\frac{\delta}{1-\delta\eta}I^{\alpha} h(\eta)-\frac{1-\delta}{1-\delta\eta}g(x)-\frac{1}{1-\delta\eta }I^{\alpha} h(1)\). Hence, we obtain
$$ x(t)=I^{\alpha}h(t)+g(x)+t \biggl[\frac{\delta}{1-\delta\eta }I^{\alpha} h(\eta)- \frac{1-\delta}{1-\delta\eta}g(x)-\frac{1}{1-\delta\eta }I^{\alpha} h(1) \biggr], $$
(6)
which after rearranging can be written as
$$x(t)= \biggl(1-\frac{t(1-\delta)}{1-\delta\eta} \biggr)g(x)+ \int _{0}^{1} G_{\alpha}(t,s)h(s)\, ds. $$
 □
In view of Lemma 3.1, solutions of the coupled systems of BVPs (1) are solutions of the following coupled systems of Fredholm integral equations:
$$ \left \{ \textstyle\begin{array}{l} x(t)= (1-\frac{t(1-\delta)}{1-\delta\eta} )g(x)+\int _{0}^{1}G_{\alpha}(t,s) \phi(t,x(s),y(s))\, ds,\quad t\in[0,1], \\ y (t)= (1-\frac{t(1-\gamma)}{1-\gamma\xi} )h(y)+\int _{0}^{1}G_{\beta}(t,s)\psi(t,x(s),y(s))\, ds,\quad t\in[0,1], \end{array}\displaystyle \right . $$
(7)
where \(G_{\beta}(t,s) \) is defined by
$$ G_{\beta}(t,s)= \frac{1}{\Gamma(\beta)}\left \{ \textstyle\begin{array}{l@{\quad}l} (t-s)^{\beta-1}+\frac{t\gamma(\xi-s)^{\beta-1}}{1-\gamma\xi }-\frac{t(1-s)^{\beta-1}}{1-\gamma\xi},& 0\leq s\leq t\leq\xi\leq1, \\ (t-s)^{\beta-1}-\frac{t(1-s)^{\beta-1}}{1-\gamma\xi},& 0\leq\xi \leq s\leq t\leq1, \\ \frac{t\gamma(\xi-s)^{\beta-1}}{1-\gamma\xi}-\frac{t(1-s)^{\beta -1}}{1-\gamma\xi},& 0\leq t\leq s\leq\xi\leq1, \\ -\frac{t(1-s)^{\beta-1}}{1-\gamma\xi}, &0\leq\xi\leq t\leq s\leq1. \end{array}\displaystyle \right . $$
(8)
Clearly
$$ \begin{aligned} &\max_{t\in[0,1]}\bigl\vert G_{\alpha}(t,s)\bigr\vert =\frac{(1-s)^{\alpha-1}}{(1-\delta\eta)\Gamma(\alpha)}, \\ &\max_{t\in[0,1]}\bigl\vert G_{\beta}(t,s)\bigr\vert =\frac{(1-s)^{\beta-1}}{(1-\gamma\xi)\Gamma (\beta)},\quad s\in [0,1]. \end{aligned} $$
(9)
Define the operators \(F_{1}:X\rightarrow X\), \(F_{2}:Y\rightarrow Y\) by
$$\begin{aligned}& F_{1}(x) (t)= \biggl(1-\frac{t(1-\delta)}{1-\delta\eta} \biggr)g(x), \\& F_{2}(y) (t)= \biggl(1-\frac{t(1-\gamma)}{1-\gamma\xi} \biggr)h(y) \end{aligned}$$
and the operators \(G_{1},G_{2}:X\times Y\rightarrow X\times Y\) by
$$\begin{aligned}& G_{1}(x,y) (t)= \int_{0}^{1}G_{\alpha}(t,s) \phi \bigl(t,x(s),y(s)\bigr)\, ds, \\& G_{2}(x,y) (t)= \int_{0}^{1}G_{\beta}(t,s)\psi \bigl(t,x(s),y(s)\bigr)\, ds. \end{aligned}$$
Further, we define \(F=(F_{1},F_{2})\), \(G=(G_{1},G_{2})\) and \(T=F+G\). Then the system of integral equations (7) can be written as an operator equation of the form
$$ (x,y)=T(x,y)=F(x,y)+G(x,y), $$
(10)
and solutions of the system (7) are fixed points of T.

Lemma 3.2

Under the assumptions (C1) and (C2), the operator F satisfies the Lipschitz condition and the following growth condition:
$$ \bigl\lVert F(x,y) \bigr\rVert \leq C \bigl\lVert (x, y) \bigr\rVert +M\quad \textit{for every } (x,y)\in X\times Y. $$
(11)

Proof

Using the assumption (C1), we obtain
$$\begin{aligned}& \bigl\lvert F(x,y) (t)-F(u,v) (t) \bigr\rvert \\& \quad = \biggl\lvert \biggl(1-\frac{t(1-\delta)}{1-\delta\eta} \biggr) \bigl( g(x)-g(u)\bigr)+ \biggl(1- \frac{t(1-\gamma)}{1-\gamma\xi} \biggr) \bigl(h(y)-h(v) \bigr) \biggr\rvert \\& \quad \leq K_{g} \lVert x-u \rVert + K_{h}\lVert y-v \rVert \leq K \bigl\lVert (x,y)-(u,v) \bigr\rVert ,\quad K=\max\{K_{g},K_{h} \}. \end{aligned}$$
(12)
By Proposition 2.4, F is also μ-Lipschitz with constant K. Now for the growth condition using (C2), we get
$$ \bigl\lVert F(x,y) \bigr\rVert = \bigl\lVert \bigl( F_{1}(x),F_{2} (y)\bigr) \bigr\rVert = \bigl\lVert F_{1}(x) \bigr\rVert + \bigl\lVert F_{2} (y) \bigr\rVert \leq C \bigl\lVert (x,y) \bigr\rVert +M, $$
where \(C=\max\{C_{g},C_{h}\}\), \(M=\max\{M_{g},M_{h}\}\). □

Lemma 3.3

The operator G is continuous and under the assumption (C3) satisfies the growth condition
$$ \bigl\lVert G(x,y) \bigr\rVert \leq\Delta \bigl\lVert (x,y) \bigr\rVert +\Lambda\quad \textit{for every }(x,y)\in X\times Y, $$
(13)
where \(\Delta=\theta(c+d)\), \(\theta=\max\{\frac{1}{(1-\delta\eta )\Gamma(\alpha)}, \frac{1}{(1-\gamma\xi)\Gamma(\beta)}\}\), \(c=\max\{c_{1}, c_{2}\}\), \(d=\max\{d_{1}, d_{2}\}\), \(\Lambda=\theta( M_{\phi}+ M_{\psi})\).

Proof

Let \(\{(x_{n},y_{n} )\}\) be a sequence of a bounded set \(U_{r}=\{\lVert(x,y) \rVert\leq r: (x,y) \in X\times Y\}\) such that \((x_{n},y_{n}) \rightarrow(x,y)\) in \(U_{r}\). We need to prove that \(\lVert G(x_{n},y_{n})-G(x,y)\rVert\rightarrow 0\). Consider
$$\begin{aligned}& \bigl\lvert G_{1}(x_{n},y_{n}) (t) -G_{1} (x,y) (t) \bigr\rvert \\& \quad \leq\frac{1}{\Gamma (\alpha)}\biggl[ \int_{0}^{t}(t-s)^{\alpha-1} \bigl\lvert \phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)- \phi\bigl(s,x(s),y(s)\bigr) \bigr\rvert \, ds \\& \qquad {}+\frac{\delta}{1-\delta\eta} \int_{0}^{\eta}(\eta-s)^{\alpha -1} \bigl\lvert \phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)- \phi\bigl(s,x(s),y(s)\bigr) \bigr\rvert \, ds \\& \qquad {}+\frac{1}{1-\delta\eta} \int_{0}^{1}(1-s)^{\alpha-1} \bigl\lvert \phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)- \phi\bigl(s,x(s),y(s)\bigr) \bigr\rvert \, ds \biggr]. \end{aligned}$$
From the continuity of ϕ and ψ, it follows that \(\phi (s,x_{n}(s),y_{n}(s))\rightarrow\phi(s,x(s),y(s))\) as \(n\rightarrow\infty\). For each \(t \in I\), using (C3), we obtain \((t-s)^{\alpha-1}\lvert\phi(s,x_{n},y_{n})-\phi(s,x,y)\rvert\leq (t-s)^{\alpha-1}2((c_{1}+c_{2})r+M_{\phi})\), which implies the integrability for \(s, t\in I\) and by using the Lebesgue dominated convergence theorem, we obtain \(\int_{0}^{t}(t-s)^{\alpha-1}\lvert\phi(s,x_{n},y_{n})-\phi(s,x,y)\rvert \, ds\rightarrow0\) as \(n\rightarrow\infty\). Similarly, the other terms approach 0 as \(n\rightarrow\infty\). It follows that
$$ \bigl\lVert G_{1}(x_{n},y_{n}) (t) -G_{1}(x,y) (t) \bigr\rVert \rightarrow0\quad \text{as } n\rightarrow \infty, $$
and similarly we can show that
$$ \bigl\lVert G_{2}(x_{n},y_{n}) (t) -G_{2}(x,y) (t) \bigr\rVert \rightarrow0 \quad \text{as } n\rightarrow \infty. $$
Now for growth condition on G, using (C3) and (9), we obtain
$$ \bigl\lvert G_{1}(x,y) (t) \bigr\rvert = \biggl\lvert \int_{0}^{1}G_{\alpha}(t,s)\phi \bigl(s,x(s),y(s)\bigr) \, ds \biggr\lvert \leq\frac{1}{(1-\delta\eta)\Gamma(\alpha)} \bigl(c_{1} \Vert x\Vert +c_{2}\|y\| +M_{\phi}\bigr) $$
and
$$ \bigl\lvert G_{2}(x,y) (t) \bigr\rvert = \biggl\lvert \int_{0}^{1}G_{\beta}(s,t)\psi \bigl(s,x(s),y(s)\bigr)\, ds \biggr\lvert \leq\frac{1}{(1-\gamma\xi)\Gamma(\beta)} \bigl(d_{1} \Vert x\Vert +d_{2}\|y\| +M_{\psi}\bigr). $$
Hence, it follows that
$$\begin{aligned} \bigl\lVert G(x,y) \bigr\rVert =& \bigl\lVert G_{1}(x,y) \bigr\rVert + \bigl\lVert G_{2}(x,y) \bigr\rVert \\ \leq&\theta\bigl(c_{1} \lVert x \rVert +c_{2} \lVert y \rVert+M_{\phi}\bigr)+\theta \bigl(d_{1} \lVert x \rVert+d_{2} \lVert y \rVert+M_{\psi}\bigr) \\ \leq&\theta(c+ d) \bigl( \lVert x \rVert+ \lVert y \rVert\bigr)+ \theta( M_{\phi}+ M_{\psi})=\Delta \bigl\lVert (x,y) \bigr\rVert +\Lambda. \end{aligned}$$
By this, we complete the proof. □

Lemma 3.4

The operator \(G:X\times Y\rightarrow X\times Y\) is compact. Consequently, G is μ-Lipschitz with zero constant.

Proof

Take a bounded set \(\mathcal{B}\subset{U_{r}}\subseteq X\times Y\) and a sequence \(\{(x_{n},y_{n})\}\) in \(\mathcal{B}\), then using (13), we have
$$ \bigl\lVert G(x_{n},y_{n}) \bigr\rVert \leq\Delta r+\Lambda \quad \text{for every }(x,y)\in X\times Y, $$
which implies that \(G(\mathcal{B})\) is bounded. Now, for equi-continuity and for given \(\epsilon>0\), take
$$\delta=\min \biggl\{ \delta_{1}=\frac{1}{2} \biggl( \frac{\epsilon \Gamma(\alpha+1)}{6 ([c_{1}+c_{2}]r+M_{\phi})} \biggr)^{\frac {1}{\alpha}}, \delta_{2}= \frac{1}{2} \biggl(\frac{\epsilon\Gamma (\beta+1)}{6 ([d_{1}+d_{2}]r+M_{\psi})} \biggr)^{\frac{1}{\beta }} \biggr\} . $$
For each \((x_{n},y_{n})\in\mathcal{B}\), we claim that if \(t,\tau\in [0,1]\) and \(0<\tau-t<\delta_{1}\), then
$$\bigl\vert G_{1}(x_{n},y_{n}) (t)-G_{1}(x_{n},y_{n}) (\tau)\bigr\vert < \frac{\epsilon}{2}. $$
Now consider
$$\begin{aligned}& \bigl\vert G_{1}(x_{n},y_{n}) (t)-G_{1}(x_{n},y_{n}) (\tau)\bigr\vert \\& \quad =\biggl\vert \frac{1}{\Gamma (\alpha)} \int_{0}^{t}\bigl[(t-s)^{\alpha-1}-( \tau-s)^{\alpha-1}\bigr]\phi \bigl(s,x_{n}(s),y_{n}(s) \bigr)\,ds \\& \qquad {}+\frac{1}{\Gamma(\alpha)} \int_{t}^{\tau }(\tau-s)^{\alpha-1}\phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)\,ds \\& \qquad {}+\frac{\delta(t-\tau)}{(1-\delta\eta) \Gamma (\alpha)} \int_{0}^{\eta}(\eta-s)^{\alpha-1}\phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)\,ds \\& \qquad {}+\frac{(\tau-t)}{(1-\delta\eta) \Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha-1}\phi \bigl(s,x_{n}(s),y_{n}(s)\bigr)\,ds\biggr\vert \\& \quad \leq \frac{(c_{1}\lvert x_{n}\rvert+c_{2}\lvert y_{n}\rvert+M_{\phi})}{\Gamma(\alpha+1)} \bigl[\bigl(t^{\alpha}-\tau^{\alpha}\bigr)+2(\tau-t)^{\alpha}\bigr] \\& \quad \leq\frac{(c_{1}+c_{2})r+M_{\phi}}{\Gamma(\alpha+1)} \bigl[\bigl(\tau ^{\alpha}-t^{\alpha}\bigr)+2(\tau-t)^{\alpha}\bigr]. \end{aligned}$$
We continue the proof with several cases.
Case 1. \(\delta_{1}\leq t<\tau<1\):
$$\begin{aligned} \bigl\vert G_{1}(x_{n},y_{n}) (t)-G_{1}(x_{n},y_{n}) (\tau)\bigr\vert < & \frac{(c_{1}+c_{2})r+M_{\phi}}{\Gamma(\alpha+1)}(2+\alpha)\delta_{1}^{\alpha-1}(\tau-t) \\ < & \frac{(c_{1}+c_{2})r+M_{\phi}}{\Gamma(\alpha+1)}(2+\alpha)\delta _{1}^{\alpha}< \frac{\epsilon}{2}. \end{aligned}$$
Case 2. \(0\leq t<\delta_{1}\), \(\tau<2\delta_{1}\):
$$\begin{aligned} \bigl\vert G_{1}(x_{n},y_{n}) (t)-G_{1}(x_{n},y_{n}) (\tau)\bigr\vert < & \frac{(c_{1}+c_{2})r+M_{\phi}}{\Gamma(\alpha+1)}\bigl[3\tau^{\alpha}\bigr] \\ < & \frac{(c_{1}+c_{2})r+M_{\phi}}{\Gamma(\alpha+1)}3(2\delta _{1})^{\alpha}= \frac{\epsilon}{2}. \end{aligned}$$
Similarly for the second part, for each \((x_{n},y_{n})\in\mathcal{B}\), we claim that if \(t,\tau\in[0,1]\) and \(0<\tau-t<\delta_{2}\), then \(|G_{2}(x_{n},y_{n})(t)-G_{2}(x_{n},y_{n})(\tau)|<\frac{\epsilon}{2}\). Then:
Case 1. \(\delta_{2}\leq t<\tau<1\):
$$\begin{aligned} \bigl\vert G_{2}(x_{n},y_{n}) (t)-G_{2}(x_{n},y_{n}) (\tau)\bigr\vert < & \frac{(d_{1}+d_{2})r+M_{\psi}}{\Gamma(\beta+1)}(2+\beta)\delta_{2}^{\beta-1}(\tau-t) \\ < & \frac{(d_{1}+d_{2})r+M_{\psi}}{\Gamma(\beta+1)}(2+\beta)\delta _{2}^{\beta}< \frac{\epsilon}{2}. \end{aligned}$$
Case 2. \(0\leq t<\delta_{2}\), \(\tau<2\delta_{2}\):
$$\begin{aligned} \bigl\vert G_{2}(x_{n},y_{n}) (t)-G_{2}(x_{n},y_{n}) (\tau)\bigr\vert < & \frac{(d_{1}+d_{2})r+M_{\psi}}{\Gamma(\beta+1)}\bigl[3\tau^{\beta}\bigr] \\ < & \frac{(d_{1}+d_{2})r+M_{\psi}}{\Gamma(\beta+1)}3(2\delta_{2})^{\beta }=\frac{\epsilon}{2}. \end{aligned}$$
Hence, we have
$$ \bigl\Vert G(x_{n},y_{n})-G(x_{n},y_{n}) \bigr\Vert < \epsilon. $$
(14)
Thus \(G(\mathcal{B})\) is equi-continuous. In view of the Arzelà-Ascoli theorem \(G(\mathcal{B})\) is compact. Furthermore, by Proposition 2.3, G is μ-Lipschitz with constant zero. □

Theorem 3.1

Under the assumptions (C1)-(C3), the system (1) has at least one solution \((x,y)\in X\times Y\) provided \(C+\Delta<1\). Moreover, the set of solutions of (1) is bounded in \(X\times Y\).

Proof

By Lemma 3.2, F is μ-Lipschitz with constant \(K\in[0,1)\) and by Lemma 3.4 G is μ-Lipschitz with constant 0. It follows by Proposition 2.2 that T is a strict μ-contraction with constant K. Define
$$ \mathbb{B}=\bigl\{ (x,y)\in X\times Y: \text{there exist } \lambda\in [0,1] \text{ such that } (x,y)=\lambda T(x,y)\bigr\} . $$
We have to prove that \(\mathbb{B}\) is bounded in \(X\times Y\). For this, choose \((x,y)\in\mathbb{B}\), then in view of the growth conditions as in Lemmas 3.2 and 3.3, we have
$$\begin{aligned} \bigl\lVert (x,y) \bigr\rVert &= \bigl\lVert \lambda T(x,y) \bigr\rVert =\lambda \bigl\lVert T(x,y) \bigr\rVert \leq\lambda \bigl( \bigl\lVert F(x,y) \bigr\rVert + \bigl\lVert G(x,y) \bigr\rVert \bigr) \\ & \leq\lambda \bigl[C \bigl\lVert (x,y) \bigr\rVert +M+\Delta \bigl\lVert (x,y) \bigr\rVert +\Lambda \bigr] \\ & =\lambda (C+\Delta) \bigl\lVert (x,y) \bigr\rVert +\lambda(M+\Lambda) , \end{aligned}$$
which implies that \(\mathbb{B}\) is bounded in \(X\times Y\). Therefore, by Theorem 2.1, T has at least one fixed point and the set of fixed points is bounded in \(X\times Y\). □

Theorem 3.2

In addition to the assumption (C1)-(C4), assume that \(K+\theta(L_{\phi}+L_{\psi})<1\), then the system (1) has a unique solution.

Proof

We use the Banach contraction theorem, for \((x,y), (u,v)\in\mathbb{R}\times\mathbb{R}\), we have from (12)
$$ \bigl\lvert F(x,y)-F(u,v) \bigr\rvert \leq K \bigl\lVert (x,y)-(u,v) \bigr\rVert . $$
(15)
Using (C4) and (9), we obtain
$$\begin{aligned} \bigl\lvert G_{1}(x,y)-G_{1}(u,v) \bigr\rvert \leq& \int_{0}^{1} \bigl\lvert G_{\alpha}(t,s) \bigr\lvert \bigl\lvert \phi\bigl(s,x(s),y(s)\bigr)-\phi\bigl(s,u(s),v(s)\bigr) \bigr\rvert \,ds \\ \leq&\theta L_{\phi}\bigl( \lvert x-u \rvert+ \lvert y-v \rvert\bigr), \end{aligned}$$
which implies that
$$\begin{aligned} \bigl\lVert G_{1}(x,y)-G_{1}(u,v) \bigr\rVert \leq& \theta L_{\phi}\bigl( \lVert x-u \rVert+\lVert y-v\rVert \bigr) \\ =&\theta L_{\phi} \bigl\lVert (x-u, y-v) \bigr\rVert \\ =&\theta L_{\phi} \bigl\lVert (x,y)-(u,v) \bigr\rVert . \end{aligned}$$
(16)
Similarly, we have
$$ \bigl\lVert G_{2}(x,y)-G_{2}(u,v) \bigr\rVert \leq\theta L_{\psi} \bigl\lVert (x,y)-(u,v) \bigr\rVert . $$
(17)
From (16) and (17), it follows that
$$\begin{aligned} \bigl\lVert G(x,y)-G(u,v) \bigr\rVert =& \bigl\lVert G_{1}(x,y)-G_{1}(u,v) \bigr\rVert + \bigl\lVert G_{2}(x,y)-G_{2}(u,v) \bigr\rVert \\ \leq&\theta(L_{\psi}+L_{\psi}) \bigl\lVert (x,y)-(u,v) \bigr\rVert . \end{aligned}$$
(18)
Hence, in view of (15) and (18), we obtain
$$\begin{aligned} \bigl\lvert T(x,y)-T(u,v)\bigr\rvert \leq& \bigl\lVert F(x,y)-F(u,v) \bigr\rVert + \bigl\lVert G(x,y)-G(u,v) \bigr\rVert \\ \leq&\bigl(K+\theta (L_{\psi}+L_{\psi})\bigr) \bigl\lVert (x,y)-(u,v) \bigr\rVert , \end{aligned}$$
which implies that T is a contraction. By the Banach contraction principle, the system (1) has a unique solution. □

4 Example

Example 4.1

Consider the following multi-point BVP:
$$ \left \{ \textstyle\begin{array}{l} D^{\frac{3}{2}} x(t)=\frac{1}{50+t^{2}}(1+\lvert x(t)\rvert+\lvert y(t) \rvert), \quad t\in[0,1], \\ D^{\frac{3}{2}} y(t)=\frac{1+\lvert x(t)\rvert+\lvert y(t) \rvert }{50+\lvert \cos x(t)\rvert+\lvert \sin y(t)\rvert} ,\quad t\in[0,1], \\ x(0)=\frac{g(x)}{2}, \qquad x(1)=\frac{1}{2}x (\frac{1}{2} ), \\ y(0)=\frac{h(y)}{2}, \qquad y(1)=\frac{1}{3}y (\frac{1}{3} ). \end{array}\displaystyle \right . $$
(19)
The solution of the BVP (19) is given by
$$\begin{aligned}& x(t)=\frac{g(x)}{2} \biggl(1-\frac{2t}{3} \biggr)+ \int_{0}^{1} G_{\alpha}(t,s)\phi \bigl(s,x(s),y(s)\bigr)\,ds, \\& y(t)=\frac{h(y)}{2} \biggl(1-\frac{3t}{4} \biggr)+ \int_{0}^{1} G_{\beta}(t,s)\psi \bigl(s,x(s),y(s)\bigr)\,ds, \end{aligned}$$
where \(G_{\alpha}\), \(G_{\beta}\) are the Green’s functions and can be obtained easily as obtained generally in (4) and (8), respectively. From the system (19) we take \(\alpha=\beta=\frac {3}{2}\), \(\delta=\eta=\frac{1}{2}\), \(\gamma=\xi=\frac{1}{3}\), and \(r=2\in (1,3)\), and let us take \(\lambda=\frac{1}{2}\in[0,1]\). Then by the use of Theorem 3.2, we have \(L_{\phi}=L_{\psi}=\frac{1}{50}\), \(M_{\phi}=M_{\psi}=\frac {1}{50}=c_{i}=d_{i}\), for \(i=1,2\), and taking \(K_{g}=\frac{1}{2}\), \(K_{h}=\frac{1}{2}\), then assumptions (C1)-(C4) are satisfied. We have
$$\begin{aligned}& F_{1}x(t)=\frac{g(x)}{2} \biggl(1-\frac{2t}{3} \biggr), \qquad G_{1}x(t)= \int_{0}^{1} G_{\alpha}(t,s)\phi \bigl(s,x(s),y(s)\bigr)\,ds, \\& F_{2}y(t)=\frac{h(y)}{2} \biggl(1-\frac{3t}{4} \biggr), \qquad G_{2}y(t)= \int_{0}^{1} G_{\beta}(t,s)\psi \bigl(s,x(s),y(s)\bigr)\,ds. \end{aligned}$$
Since \(F_{1}\), \(F_{2}\), \(G_{1}\), \(G_{2}\) are continuous and bounded, also \(F=(F_{1},F_{2})\), \(G=(G_{1},G_{2})\), which further implies that \(T=F+G\) is continuous and bounded. Further
$$ \bigl\lVert F(x,y)-F(u,v)\bigr\rVert \leq\frac{1}{2}\bigl\lVert (x,y)-(u,v) \bigr\rVert , $$
that is, if F is μ-Lipschitz with Lipschitz constant \(\frac {1}{2}\) and G is μ-Lipschitz with zero constant, this implies that T is a strict-μ-contraction with constant \(\frac{1}{2}\). Further it is easy to calculate \(\theta =1.5045\). As
$$ \mathbb{B}=\biggl\{ (x,y)\in C(I\times\mathbb{R}\times\mathbb{R},\mathbb {R}), \exists\lambda\in[0,1] : (x,y)=\frac{1}{2}T(x,y)\biggr\} . $$
Then the solution
$$ \bigl\lVert (x,y) \bigr\rVert \leq\frac{1}{2} \bigl\lVert T(x,y) \bigr\rVert \leq1, $$
implies that \(\mathbb{B}\) is bounded and by Theorem 3.1 the BVP (19) has at least a solution \((x,y)\) in \(C(I\times\mathbb{R}\times\mathbb {R},\mathbb{R})\). Furthermore \(K+\theta(L_{\phi}+L_{\psi})=0.56018<1\). Hence by Theorem 3.2 the boundary value problem (19) has a unique solution.

Declarations

Acknowledgements

We are thankful to the reviewver for his/her valuable suggestions, which improved the manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan

References

  1. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010) View ArticleMathSciNetMATHGoogle Scholar
  2. Agarwal, RP, Belmekki, M, Benchohra, M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009, Article ID 981728 (2009) MathSciNetGoogle Scholar
  3. Balachandran, K, Kiruthika, S, Trujillo, JJ: Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 1970-1977 (2011) View ArticleMathSciNetMATHGoogle Scholar
  4. Benchohra, M, Graef, JR, Hamani, S: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87, 851-863 (2008) View ArticleMathSciNetMATHGoogle Scholar
  5. El-Sayed, AMA, Bin-Taher, EO: Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order. Electron. J. Differ. Equ. 2013, 64 (2013) View ArticleMathSciNetGoogle Scholar
  6. El-Shahed, M, Nieto, JJ: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 59(11), 3438-3443 (2010) View ArticleMathSciNetMATHGoogle Scholar
  7. Han, X, Wang, T: The existence of solutions for a nonlinear fractional multi-point boundary value problem at resonance. Int. J. Differ. Equ. 2011, Article ID 401803 (2011) Google Scholar
  8. Khan, RA, Shah, K: Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Commun. Appl. Anal. 19, 515-526 (2015) Google Scholar
  9. Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363-1375 (2010) View ArticleMathSciNetMATHGoogle Scholar
  10. Lv, L, Wang, J, Wei, W: Existence and uniqueness results for fractional differential equations with boundary value conditions. Opusc. Math. 31, 629-643 (2011) View ArticleMathSciNetMATHGoogle Scholar
  11. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  12. Rehman, M, Khan, RA: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 23(9), 1038-1044 (2010) View ArticleMathSciNetMATHGoogle Scholar
  13. Yang, W: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Comput. Math. Appl. 63, 288-297 (2012) View ArticleMathSciNetMATHGoogle Scholar
  14. Caputo, M: Linear models of dissipation whose Q is almost frequency independent. Geophys. J. R. Astron. Soc. 13(5), 529-539 (1967) View ArticleGoogle Scholar
  15. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) View ArticleMATHGoogle Scholar
  16. Kilbas, AA, Marichev, OI, Samko, SG: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993) MATHGoogle Scholar
  17. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
  18. Lakshmikantham, V, Leela, S, Vasundhara, J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009) MATHGoogle Scholar
  19. Podlubny, I: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999) Google Scholar
  20. Cai, L, Wu, J: Analysis of an HIV/AIDS treatment model with a nonlinear incidence rate. Chaos Solitons Fractals 41(1), 175-182 (2009) View ArticleMathSciNetMATHGoogle Scholar
  21. Ahmad, B, Nieto, JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838-1843 (2009) View ArticleMathSciNetMATHGoogle Scholar
  22. Su, X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64-69 (2009) View ArticleMathSciNetMATHGoogle Scholar
  23. Shah, K, Khan, RA: Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti-periodic boundary conditions. Differ. Equ. Appl. 7(2), 245-262 (2015) MathSciNetGoogle Scholar
  24. Zeidler, E: Nonlinear Functional Analysis an Its Applications. I: Fixed Point Theorems. Springer, New York (1986) View ArticleGoogle Scholar
  25. Ahmad, B, Nieto, JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295-304 (2010) MathSciNetMATHGoogle Scholar
  26. Chen, T, Liu, W, Hu, Z: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210-3217 (2012) View ArticleMathSciNetMATHGoogle Scholar
  27. Wang, X, Wang, L, Zeng, Q: Fractional differential equations with integral boundary conditions. J. Nonlinear Sci. Appl. 8, 309-314 (2015) MathSciNetGoogle Scholar
  28. Wang, J, Zhou, Y, Wei, W: Study in fractional differential equations by means of topological degree methods. Numer. Funct. Anal. Optim. 33(2), 216-238 (2012) View ArticleMathSciNetMATHGoogle Scholar
  29. Yang, A, Ge, W: Positive solutions of multi-point boundary value problems of nonlinear fractional differential equation at resonance. J. Korean Soc. Math. Educ., Ser. B Pure Appl. Math. 16, 181-193 (2009) MathSciNetGoogle Scholar
  30. Shah, K, Khalil, H, Khan, RA: Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos Solitons Fractals 77, 240-246 (2015) View ArticleMathSciNetGoogle Scholar
  31. Yao, ZJ: New results of positive solutions for second-order nonlinear three-point integral boundary value problems. J. Nonlinear Sci. Appl. 8, 93-98 (2015) MathSciNetGoogle Scholar
  32. Nanware, A, Dhaigude, DB: Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions. J. Nonlinear Sci. Appl. 7, 246-254 (2014) MathSciNetMATHGoogle Scholar
  33. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003) View ArticleMATHGoogle Scholar
  34. Isaia, F: On a nonlinear integral equation without compactness. Acta Math. Univ. Comen. 75, 233-240 (2006) MathSciNetMATHGoogle Scholar
  35. Deimling, K: Nonlinear Functional Analysis. Springer, New York (1985) View ArticleMATHGoogle Scholar

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