Existence and approximation of solutions to fractional order hybrid differential equations
- Dussadee Somjaiwang^{1} and
- Parinya Sa Ngiamsunthorn^{1, 2}Email author
https://doi.org/10.1186/s13662-016-0999-8
© Somjaiwang and Sa Ngiamsunthorn 2016
Received: 27 May 2016
Accepted: 14 October 2016
Published: 28 October 2016
Abstract
We consider fractional hybrid differential equations involving the Caputo fractional derivative of order \(0<\alpha<1\). Using fixed point theorems developed by Dhage et al. in Applied Mathematics Letters 34, 76-80 (2014), we prove the existence and approximation of mild solutions. In addition, we provide a numerical example to illustrate the results obtained.
Keywords
hybrid differential equations Caputo fractional derivative hybrid fixed point results approximation of solutionsMSC
34A08 34A38 34A451 Introduction
Fractional differential equations are of interest in many areas of applications, such as economics, signal identification and image processing, optical systems, aerodynamics, biophysics, thermal system materials and mechanical systems, control theory (see [2–6]). There are several results that investigate the existence of solutions of various classes of fractional differential equations. Much attention has been focused on the study of the existence and multiplicity of solutions as well as positive solutions for boundary value problem of fractional differential equations [7–9]. The main techniques used in these studies are fixed point techniques, Leray-Schauder theory, or upper and lower solutions methods (see, for example, [10–12] and the above references).
Apart from the study of the existence of solutions, an approximation of the solution is also of interest. Dhange et al. in [1] imposed the concept of partial continuity and partial compactness to generalize the approach of Kranoselskii fixed point theorem and obtained the approximation of solutions to hybrid differential equation. Other results on the approximation of solutions to various types of equations can be found, for example, in [17, 18].
The main objective of this work is to extend the existence results in Herzallah and Beleanu [15] by following the approach in [17] to construct an iterative sequence that approximates the solution based on some fixed point theorem. We note that in [15], only the existence of a solution is proved. Our result gives both the existence and approximation of solutions to Caputo fractional order hybrid differential equations and also extends the existence results for ordinary hybrid differential equations. Moreover, the procedure in this paper allows us to approximate the solutions numerically.
This paper is organized as follows. In the next section, we introduce the notation and concepts of fractional order hybrid differential equations and discuss the frameworks of our problem. Section 3 is devoted to a proof of the existence and approximation of mild solutions of fractional order hybrid differential equations (1.1). Finally, in Section 4, we provide numerical example to illustrate the obtained results.
2 Preliminaries and framework
We consider the fractional order nonlinear hybrid ordinary differential equation with initial value problem given by (1.1), where \(f \in C(J \times \mathbb{R},\mathbb{R}), g \in C(J \times \mathbb{R},\mathbb{R})\) and the initial data \(x_{0} \in \mathbb {R}\). The fractional order derivative used in this paper is taken in the sense of Caputo, which is defined as follows.
Definition 2.1
Note that the integrability of the nth order derivative of f is required for the Caputo fractional derivative.
We shall study the existence and approximation of mild solutions in the following sense.
Definition 2.2
In this paper, we consider the Banach space \(C(J,\mathbb{R})\) together with a partial order relation. For any \(x,y \in C(J, \mathbb {R})\), it is well known that the order relation \(x \leq y\) given by \(x(t)\leq y(t)\) for all \(t \in J\) gives a partial ordering in \(C(J,\mathbb{R})\). We shall mention the following important properties of the partially ordered Banach space \((C(J,\mathbb{R}),\leq)\), necessary for our study, from the work of [1, 18–20].
Let \(E=(E,\preceq, \Vert \cdot \Vert )\) be a normed linear space equipped with a partial order relation ⪯. The space E is said to be regular if, for any nondecreasing sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in E such that \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\), we have \(x_{n}\preceq x^{*}\) for all \(n\in\mathbb{N}\). In particular, the space \(C(J,\mathbb{R})\) is regular [1].
Definition 2.3
[19]
An operator \(T:E\rightarrow E\) is called nondecreasing if the order relation is preserved under T, that is, for any \(x,y \in E\) such that \(x\leq y\), we have \(Tx \leq Ty\).
Definition 2.4
[20]
An operator \(T:E \rightarrow E\) is called partially continuous at \(a \in E\) if for any \(\varepsilon>0\), there exists \(\delta>0\) such that \(\Vert Tx-Ta\Vert <\varepsilon\) for all x comparable to a in E with \(\Vert x-a\Vert <\delta\). T is called partially continuous on E if it is partially continuous at every \(a \in E\). In particular, if T is partially continuous on E, then it is continuous on every chain \(\mathcal{C}\) in E. An operator T is called partially bounded if \(T(\mathcal{C})\) is bounded for every chain \(\mathcal{C}\) in E. An operator T is said to be uniformly partially bounded if all chains \(T(\mathcal{C})\) in E are bounded by the same constant.
Definition 2.5
[20]
An operator \(T:E\rightarrow E\) is called partially compact if for any chains \(\mathcal{C}\) in E, the set \(T(\mathcal{C})\) is a relatively compact subset of E. An operator T is said to be partially totally bounded if for any totally ordered and bounded subset \(\mathcal{C}\) of E, the set \(T(\mathcal{C})\) is a relatively compact subset of E. If T is partially continuous and partially totally bounded, then we call it a partially completely continuous operator on E.
Definition 2.6
[19]
Let E be a nonempty set equipped with an order relation ⪯ and a metric d. We say that the order relation ⪯ and the metric d are compatible if the following property is satisfied: if \(\{x_{n}\}_{n\in \mathbb{N}}\) is a monotone sequence in E for which a subsequence \(\{x_{n_{k}}\}_{k\in \mathbb{N}}\) of \(\{x_{n}\}_{n\in \mathbb{N}}\) converges to \(x^{*}\), then the whole sequence \(\{x_{n}\}_{n\in \mathbb{N}}\) converges to \(x^{*}\). Similarly, if \((E,\preceq, \Vert \cdot \Vert )\) is a partially ordered normed linear space, we say that the order relation ⪯ and the norm \(\Vert \cdot \Vert \) are compatible whenever the order relation ⪯ and the metric d induced by the norm \(\Vert \cdot \Vert \) are compatible.
We point out that the order relations and norms of \((\mathbb{R},\leq,|\cdot|)\) and \((C(J,\mathbb{R}),\leq, \Vert \cdot \Vert )\) are compatible.
Definition 2.7
[19]
An upper semi-continuous and nondecreasing function \(\psi:\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) is called a \(\mathcal{D}\)-function if \(\psi(0)=0\).
Definition 2.8
[20]
The following hybrid fixed point result of [19] is often applied to establish the existence and approximation of solutions of various differential and integral equations.
Theorem 2.1
[20]
- (a)
\(\mathcal{P}\) is a partially nonlinear \(\mathcal{D}\)-contraction.
- (b)
\(\mathcal{Q}\) partially continuous and partially compact.
- (c)
There exists an element \(x_{0}\in E\) such that \(x_{0}\preceq \mathcal{P}x_{0}+\mathcal{Q}x_{0}\).
We shall state the framework and assumptions for our study now.
Assumption 1
- (A0)
The functions \(f: J\times\mathbb{R}\rightarrow\mathbb{R}\) and \(g:J\times\mathbb{R}\rightarrow\mathbb{R}\) are continuous.
- (A1)
f is nondecreasing in x for each \(t\in J\) and \(x\in \mathbb{R}\).
- (A2)
There exists a constant \(M_{f}>0\) such that \(0 \leq |f(t,x)|\leq M_{f}\) for all \(t\in J\) and \(x\in \mathbb{R} \).
- (A3)
There exists a \(\mathcal {D}\)-contraction ϕ such that \(0\leq f(t,x)-f(t,y)\leq \phi(x-y)\) for \(t\in J\), and \(x,y\in\mathbb{R}\) with \(x\geq y\).
- (B1)
g is nondecreasing in x for each \(t\in J\) and \(x\in \mathbb{R}\).
- (B2)
There exists a constant \(M_{g}>0\) such that \(0\leq |g(t,x)|\leq M_{g}\) for all \(t\in J\) and \(x\in \mathbb{R} \).
- (B3)There exists a function \(u\in C(J,\mathbb{R})\) such that u is a lower solution the problem (1.1), that is,$$ \begin{aligned} &\frac{d^{\alpha}}{dt^{\alpha}} \bigl[u(t)-f \bigl(t,u(t) \bigr) \bigr] \leq g \bigl(t,u(t) \bigr),\quad t\in J=[t_{0},t_{0}+a], \\ &u(t_{0}) \leq x_{0} \in \mathbb{R.} \end{aligned} $$(2.2)
3 Existence and approximation of mild solutions
This section is devoted to a proof of our main result on the existence and approximation of mild solutions of fractional order hybrid differential equations.
Theorem 3.1
Proof
Thus, we conclude that the operators \(\mathcal{P}\) and \(\mathcal{Q}\) satisfy all conditions in Theorem 2.1. Then the operator equation \(\mathcal{P}x+\mathcal{Q}x=x\) has a solution. Moreover, we have the approximation of solutions \(x_{n}\) as \(n=1, 2,\ldots \) for equation (1.1). □
4 Numerical examples
In this section, we give an example of hybrid fractional differential equation and show that our main result can be applied to construct an approximate sequence for a solution. We also illustrate it by showing a numerical result.
Example 4.1
Declarations
Acknowledgements
The authors would like to thank referees for valuable comments and suggestions to improve the manuscript. This project was supported by the Theoretical and Computational Science Center (TaCS) under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.
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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