Open Access

Existence results for fractional-order differential equations with nonlocal multi-point-strip conditions involving Caputo derivative

Advances in Difference Equations20152015:348

Received: 10 October 2015

Accepted: 2 November 2015

Published: 11 November 2015


In this paper, we investigate the existence and uniqueness of solutions for a differential equation of fractional-order \(q \in(1, 2]\) subject to nonlocal boundary conditions involving Caputo derivative of the form
$$x(0)=\delta x(\sigma),\qquad a {}^{c}D^{\mu} x( \varrho_{1})+b {}^{c}D^{\mu} x(\varrho_{2})=c \int_{\beta_{1}}^{\beta_{2}} {}^{c}D^{\mu} x(s) \,ds, $$
\(0 < \varrho_{1} < \sigma< \beta_{1} < \beta_{2} < \varrho_{2} <1\), \(0<\mu<1\), and δ, a, b, c are real constants. We make use of some standard tools of fixed point theory to obtain the desired results which are well illustrated with the aid of examples.


fractional order derivative nonlocal conditions strip existence fixed point


34A08 34B15

1 Introduction

The study of fractional-order differential equations supplemented with a variety of initial and boundary conditions, such as classical, nonlocal, multi-point, periodic/anti-periodic, and integral boundary conditions, has attracted significant attention in recent years. In consequence, the literature on the topic is now much enriched and covers theoretical aspects as well as analytic/numerical methods for solving fractional-order initial and boundary value problems. The widespread applications of fractional calculus modeling techniques in several disciplines of applied and technical sciences have played a key role in the popularity of the subject. Examples include viscoelasticity, control theory, biological sciences, ecology, aerodynamics, electro-dynamics of complex medium, environmental issues, etc. An important and useful feature characterizing fractional-order differential and integral operators (in contrast to integer-order operators) is their nonlocal nature that accounts for the past and hereditary behavior of materials and processes involved in the real world problems. For examples and details, we refer the reader to the works [15].

Nonlocal conditions, introduced by Bitsadze and Samarskii [6], are regarded as more plausible than the classical initial/boundary conditions in view of their ability to describe certain peculiarities of chemical, physical or other processes happening inside the domain. Computational fluid dynamics (CFD) studies of blood flow indicate that it is not always possible to assume circular cross-section of blood arteries. Several approaches have been proposed to resolve this issue. However, the idea of introducing integral boundary conditions [7] is found to be quite a productive one. Also, integral boundary conditions are applied to regularize ill-posed parabolic backward problems in time partial differential equations, see, for example, mathematical models for bacterial self-regularization [8]. Some recent results on fractional-order boundary value problems involving nonlocal and integral boundary conditions can be found in [920] and the references cited therein.

In this paper, motivated by the utility of nonlocal integral boundary conditions in several diverse disciplines, we propose a new class of Caputo type nonlocal boundary value problems supplemented with integral boundary conditions. In precise terms, we consider the following problem:
$$\begin{aligned}& {}^{c}D^{q}x(t)=f(t,x),\quad 1< q\leq2, t\in[0,1], \end{aligned}$$
$$\begin{aligned}& x(0)=\delta x(\sigma),\qquad a {}^{c}D^{\mu} x(\varrho_{1})+b {}^{c}D^{\mu} x(\varrho_{2})=c \int_{\beta_{1}}^{\beta_{2}} {}^{c}D^{\mu}x(s) \,ds, \end{aligned}$$
where \(f: [0, 1] \times\mathbb{R} \to\mathbb{R}\) is a given continuous function, \(0<\varrho_{1}< \sigma< \beta_{1} < \beta_{2} < \varrho_{2} < 1\), \(0<\mu<1\), and δ, a, b, c are real constants. The integral boundary condition in (1.2) can be interpreted as the linear combination of the values of Caputo derivative of the unknown function of order \(\mu\in(0, 1)\) at nonlocal positions \(\varrho_{1}\) and \(\varrho_{2}\) (off the strip) is proportional to the strip contribution of the Caputo derivative of the unknown function, occupying the position \((\beta_{1}, \beta_{2})\).

The content of the paper is organized as follows. Section 2 is devoted to some basic concepts and a lemma concerning the unique solution of a linear variant of problem (1.1)-(1.2). In Section 3, we present our main results which are obtained via Krasnoselskii’s fixed point theorem, Schauder type fixed point theorem, nonlinear alternative for single-valued maps and Banach’s theorem. It is imperative to note that the exposition of indicated tools of fixed point theory is new in the context of problem (1.1)-(1.2). Finally, we discuss some examples for illustration of the main results.

2 Preliminaries

First of all, we recall some basic definitions.

Definition 2.1


For at least n-times absolutely continuously differentiable function \(h : [0,\infty) \to \mathbb{R}\), the Caputo derivative of fractional order q is defined as
$${}^{c}D^{q} h(t)=\frac{1}{\Gamma(n-q)} \int_{0}^{t}(t-s)^{n-q-1}h^{(n)}(s) \,ds,\quad n-1 < r < n, n=[q]+1, $$
where \([q]\) denotes the integer part of the real number q.

Definition 2.2


The Riemann-Liouville fractional integral of order q for a continuous function h is defined as
$$I^{q} h(t)=\frac{1}{\Gamma(q)} \int_{0}^{t} \frac{h(s)}{(t-s)^{1-q}}\,ds,\quad q>0, $$
provided the integral exists.

Next we present an auxiliary lemma to define the solution for problem (1.1)-(1.2).

Lemma 2.3

Let \(y \in L[0,1]\) and \(x \in AC^{2}[0, 1]\). Then the solution of the linear fractional differential equation
$$ {}^{c}D^{q}x(t)=y(t),\quad 1< q\leq2, t\in[0,1], $$
supplemented with boundary conditions (1.2) is equivalent to the integral equation
$$\begin{aligned} x(t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}y(s)\,ds + \frac{\delta}{1-\delta} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma (q)}y(s)\,ds \\ &{}+ \biggl(\frac{\delta\sigma}{\mathcal{A}(1-\delta)}+\frac {t}{\mathcal {A}} \biggr) \biggl(-a \int_{0}^{\varrho_{1}}\frac{(\varrho_{1}-s)^{q-\mu-1}}{\Gamma (q-\mu)} y(s)\,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} y(s)\,ds+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s} \frac{(s-u)^{q-\mu-1}}{\Gamma(q-\mu)}y(u)\,du \,ds \biggr), \end{aligned}$$
$$ \mathcal{A}=a\frac{\varrho_{1} ^{1-\mu}}{\Gamma(2-\mu)} +b\frac{\varrho_{2} ^{1-\mu}}{\Gamma(2-\mu)} - c \frac{({\beta_{2}}^{2-\mu}-\beta_{1}^{2-\mu})}{\Gamma(3-\mu)}\neq0. $$


It is well known that the general solution of the fractional differential equation (2.1) can be written as
$$ x(t)= \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}y(s) \,ds+c_{0}+c_{1}t, $$
where \(c_{0},c_{1} \in\mathbb{R}\) are arbitrary constants. Applying the boundary conditions (1.2) and using (2.3), we find that
$$\begin{aligned} c_{0} =&\frac{\delta}{1-\delta} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma(q)}y(s)\,ds +\frac{\delta\sigma}{\mathcal{A}(1-\delta)} \biggl(-a \int _{0}^{\varrho _{1}}\frac{(\varrho_{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} y(s)\,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} y(s)\,ds+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s} \frac{(s-u)^{q-\mu-1}}{\Gamma(q-\mu)}y(u)\,du \,ds \biggr) \end{aligned}$$
$$ \begin{aligned}[b] c_{1}={}&\frac{1}{\mathcal{A}} \biggl(-a \int_{0}^{\varrho_{1}}\frac {(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} y(s)\,ds\\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} y(s)\,ds+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s} \frac{(s-u)^{q-\mu-1}}{\Gamma(q-\mu)}y(u)\,du \,ds \biggr). \end{aligned} $$

Substituting the values of \(c_{0}\), \(c_{1}\) in (2.4), we get (2.2). This completes the proof. □

3 Main results

In relation to problem (1.1)-(1.2), we define an operator \(\mathcal{S}: \mathcal{C}\longrightarrow\mathcal{C} \) via Lemma 2.3 as follows:
$$\begin{aligned} (\mathcal{S}x) (t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}f\bigl(s,x(s)\bigr)\,ds +\frac{\delta}{1-\delta} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma (q)}f\bigl(s,x(s)\bigr) \,ds \\ &{}+ \biggl(\frac{\delta\sigma}{\mathcal{A}(1-\delta)}+\frac {t}{\mathcal {A}} \biggr) \biggl(-a \int_{0}^{\varrho_{1}}\frac{(\varrho_{1}-s)^{q-\mu -1}}{\Gamma (q-\mu)} f\bigl(s,x(s)\bigr) \,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} f\bigl(s,x(s)\bigr) \,ds \\ &{}+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu-1}}{\Gamma(q-\mu)}f\bigl(u,x(u)\bigr) \,du \,ds \biggr), \end{aligned}$$
where \(\mathcal{C}= C([0,1], \mathbb{R}) \) denote the Banach space of all continuous functions from [0,1] to \(\mathbb{R}\) endowed with the norm: \(\|x\|=\sup\{|x(t)|,t\in[0,1]\}\) and \(\mathcal{A}\) is given by (2.3). Observe that problem (1.1)-(1.2) has solutions if and only if the operator \(\mathcal{S}\) has fixed points. For computational convenience, we set
$$\begin{aligned}& \begin{aligned}[b] \vartheta_{1}={}&\frac{1}{\Gamma(q+1)}+ \frac{|\delta|\sigma ^{q}}{|1-\delta |\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &{} \times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c| \frac{({\beta_{2}}^{q-\mu+1}-{\beta _{1}}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr), \end{aligned} \end{aligned}$$
$$\begin{aligned}& \vartheta_{2}=\vartheta_{1}-\frac{1}{\Gamma(q+1)}. \end{aligned}$$

Now we state the known fixed point results which we need in the forthcoming analysis.

Lemma 3.1

(Krasnoselskii [21])

Let \(\mathcal{Q}\) be a closed, convex, bounded and nonempty subset of a Banach space Y. Let \(\mathcal{\phi}_{1}\), \(\mathcal{\phi}_{2}\) be operators such that (a) \(\mathcal{\phi}_{1}\nu_{1}+\mathcal{\phi}_{2}\nu_{2} \in\mathcal{Q}\) whenever \(\nu_{1},\nu_{2} \in\mathcal{Q}\); (b) \(\mathcal{\phi}_{1}\) is compact and continuous; and (c) \(\mathcal{\phi}_{2}\) is a contraction mapping. Then there exists \(\nu\in\mathcal{Q}\) such that \(\nu=\mathcal{\phi}_{1}\nu+\mathcal{\phi}_{2}\nu\).

Lemma 3.2


Let X be a Banach space. Assume that \(T:X\longrightarrow X\) is a completely continuous operator and the set \(V=\{u\in X|u=\epsilon Tu,0<\epsilon<1\} \) is bounded. Then T has a fixed point in X.

Lemma 3.3

(Nonlinear alternative for single-valued maps [22])

Let E be a Banach space, \(E_{1} \) be a closed, convex subset of E, V be an open subset of \(E_{1} \), and \(0\in V \). Suppose that \(\mathcal{U}:\overline{V}\longrightarrow E_{1} \) is a continuous, compact (that is, \(\mathcal{U}(\overline{V})\) is a relatively compact subset of \(E_{1} \)) map. Then either
  1. (i)

    \(\mathcal{U}\) has a fixed point in , or

  2. (ii)

    there is \(x\in\partial V \) (the boundary of V in \(E_{1} \)) and \(\kappa\in(0,1) \) with \(x=\kappa\) \(\mathcal{U} (x) \).


Our first existence result is based on Lemma 3.1.

Theorem 3.4

Let \(f:[0,1] \times\mathbb{R} \rightarrow\mathbb{R} \) be a continuous function satisfying the following conditions:

\(|f(t,x) -f(t,y)|\leq \ell|x-y|\), \(\ell >0\), \(\forall t\in[0,1]\), \(x,y\in\mathbb{R}\);


\(|f(t,x)|\leq \omega(t) \), \(\forall(t,x)\in[0,1] \times\mathbb{R}\), \(\omega\in C([0,1], \mathbb{R^{+}})\).

Then problem (1.1)-(1.2) has at least one solution on \([0,1] \) if \(\ell\vartheta_{2} < 1 \), where \(\vartheta_{2}\) is given by (3.3).


Let us introduce a set \(B_{\rho}=\{x\in\mathcal{C}:\|x\|\leq\rho\}\) with \(\rho\geq \|\omega\| \vartheta_{1}\), where \(\|\omega\|=\sup\{|\omega(t)|,t\in[0,1]\}\) and \(\vartheta_{1}\) is given by (3.2). Define the operators \(\mathcal{S}_{1} \) and \(\mathcal{S}_{2} \) on \(B_{\rho}\) as
$$\begin{aligned} (\mathcal{S}_{1}x) (t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma (q)}f\bigl(s,x(s)\bigr) \,ds , \\ (\mathcal{S}_{2}x) (t) =&\frac{\delta}{1-\delta} \int_{0}^{\sigma}\frac {(\sigma -s)^{q-1}}{\Gamma(q)}f\bigl(s,x(s)\bigr) \,ds \\ &{}+ \biggl(\frac{\delta\sigma}{\mathcal{A}(1-\delta)}+\frac {t}{\mathcal {A}} \biggr) \biggl(-a \int_{0}^{\varrho_{1}}\frac{(\varrho_{1}-s)^{q-\mu -1}}{\Gamma (q-\mu)} f\bigl(s,x(s)\bigr) \,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} f\bigl(s,x(s)\bigr) \,ds+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu-1}}{\Gamma(q-\mu)}f\bigl(u,x(u)\bigr) \,du \,ds \biggr). \end{aligned}$$
For \(x,y \in B_{\rho}\), it is easy to show that \(\|(\mathcal{S}_{1}x) +(\mathcal{S}_{2}y) \| \leq\|\omega\| \vartheta _{1} \leq\rho\), which implies that \(\mathcal{S}_{1} x +\mathcal{S}_{2} y \in B_{\rho}\). Applying the condition (\(\mathcal{H}_{1}\)), we find that
$$\begin{aligned} &\bigl\| (\mathcal{S}_{2} x)-(\mathcal{S}_{2} y)\bigr\| \\ &\quad\leq \sup_{t\in[0,1]} \biggl\{ \frac{|\delta|}{|1-\delta|} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma(q)}\bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &\qquad{}+ \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {t}{|\mathcal{A}|} \biggr) \biggl(|a| \int_{0}^{\varrho_{1}}\frac{(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &\qquad{}+|b| \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)}\bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &\qquad{}+|c| \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu -1}}{\Gamma (q-\mu)}\bigl|f\bigl(u,x(u)\bigr)-f \bigl(u,y(u)\bigr)\bigr|\,du \,ds \biggr) \biggr\} \\ &\quad\leq\ell\sup_{t\in[0,1]} \biggl\{ \frac{|\delta|\sigma ^{q}}{|1-\delta|\Gamma (q+1)}+ \biggl( \frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|} +\frac{t}{|\mathcal{A}|} \biggr) \biggl(|a|\frac{\varrho_{1}^{q-\mu }}{\Gamma (q-\mu+1)} \\ &\qquad{}+|b|\frac{\varrho_{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c|\frac{(\beta _{2}^{q-\mu +1}-\beta_{1}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr) \biggr\} \|x-y\| \\ &\quad\leq\ell\vartheta_{2} \|x-y\|, \end{aligned}$$
which, in view of the condition \(\ell\vartheta_{2}<1\), implies that the operator \(\mathcal{S}_{2} \) is a contraction. Further \(\mathcal{S}_{1}\) is continuous in view of the continuity of f. Also, \(\mathcal{S}_{1} \) is uniformly bounded on \(B_{\rho}\) as
$$\begin{aligned} \|\mathcal{S}_{1} x\| \leq& \sup_{t\in[0,1]} \biggl\{ \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}\bigl|f\bigl(s,x(s)\bigr)\bigr| \,ds \biggr\} \le \sup_{t\in[0,1]} \biggl\{ \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}\omega(s)\,ds \biggr\} \\ \leq& \|\omega\|\sup_{t\in[0,1]} \biggl\{ \frac{t^{q}}{\Gamma (q+1)} \biggr\} \le\frac{\|\omega\|}{\Gamma(q+1)}. \end{aligned}$$
With \(\sup_{(t,x) \in[0,1]\times B_{\rho}} |f(t,x)|=f_{m} <\infty\) and \(0< t_{1} < t_{2}<T\), we have
$$\begin{aligned} &\bigl|(\mathcal{S}_{1} x) (t_{2})-(\mathcal{S}_{1} x) (t_{1})\bigr| \\ &\quad= \biggl| \int _{0}^{t_{2}} \frac{(t_{2}-s)^{q-1}}{\Gamma(q)} f\bigl(s,x(s) \bigr)\,ds- \int_{0}^{t_{1}} \frac{(t_{1}-s)^{q-1}}{\Gamma (q)}f\bigl(s,x(s)\bigr) \,ds \biggr| \\ &\quad\leq \int_{0}^{t_{1}} \frac{[(t_{2}-s)^{q-1}-(t_{1}-s)^{q-1}]}{\Gamma(q)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds \\ &\qquad{}+ \int_{t_{1}}^{t_{2}} \frac{(t_{2}-s)^{q-1}}{\Gamma(q)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds \leq\frac{f_{m}}{\Gamma(q+1)}\bigl[2|t_{2}-t_{1}|^{q} +\bigl|t_{2}^{q} -t_{1}^{q}\bigr|\bigr], \end{aligned}$$
which tends to zero independent of x as \(t_{2}-t_{1} \longrightarrow0\). This shows that \(\mathcal{S}_{1}\) is relatively compact on \(B_{\rho}\). Hence, by the Arzelá-Ascoli theorem, \(\mathcal{S}_{1} \) is compact on \(B_{\rho}\). Thus all the assumptions of Lemma 3.1 are satisfied. So problem (1.1)-(1.2) has at least one solution on \([0,1]\). This completes the proof. □

In the next result, we make use of Lemma 3.2.

Theorem 3.5

Assume that there exists a positive constant \(L_{1}\) such that \(|f(t,x)|\leq L_{1} \) for all \(t\in[0,1] \), \(x\in\mathcal{C} \). Then there exists at least one solution for problem (1.1)-(1.2).


In the first step, it will be shown that the operator \(\mathcal{S} \) is completely continuous. Clearly the continuity of \(\mathcal{S}\) follows from the continuity of f. Let \(\mathcal{V}\subset C([0,1], \mathbb{R^{+}})\) be bounded. Then \(\forall x\in\mathcal{V}\), it is easy to establish that \(|(\mathcal{S}x)(t)|\leq L_{1}\vartheta_{1} =L_{2} \). Furthermore, we find that
$$\begin{aligned} \bigl|(\mathcal{S}x)'(t)\bigr| =& \biggl| \int_{0}^{t}\frac{(t-s)^{q-2}}{\Gamma (q-1)}f\bigl(s,x(s)\bigr) \,ds \\ &{}+\frac{1}{\mathcal{A}} \biggl(-a \int_{0}^{\varrho_{1}}\frac{(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} f\bigl(s,x(s)\bigr) \,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} f\bigl(s,x(s)\bigr) \,ds\\ &{}+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu-1}}{\Gamma (q-\mu )}f\bigl(u,x(u)\bigr) \,du \,ds \biggr) \biggr| \\ \leq& L_{1} \biggl(\frac{1}{\Gamma(q)}+\frac{1}{|\mathcal{A}|} \biggl[|a| \frac {\varrho _{1}^{q-\mu}}{\Gamma(q-\mu+1)} \\ &{}+|b|\frac{\varrho_{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c|\frac{(\beta _{2}^{q-\mu +1}-\beta_{1}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr] \biggr)=L_{3}. \end{aligned}$$
Hence, for \(t_{1},t_{2}\in[0,1] \), it follows that
$$\bigl|(\mathcal{S}x) (t_{1})-(\mathcal{S}x) (t_{2})\bigr|\leq \int _{t_{1}}^{t_{2}}\bigl|(\mathcal {S}x)'(s)\bigr|\,ds \leq L_{3}(t_{2}-t_{1}). $$
Therefore, \(\mathcal{S}\) is equicontinuous on \([0,1]\). Thus, by the Arzelá-Ascoli theorem, the operator \(\mathcal{S}\) is completely continuous.
Next, we consider the set \(\mathcal{U}=\{x\in\mathcal{C}:\xi\mathcal{S}x,0<\xi<1\}\) and show that \(\mathcal{U}\) is bounded. Then, for \(t\in[0,1]\), we have
$$\begin{aligned} x(t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}f\bigl(s,x(s)\bigr)\,ds +\frac{\delta}{1-\delta} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma (q)}f\bigl(s,x(s)\bigr) \,ds \\ &{}+ \biggl(\frac{\delta\sigma}{\mathcal{A}(1-\delta)}+\frac {t}{\mathcal {A}} \biggr) \biggl(-a \int_{0}^{\varrho_{1}}\frac{(\varrho_{1}-s)^{q-\mu -1}}{\Gamma (q-\mu)} f\bigl(s,x(s)\bigr) \,ds \\ &{}-b \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} f\bigl(s,x(s)\bigr) \,ds+c \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu-1}}{\Gamma (q-\mu )}f\bigl(u,x(u)\bigr) \,du \,ds \biggr) \end{aligned}$$
and \(|x(t)|=\xi| (\mathcal{S}x)(t)|\leq L_{1}\vartheta_{1}=L_{2} \). In consequence, we get \(\|x\|\leq L_{2}\), \(\forall x\in\mathcal{U}\), \(t\in[0,1]\). So \(\mathcal{U}\) is bounded. Thus, by the application of Lemma 3.2, problem (1.1)-(1.2) has at least one solution. This completes the proof. □

The following theorem deals with the uniqueness of solutions for problem (1.1)-(1.2).

Theorem 3.6

Let \(f:[0,1]\times\mathbb {R}\longrightarrow \mathbb{R}\) be a continuous function satisfying the condition (\({\mathcal{H}}_{1}\)) and that \(\ell\vartheta_{1}<1\), where \(\vartheta_{1}\) is given by (3.2). Then there exists a unique solution for problem (1.1)-(1.2) on \([0, 1]\).


In the first step, we consider the operator \(\mathcal {S}: \mathcal{C} \longrightarrow\mathcal{C}\) defined by (3.1) and show that \(\mathcal{S}\mathcal{E}_{r}\subset\mathcal{E}_{r}\), where \(\mathcal{E}_{r}=\{x\in\mathcal{C}:\|x\|\leq r\} \) with \(\sup_{t\in[0,1]} |f(t,0)|=\alpha\) and \(r>\frac{\vartheta_{1}\alpha}{1-\vartheta_{1}\ell}\) for \(x\in \mathcal{E}_{r}\), \(t\in[0,1]\). Using the fact that \(|f(s,x(s))|=|f(s,x(s))-f(s,0)+f(s,0)| \leq\ell r+\alpha\), we get
$$\begin{aligned} \bigl\| (\mathcal{S}x)\bigr\| \leq& \sup_{t\in[0,1]} \biggl\{ \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}\bigl|f\bigl(s,x(s)\bigr)\bigr| \,ds +\frac{|\delta|}{|1-\delta|} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma (q)}\bigl|f\bigl(s,x(s)\bigr)\bigr| \,ds \\ &{}+ \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {t}{|\mathcal{A}|} \biggr) \biggl(|a| \int_{0}^{\varrho_{1}}\frac{(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds \\ &{}+|b| \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds \\ &{}+|c| \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu -1}}{\Gamma (q-\mu)}\bigl|f\bigl(u,x(u)\bigr)\bigr| \,du \,ds \biggr) \biggr\} \\ \leq& (\ell r+\alpha)\vartheta_{1} \leq r. \end{aligned}$$
This shows that \(\mathcal{S}\mathcal{E}_{r} \subset\mathcal{E}_{r} \), where we have used (3.2).
Now, for \(x,y\in\mathcal{C}\) and for each \(t\in[0,1]\), we obtain
$$\begin{aligned} \|\mathcal{S}x - \mathcal{S}y\| \leq& \sup_{t\in[0,1]} \biggl\{ \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma (q)}\bigl|f\bigl(s,x(s)\bigr)-f \bigl(s,y(s)\bigr)\bigr|\,ds \\ &{}+\frac{|\delta|}{|1-\delta|} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma(q)}\bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &{}+ \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {t}{|\mathcal{A}|} \biggr) \biggl(|a| \int_{0}^{\varrho_{1}}\frac{(\varrho_{1}-s)^{q-\mu-1}}{\Gamma (q-\mu)} \bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &{}+|b| \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)-f\bigl(s,y(s)\bigr)\bigr|\,ds \\ &{}+|c| \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu -1}}{\Gamma (q-\mu)}\bigl|f\bigl(u,x(u)\bigr)-f \bigl(u,y(u)\bigr)\bigr|\,du \,ds \biggr) \biggr\} \\ \leq&\ell\vartheta_{1} \|x-y\|, \end{aligned}$$
which, by the assumption \(\ell\vartheta_{1} < 1\), implies that the operator \(\mathcal{S} \) is a contraction. Thus, Banach’s contraction mapping principle applies and there exists a unique solution for problem (1.1)-(1.2). This completes the proof. □

Our final result relies on Lemma 3.3 (nonlinear alternative for single-valued maps).

Theorem 3.7

Let \(f:[0,1]\times\mathbb{R}\longrightarrow\mathbb{R} \) be a continuous function, and assume that

there exist a function \(p\in\mathcal{C} ([0,1], \mathbb{R^{+}})\) and a nondecreasing function \(\varphi:\mathbb{R^{+}}\longrightarrow\mathbb{R^{+}}\) such that \(|f(t,x)|\leq p(t)\varphi(\|x\|) \), \(\forall(t,x) \in[0,1]\times\mathbb{R}\);

there exists a constant \(M>0 \) such that
$$\begin{aligned} &M \biggl[\varphi(M)\|p\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{|\delta |\sigma ^{q}}{|1-\delta|\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &\quad{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c| \frac{({\beta_{2}}^{q-\mu+1} -{\beta_{1}}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr) \biggr\} \biggr]^{-1}>1. \end{aligned}$$
Then problem (1.1)-(1.2) has at least one solution on \([0,1]\).


We complete the proof in several steps. As a first step, we show that the operator \(\mathcal{S}:\mathcal{C}\longrightarrow\mathcal{C} \) defined by (3.1) maps bounded sets into bounded sets in \(\mathcal{C}\). For a positive number ν, let \(\mathbf{B}_{\nu}=\{x\in\mathcal{C}:\|x\|\leq\nu\} \) be a bounded set in \(\mathcal{C}\). Then, for \(x\in\mathbf{B}_{\nu}\), using (\({\mathcal{H}}_{3}\)) we obtain
$$\begin{aligned} \bigl|(\mathcal{S}x) (t)\bigr| \leq& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma (q)}p(s)\varphi\bigl(\|x\|\bigr)\,ds +\frac{|\delta|}{|1-\delta|} \int_{0}^{\sigma}\frac{(\sigma -s)^{q-1}}{\Gamma (q)}p(s)\varphi\bigl(\|x\|\bigr)\,ds \\ &{}+ \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {t}{|\mathcal{A}|} \biggr) \biggl(|a| \int_{0}^{\varrho_{1}}\frac{(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} p(s)\varphi\bigl(\|x\|\bigr) \,ds \\ &{}+ |b| \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)}p(s)\varphi\bigl(\|x\|\bigr) \,ds \\ &{}+|c| \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu -1}}{\Gamma (q-\mu)}p(u)\varphi\bigl(\|x\|\bigr)\,du \,ds \biggr) \\ \leq&\varphi(\nu)\|p\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{|\delta |\sigma ^{q}}{|1-\delta|\Gamma(q+1)} + \biggl( \frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c| \frac{({\beta_{2}}^{q-\mu+1}-\beta _{1}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr) \biggr\} . \end{aligned}$$
Next, it will be shown that the operator \(\mathcal{S} \) maps bounded sets into equicontinuous sets of \(\mathcal{C}\). Let \(t_{1}, t_{2}\in[0,1] \) with \(t_{1}< t_{2}\). Then, for \(x\in\mathbf{B}_{\nu}\), we have
$$\begin{aligned} &\bigl|(\mathcal{S}x) (t_{2})-(\mathcal{S}x) (t_{1})\bigr| \\ &\quad\leq \biggl| \int _{0}^{t_{1}}\frac{[(t_{2}-s)^{q-1}-(t_{1}-s)^{q-1}]}{\Gamma(q)}f\bigl(s,x(s)\bigr) \,ds + \int_{t_{1}}^{t_{2}}\frac{(t_{2}-s)^{q-1}}{\Gamma(q)}f\bigl(s,x(s)\bigr) \biggr| \\ &\qquad{}+\frac{|t_{2}-t_{1}|}{|A|} \biggl(|a| \int_{0}^{\varrho_{1}}\frac{(\varrho _{1}-s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds \\ &\qquad{}+|b| \int_{0}^{\varrho_{2}}\frac{(\varrho_{2} -s)^{q-\mu-1}}{\Gamma(q-\mu)} \bigl|f\bigl(s,x(s) \bigr)\bigr|\,ds+|c| \int_{\beta_{1}}^{\beta_{2}} \int_{0}^{s}\frac{(s-u)^{q-\mu -1}}{\Gamma (q-\mu)}\bigl|f\bigl(u,x(u)\bigr)\bigr| \,du \,ds \biggr) \\ &\quad\leq\varphi(\nu) \|p\| \biggl[\frac{[2|t_{2}-t_{1}|^{q} +|t_{2}^{q}-t_{1}^{q}|]}{\Gamma(q+1)}+\frac {|t_{2}-t_{1}|}{|\mathcal{A}|} \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma (q-\mu +1)} \\ &\qquad{}+|b|\frac{\varrho_{2}^{q-\mu}}{\Gamma(q-\mu+1)}+|c|\frac{({\beta _{2}}^{q-\mu+1}-{\beta_{1}}^{q-\mu+1})}{\Gamma(q-\mu+2)} \biggr) \biggr]. \end{aligned}$$
Clearly, the right-hand side tends to zero independent of \(x\in\mathbf{B}_{\nu}\) as \(t_{2}\longrightarrow t_{1}\). Thus, by the Arzelá theorem, the operator \(\mathcal{S} \) is completely continuous.
Let x be a solution of problem (1.1)-(1.2). Then, for \(\lambda\in(0,1)\), using the method of computation employed to show the boundedness of the operator \(\mathcal{S}\), it can be found that
$$\begin{aligned} \bigl|x(t)\bigr| =& \bigl|\lambda(\mathcal{S}x) (t)\bigr|\leq\varphi \bigl(\|x\|\bigr)\|p\| \biggl[ \frac {1}{\Gamma(q+1)}+\frac{|\delta|\sigma^{q}}{|1-\delta|\Gamma(q+1)} \\ &{}+ \biggl(\frac{1}{|\mathcal{A}|}+\frac{|\delta|\sigma}{|\mathcal {A}(1-\delta)|} \biggr) \biggl(|c| \frac{({\beta_{2}}^{q-\mu+1}-{\beta _{1}}^{q-\mu +1})}{\Gamma(q+1)} +\frac{(|a|\varrho_{1}^{q-\mu}+|b|\varrho_{2}^{q-\mu})}{\Gamma(q-\mu +1)} \biggr) \biggr]. \end{aligned}$$
In consequence, we get
$$\begin{aligned} &\|x\| \biggl[\varphi\bigl(\|x\|\bigr)\|p\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{|\delta |\sigma ^{q}}{|1-\delta|\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &\quad{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)} +|c| \frac{({\beta_{2}}^{q-\mu+1}-{\beta_{1}}^{q-\mu+1})}{\Gamma(q-\mu +2)} \biggr) \biggr\} \biggr]^{-1}\leq 1. \end{aligned}$$
In view of (\({\mathcal{H}}_{4}\)), there exists M such that \(\|x\|\neq M\). Let us select a set \(\mathcal{N}=\{x\in\mathcal{C}:\|x\|< M+1\}\). Observe that the operator \(\mathcal{S}:\mathcal{\overline{N}}\longrightarrow\mathcal{C}\) is continuous and completely continuous. From the choice of \(\mathcal{N}\), there is no \(x\in\partial\mathcal{N} \) such that \(x=\lambda\mathcal{S}(x)\) for some \(\lambda\in(0,1)\). Thus, by Lemma 3.3, we deduce that the operator \(\mathcal{S} \) has a fixed point \(x\in\mathcal{\overline{N}} \) which is a solution of problem (1.1)-(1.2). This completes the proof. □

Example 3.8

Consider a fractional boundary value problem given by
$$ \left \{ \textstyle\begin{array}{@{}l} {}^{c}D^{3/2}x(t)= \frac{t}{9}\sin x+\frac{e^{-t}}{9} \tan^{-1} x +\frac {1}{2},\quad t\in[0,1], \\ x(0)=\frac{1}{2} x(1/3),\qquad 2 {}^{c}D^{1/2} x(1/6)+3 {}^{c}D^{1/2} x(3/4)=\int_{3/7} ^{1/2} {}^{c}D^{1/2} x(s)\,ds. \end{array}\displaystyle \right . $$
Here, \(\delta=1/2\), \(q=3/2\), \(a=2\), \(b=3\), \(c=1\), \(\sigma=1/3\), \(\varrho_{1}=1/6\), \(\varrho_{2}=3/4\), \(\beta_{1}=3/7\), \(\beta_{2}=1/2\), \(\mu=1/2\) and \(f(t,x)=\frac{t}{9}\sin x+\frac{e^{-t}}{9} \tan^{-1}x\). With the given data, \(\ell=\frac{2}{9} \),
$$\begin{aligned}& |\mathcal{A}|=\biggl|a\frac{\varrho_{1}^{1-\mu}}{\Gamma (2-\mu)} +b\frac{\varrho_{2} ^{1-\mu}}{\Gamma(2-\mu)} -c\frac{(\beta _{2}^{2-\mu }-\beta_{1}^{2-\mu})}{\Gamma(3-\mu)}\biggr|=3.797291, \\& \begin{aligned}[b] \vartheta_{1}={}&\frac{1}{\Gamma(q+1)}+ \frac{|\delta|\sigma ^{q}}{|1-\delta |\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)} +|c| \frac{(\xi^{q-\mu+1}-\beta_{1}^{q-\mu+1})}{\Gamma(q-\mu +2)} \biggr) =1.815607. \end{aligned} \end{aligned}$$
Clearly \(\ell<\frac{1}{\vartheta_{1}}\). Thus all the conditions of Theorem 3.6 are satisfied and, consequently, there exists a unique solution for problem (3.4) on \([0,1]\).

Example 3.9

Consider problem (3.4) with
$$ f(t,x)=(t/3+1) (\cos x +\sin x/5). $$
Clearly, \(| f(t,x)|\leq p(t)\varphi(|x|)\), where \(p(t)=\frac{t}{3}+1 \), \(\varphi(|x|)=1+\frac{|x|}{5} \). By the assumption (\({\mathcal{H}}_{4}\)):
$$\begin{aligned} &M \biggl\{ \varphi(M)\|p\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{|\delta |\sigma ^{q}}{|1-\delta|\Gamma(q+1)} + \biggl(\frac{|\delta|\sigma}{|\mathcal{A}(1-\delta)|}+\frac {1}{|\mathcal {A}|} \biggr) \\ &\quad{}\times \biggl(|a|\frac{\varrho_{1}^{q-\mu}}{\Gamma(q-\mu+1)}+|b|\frac {\varrho _{2}^{q-\mu}}{\Gamma(q-\mu+1)} +|c| \frac{(\xi^{q-\mu+1}-\beta_{1}^{q-\mu+1})}{\Gamma(q-\mu +2)} \biggr) \biggr\} \biggr\} ^{-1}> 1, \end{aligned}$$
we find that \(M>4.692964\). As the hypothesis of Theorem 3.7 is satisfied, therefore, its conclusion implies that there exists at least one solution for problem (3.4) with \(f(t,x) \) given by (3.5).



The paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR technical and financial support.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University


  1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  2. Zaslavsky, GM: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005) MATHGoogle Scholar
  3. 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
  4. Sabatier, J, Agrawal, OP, Machado, JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007) Google Scholar
  5. Konjik, S, Oparnica, L, Zorica, D: Waves in viscoelastic media described by a linear fractional model. Integral Transforms Spec. Funct. 22, 283-291 (2011) MathSciNetView ArticleMATHGoogle Scholar
  6. Bitsadze, A, Samarskii, A: On some simple generalizations of linear elliptic boundary problems. Dokl. Math. 10, 398-400 (1969) MATHGoogle Scholar
  7. Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl. 9, 1727-1740 (2008) MathSciNetView ArticleMATHGoogle Scholar
  8. Čiegis, R, Bugajev, A: Numerical approximation of one model of the bacterial self-organization. Nonlinear Anal., Model. Control 17, 253-270 (2012) MathSciNetMATHGoogle Scholar
  9. Ahmad, B, Eloe, P: A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices. Commun. Appl. Nonlinear Anal. 17, 69-80 (2010) MathSciNetMATHGoogle Scholar
  10. Bai, ZB, Sun, W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, 1369-1381 (2012) MathSciNetView ArticleMATHGoogle Scholar
  11. Ahmad, B, Malar, K, Karthikeyan, K: A study of nonlocal problems of impulsive integrodifferential equations with measure of noncompactness. Adv. Differ. Equ. 2013, 205 (2013) MathSciNetView ArticleGoogle Scholar
  12. Zhai, C, Xu, L: Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 19, 2820-2827 (2014) MathSciNetView ArticleGoogle Scholar
  13. Punzo, F, Terrone, G: On the Cauchy problem for a general fractional porous medium equation with variable density. Nonlinear Anal. 98, 27-47 (2014) MathSciNetView ArticleMATHGoogle Scholar
  14. Zhang, L, Ahmad, B, Wang, G: The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative. Appl. Math. Lett. 31, 1-6 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Henderson, J, Kosmatov, N: Eigenvalue comparison for fractional boundary value problems with the Caputo derivative. Fract. Calc. Appl. Anal. 17, 872-880 (2014) MathSciNetView ArticleMATHGoogle Scholar
  16. Ahmad, B, Agarwal, RP: Some new versions of fractional boundary value problems with slit-strips conditions. Bound. Value Probl. 2014, 175 (2014) MathSciNetView ArticleGoogle Scholar
  17. Henderson, J, Luca, R, Tudorache, A: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18, 361-386 (2015) MathSciNetView ArticleMATHGoogle Scholar
  18. Ahmad, B, Ntouyas, SK, Alsaedi, A, Alzahrani, F: New fractional-order multivalued problems with nonlocal nonlinear flux type integral boundary conditions. Bound. Value Probl. 2015, 83 (2015) MathSciNetView ArticleGoogle Scholar
  19. Ding, Y, Wei, Z, Xu, J, O’Regan, D: Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian. J. Comput. Appl. Math. 288, 151-158 (2015) MathSciNetView ArticleGoogle Scholar
  20. Zhang, L, Ahmad, B, Wang, G: Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half line. Bull. Aust. Math. Soc. 91, 116-128 (2015) MathSciNetView ArticleMATHGoogle Scholar
  21. Smart, DR: Fixed Point Theorems. Cambridge University Press, Cambridge (1980) MATHGoogle Scholar
  22. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003) View ArticleMATHGoogle Scholar


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