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Boundary value problems with four orders of RiemannLiouville fractional derivatives
 Somboon Niyom^{1},
 Sotiris K Ntouyas^{2, 3},
 Sorasak Laoprasittichok^{4} and
 Jessada Tariboon^{4, 5}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201608970
© Niyom et al. 2016
Received: 17 March 2016
Accepted: 13 June 2016
Published: 23 June 2016
Abstract
In this paper we study a new class of boundary value problems for fractional differential equations which contains RiemannLiouville fractional derivatives of four orders, two in a fractional differential equation and two in boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.
Keywords
 fractional differential equation
 boundary value problem
 existence
 fixed point theorems
MSC
 34A08
 34A12
1 Introduction
Observe that the RiemannLiouville fractional derivatives appearing in the differential equation and in the boundary conditions depend on the values of the constants λ and μ, respectively. If \(\lambda=1\), then the first equation of (1.1) is reduced to a single order fractional derivative. Also in boundary conditions, the value of constant μ has an effect for fractional derivative of order \(\gamma_{1}\) and \(\gamma_{2}\). In applications, it seems that the values of λ and μ can be interpreted as the adjustable instruments for a suitable real world phenomenon.
Fractional calculus has found numerous miscellaneous applications connected with real world problems as they appear in many fields of science and engineering, including fluid flow, signal and image processing, fractal theory, control theory, electromagnetic theory, fitting of experimental data, optics, potential theory, biology, chemistry, diffusion, and viscoelasticity. For a detailed account of applications and recent results on initial and boundary value problems of fractional differential equations, we refer the reader to [1–17] and the references therein.
The paper is organized as follows. In Section 2, we present the framework in which the boundary value problem (1.1), is formulated in a fixed point equation. Section 3 is devoted to the main results. Illustrative examples are also presented.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [1, 2] and we present preliminary results needed in our proofs later.
Definition 2.1
Definition 2.2
From the definition of the RiemannLiouville fractional derivative, we can obtain the following lemmas.
Lemma 2.1
(see [2])
Lemma 2.2
(see [2])
Lemma 2.3
Proof
3 Main results
Let \(\mathcal{C} = C(J,\mathbb{R})\) denotes the Banach space of all continuous functions from \(J:=[0,T]\) to \(\mathbb{R}\) endowed with the usual supnorm \(\u\ = \sup_{t\in J} u(t)\).
3.1 Existence and uniqueness result via Banach’s fixed point theorem
In the first result we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.
Theorem 3.1
 (H_{1}):

there exists a constant \(L>0\) such that \(f(t,x)f(t,y)\leq Lxy\), for each \(t \in J\) and \(x,y \in\mathbb{R}\).
Proof
Example 3.1
Here \(\alpha= 17/9\), \(\beta= 15/8\), \(\lambda= 26/27\), \(\mu= 2/67\), \(\gamma_{1} = 1/100\), \(\gamma_{2} = 1/101\), \(\gamma_{3} = 2/9\), \(T = 3\), and \(f(t,x) = (e^{t}/2(7t)^{2})((x^{2}+2x)/(x+1))\). Since \(f(t,x)  f(t,y) \leq(1/16)xy\), then (H_{1}) is satisfied with \(L =1/16\). By direct computation, we have \(\Lambda\approx2.63552 \neq0\), \(\Omega _{1}\approx 0.07837\), and \(\Omega_{2}\approx9.16679\). Thus \(L\Omega_{2} + \Omega_{1} \approx0.65129 <1\). Hence, by Theorem 3.1, the problem (3.6) has a unique solution on \([0,3]\).
3.2 Existence result via Krasnoselskii’s fixed point theorem
Next, we prove an existence result based on Krasnoselskii’s fixed point theorem.
Theorem 3.2
(Krasnoselskii’s fixed point theorem [18])
 (a)
\(Ax+By \in M\) where \(x,y \in M\);
 (b)
A is compact and continuous;
 (c)
B is a contraction mapping.
Theorem 3.3
 (H_{2}):

\(f(t,x)\leq\nu(t)\), \(\forall(t,x) \in J\times \mathbb{R}\) and \(\nu\in C(J,\mathbb{R}^{+})\).
Proof
Example 3.2
3.3 Existence result via LeraySchauder’s nonlinear alternative
Our final existence result is based on LeraySchauder’s nonlinear alternative.
Theorem 3.4
(Nonlinear alternative for single valued maps [19])
 (i)
\(\mathcal{F}\) has a fixed point in U̅, or
 (ii)
there is a \(x\in\partial U\) (the boundary of U in C) and \(\lambda\in(0,1)\) with \(x = \lambda \mathcal{F}(x)\).
Theorem 3.5
 (H_{3}):

there exist a continuous nondecreasing function \(\psi: [0,\infty) \to(0,\infty)\) and a function \(\varphi\in C(J,\mathbb {R}^{+})\) such that$$\bigl\vert f(t,x)\bigr\vert \le\varphi(t)\psi\bigl(\Vert x\Vert \bigr) \quad \textit{for each } (t,x) \in J \times\mathbb{R}; $$
 (H_{4}):
Proof
The result will follow from the LeraySchauder nonlinear alternative (Theorem 3.4) once we have proved the boundedness of the set of the solutions to equations \(x=\theta\mathcal{F} x\) for \(\theta\in(0,1)\).
Example 3.3
3.4 Special cases
Corollary 3.1
Assume that the condition (H_{1}) holds. If \(L\Omega_{2}<1\), then the boundary value problem (3.11) has a unique solution on J.
Corollary 3.2
Let the conditions (H_{1}) and (H_{2}) be satisfied. Then the boundary value problem (3.11) has at least one solution on J, provided \(\Omega_{1}<1\).
Corollary 3.3
Corollary 3.4
Suppose that the condition (H_{1}) holds. If \(L\Omega_{2}+\Omega_{1}<1\), where \(\Omega_{1}\), \(\Omega_{2}\) are defined by (3.2) and (3.3) and Λ by (3.13), then the boundary value problem (3.12) has a unique solution on J.
Declarations
Acknowledgements
The authors thank the reviewers for their constructive comments, which led to the improvement of the original manuscript. This research is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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
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