Existence results for a class of generalized fractional boundary value problems
 Wen Cao†^{1},
 Y Xu†^{1} and
 Zhoushun Zheng†^{1}Email author
https://doi.org/10.1186/s1366201713740
© The Author(s) 2017
Received: 2 June 2017
Accepted: 22 September 2017
Published: 27 October 2017
Abstract
In this paper, we study a class of generalized fractional order threepoint boundary value problems that involve fractional derivative defined in terms of weight and scale functions. Using several fixed point theorems, the existence and uniqueness results are obtained.
Keywords
1 Introduction
Fractional calculus is the subject of studying fractional integrals and fractional derivatives, which means that the orders of integration and differentiation are not integers but nonintegers, and even complex numbers. The history of fractional calculus is more than three hundreds years. However, only in the recent forty years, it was realized that these fractional integrals and derivatives may have many potential applications. Fractional differential equation is a differential equation which involves fractional derivatives, and it has been successfully used to model many realworld phenomena such as heat conduction [1], diffusion process [2], and quantum mechanics [3]. More applications can be seen in [4], Chapter 10.
Fractional boundary value problems (FBVPs) appear in many of these applications. In recent twenty years, considerable work has been done in the field of FBVPs. To verify the existence result of a solution and to study the behavior of the solution of FBVPs have become more and more popular. There are several methods to verify the existence of FBVPs, in which the topological degree method is one of the most effective techniques. By using the fixed point theorems, FBVPs with different types of boundary conditions have been studied. More specifically, in [5], the existence and multiplicity of positive solutions for a nonlinear FBVP with twopoint boundary condition are studied. In [6], the existence of solutions for a class of threepoint FBVPs involving nonlinear impulsive fractional differential equations is considered. In [7], the existence and uniqueness of solutions for a fourpoint nonlocal FBVP are derived. In [8], the positive solution of FBVP with integral boundary condition is obtained. In [9], the existence theory of FBVP with antiperiodic boundary condition is discussed. Furthermore, for the existence results of FBVPs with some mixedtype boundary conditions, the readers are referred to [10–16] and the references therein.
The literature above only focuses on the FBVP with classical fractional derivatives, i.e., RiemannLiouville or Caputo derivatives. Fractional derivative also has some limit, since it can be regarded as a convolution between a function and a fractional power kernel. The fractional power kernel puts much weight in the present and less weight in the past, which causes the nice property of fractional derivative called nonlocal property or short memory property. The short memory property is very effective in modeling some physical processes such as diffusion phenomenon in material with memory. However, some realworld phenomena cannot be modeled by such a fractional power kernel properly. For example, an old man and a child have different memory ability. The old man may remember things that happened several decades ago, but forget what happened yesterday. The child has an opposite ability, i.e., he may have no idea about things that happened in his early years, but remember almost everything in the recent week. To model this phenomenon, we need different kernels to weight the function differently. Hence in 2012, a new class of generalized fractional integrals and derivatives defined by using a weight function and a scale function was introduced in [17]. The new fractional operators contain many existing fractional integrals and derivatives as special cases. It is shown that many integral equations can be written and solved in an elegant way using the new operators. Therefore, using different weight functions and scale functions, many fractional problems are significantly generalized. It is also possible that the new generalized fractional integrals and derivatives will bring some interest in the near future, although the theoretical study and applications of them are in the very first stage right now.
2 Preliminaries
We introduce the generalized fractional integral and derivative directly, and for more details about the classical fractional integral and derivative, such as RiemannLiouville, Caputo, and Riesz operators, we refer to [19], Chapter 2.
Definition 1
([17])
Definition 2
([17])
Definition 3
([17])
Definition 4
([17])
In the above definitions, we only present the ‘leftsided’ sense of generalized fractional integrals and derivatives. The ‘rightsided’ sense of generalized fractional integrals and derivatives and their properties are discussed in [17]. We will not repeat them here since the derivative we use in this paper is defined in the leftsided sense. For simplicity, in what follows, we remove the term ‘\([z;w]\)’ from the subscript in equation (6).
Remark 1
To be more specific, we assume that the weight function is positive and the scale function \(z ( t) \) is monotone increasing over \([0,1]\). Moreover, both \(w(t)\) and \(z(t)\) are continuously differentiable.
Remark 2
To solve problem (7), we have the following lemma.
Lemma 1
Proof
Now we can solve equations (14) and (15) to get \(C_{0}\) and \(C_{1}\). Since \(L:= \gamma \frac{z(p)}{w(p)}  \frac{z(1)}{w(1)} + \frac{z(0)}{w(1)}  \gamma \frac{z(0)}{w(p)}\) is the determinant of the coefficient matrix of equations (14) and (15), when \(L\neq 0\), equations (14) and (15) have a unique nonzero solution. This completes the proof. □
The following theorems play important roles in studying the existence and uniqueness of fractional boundary value problems.
Theorem 1
(Contraction mapping principle, see [21])
Let E be a Banach space, \({D}\subset {E}\) be closed, and \(F:{D}\rightarrow {D}\) be a strict contraction, i.e., \(\vert FxFy\vert \leq k\vert xy\vert \) for some \(k\in (0,1)\) and all \(x,y\in {D}\). Then F has a unique fixed point \(x^{*}\). Furthermore, the successive approximations \(x_{n+1}=Fx_{n}=F^{n}x_{0}\), starting at any \(x_{0}\in {D}\), converge to \(x^{*}\) and satisfy \(\vert x_{n}x^{*}\vert \leq (1k)^{1}k^{n}\vert Fx_{0}x_{0}\vert \).
Theorem 2
(ArzelàAscoli, see [21])
If a sequence \(\{x_{n}\} ^{\infty }_{n=0}\) in a compact subset of X is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence.
Theorem 3
([22])
Theorem 4
(Krasnosel’skii, see [22])
 (H1):

\(Ax+By\in {M}\), wherever \(x,y\in {M}\);
 (H2):

A is compact and continuous; and
 (H3):

B is a contraction mapping.
3 Main results
In this section, we present some existence results of boundary value problem (1). Let \(C=C ( [0,1],R ) \) denote the Banach space of all continuous functions mapping \([0,1]\) to R equipped with the norm defined by \(\Vert u\Vert =\sup_{0\leq t\leq 1}\{\vert u(t)\vert \}\).
If the operator \(T:C\rightarrow {C}\) defined by equation (16) has a fixed point, then the fixed point coincides with the solution of fractional boundary problem (1). In what follows, we prove the complete continuity property of operator T.
Lemma 2
The operator \(T:C\rightarrow {C}\) defined by equation (16) is completely continuous.
Proof
Remark 3
Theorem 5
Assume that \(f:[0,1]\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) and \(\lim_{u\rightarrow {0}}f(t,u)=0\). Then the boundary value problem (1) has at least one solution.
Proof
Therefore, by Theorem 3, the operator T has at least one fixed point, which implies that the boundary value problem (1) has at least one solution. □
Theorem 6
Proof
Theorem 7
 (H1):

\(\vert f(t,u_{1})f(t,u_{2})\vert \leq L_{5} \vert u_{1}u_{2}\vert \), \(u_{1},u_{2}\in {X}\),
 (H2):

\(\vert f(t,u)\vert \leq \lambda (t)\), \(\forall (t,u)\in [0,1] \times {X}\), and \(\lambda \in L^{1}([0,1],R^{+})\).
Proof
4 Examples
We present three examples to demonstrate the main results discussed in the last section.
Example 1
Example 2
Example 3
5 Conclusion remark
The existence results of generalized fractional boundary value problem are discussed in this paper by using several fixed point theorems. The generalized fractional derivative is defined upon a weight function and a scale function, which contains many fractional derivatives in the literature as special cases. Hence, the boundary value problems studied in this paper are more general, and it is important to develop certain methods for investigating the existence results of them. In fact, equation (10) provides us with an effective transform, under which the generalized FBVP can be regarded as a regular FBVP defined in a general time scale \(z(t)\) and weighted by a weight function \(w(t)\). We hope that our work will bring much attention into this field in the near future.
Notes
Declarations
Acknowledgements
This work was partly supported by the NSFC (No. 11501581, No. 51134003, No. 51174236), the Research Project (No. 502042032) of Central South University, the Project funded by China Postdoctoral Science Foundation (No. 2015M570683), the Project supported by the National Basic Research Development Program of China (No. 2011CB606306), the project supported by the National Key Laboratory Open Program of Porous Metal Material of China (No. PMMSKL42012).
Authors’ contributions
The authors have made the same contribution. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interest.
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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