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The Cauchy type problem for interval-valued fractional differential equations with the Riemann-Liouville gH-fractional derivative
- Yonghong Shen^{1, 2}Email author
Received: 17 November 2015
Accepted: 30 March 2016
Published: 8 April 2016
Abstract
In this paper, we establish the relationship between the Cauchy type problem for interval-valued fractional differential equations with the Riemann-Liouville gH-fractional derivative and the corresponding interval-valued integral equation. Moreover, we also consider the existence of the solutions to the interval-valued integral equation. Furthermore, we obtain the solutions to the Cauchy type problem under certain conditions.
Keywords
- generalized Hukuhara difference
- interval-valued function
- interval-valued Riemann-Liouville fractional integral
- Riemann-Liouville gH-fractional derivative
- Cauchy type problem
1 Introduction
Fractional calculus can be regarded as a generalization of ordinary differentiation and integration to any real or complex order. In the past few decades, the subject has gained considerable popularity and importance due mainly to its demonstrated applications in many fields, such as rheology, viscoelasticity, electrochemistry, electromagnetism, diffusion processes, and so on. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics. For more details, the reader can refer to several important monographs, such as Oldham and Spanier [1], Miller and Ross [2], Podlubny [3], Kilbas et al. [4], Laksmikantham et al. [5], etc.
In practice, many problems are often associated with different types of imprecision, for instance, randomness and uncertainty. Accordingly, it is necessary to take into account imprecision to study some dynamical systems. Interval numbers and fuzzy numbers are two important tools to deal with uncertainty problems. In 2010, Agarwal et al. [6] first introduced the concept of solution for fractional differential equations in the space of fuzzy numbers. In the following year, Arshad and Lupulescu [7] defined the concepts of fuzzy fractional integral and fuzzy fractional derivative by means of level sets of fuzzy numbers. Meantime, they also proved the existence and uniqueness to the initial value problem for fuzzy fractional differential equations. Hereafter, Allahviranloo et al. [8] introduced the notion of fuzzy Riemann-Liouville fractional derivative (or Riemann-Liouville H-derivative) based on the Hukuhara difference (or H-difference) of fuzzy numbers. In essence, this definition is based on the strongly generalized derivative (G-derivative) of fuzzy number-valued functions introduced by Bede and Gal [9]. Subsequently, Salahshour et al. [10] considered the solutions of fuzzy fractional differential equations under Riemann-Liouville H-derivative by using the fuzzy Laplace transform method. In the same year, they together with Baluanu [11] defined the concept of Caputo H-derivative in a similar way, and further studied the existence, uniqueness and approximate solutions of fuzzy fractional differential equations. Later, Malinowski [12] studied the existence and uniqueness of the solutions of two types of random fuzzy fractional integral equations. Meantime, the author established the boundedness of solutions and the insensitivity to small changes of parameters. Unlike previous methods, Takači et al. [13] analyzed fractional differential equations with fuzzy coefficients by Mikusińki fuzzy operators. Recently, Allahviranloo et al. [14] and Hoa [15] independently introduced the concept of Caputo gH-derivative by using the generalized Hukuhara difference (or gH-difference). In fact, the gH-difference is considered as an improvement of the H-difference of fuzzy numbers. But the gH-difference of two fuzzy numbers does not always exist. However, the gH-difference for interval numbers is well defined. Interval analysis emerged as a special case of set-valued analysis has a long history [16]. To a certain degree, interval analysis was introduced as an effective method to deal with interval uncertainty that appears in many practical problems. For this reason, it is very necessary to study interval-valued differential equations.
In a recent paper [17], the author introduced fractional calculus for interval-valued functions based on gH-difference of interval numbers. Based on these concepts, Lupulescu and Hoa [18] considered the solvability of the interval Abel integral equation. However, the purpose of the present paper is to establish the relationship between the Cauchy type problem for interval-valued fractional differential equations and the corresponding interval-valued integral equation. Furthermore, we shall characterize the solutions to the Cauchy type problem by the interval-valued integral equation under certain conditions.
2 Preliminaries
Let \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathcal{K}\) denote the set of all natural numbers, the set of all real numbers and the set of all nonempty compact convex subsets of the real line \(\mathbb{R}\), respectively. Moreover, let \(T=[a,b]\), \(-\infty< a< b<\infty\), denote a finite interval on the real line \(\mathbb{R}\).
Let \(A,B\in\mathcal{K}\). If there exists \(C\in\mathcal{K}\) such that \(A=B+C\), then C is called the Hukuhara difference (H-difference for short) of A and B, and it is denoted by \(C:=A\ominus B\). Note that the H-difference is unique, but it does not always exist. A necessary condition for \(A\ominus B\) to exist is that A contains a translation of B, i.e., there exists an element c such that \(\{c\}+B\subseteq A\). To overcome this shortcoming, a generalized Hukuhara difference (gH-difference for short) is introduced by Stefanini [19].
Definition 2.1
Now we define a functional \(\Vert \cdot \Vert :\mathcal{K}\rightarrow[0,\infty)\) by \(\Vert A\Vert =\max\{\vert a^{-}\vert ,\vert a^{+}\vert \}\) for every \(A=[a^{-},a^{+}]\in\mathcal{K}\). It can easily be shown that \(\Vert \cdot \Vert \) is a norm on \(\mathcal{K}\), and thus the quadruple \((\mathcal{K},+,\cdot, \Vert \cdot \Vert )\) is a normed quasilinear space [20].
Let \(F:T\rightarrow\mathcal{K}\) be an interval-valued function. We say that F is w-increasing (w-decreasing) on T if \(w(F(t))\) is increasing (decreasing) on T. Especially, we call F is w-monotonic on T if \(w(F(t))\) is increasing or decreasing on T.
Definition 2.2
(See Stefanini [21])
Definition 2.3
(See Allahviranloo et al. [8], Lupulescu [17])
Definition 2.4
(See Allahviranloo et al. [8], Lupulescu [17])
In particular, when \(\alpha=0\) and \(\alpha=1\), we have \(\mathcal {D}_{a+}^{0}F(t)=F(t)\), \(\mathcal{D}_{a+}^{1}F(t)=F'(t)\).
Lemma 2.1
(See Markov [22])
Let \(f:T\rightarrow\mathbb{R}\) be a differentiable real valued function and let \(C\in\mathcal{K}\). Then the interval-valued function \(f\cdot{C}:T\rightarrow\mathcal{K}\) is gH-differentiable and \({(f(t)\cdot{C})'}=f'(t)\cdot{C}\).
Lemma 2.2
(See Lupulescu [17])
Based on Lemma 1.1 in [4], we can obtain the following characterization of the space \(AC([a,b],\mathcal{K})\).
Lemma 2.3
Proof
Lemma 2.4
Proof
Lemma 2.5
Let \(C=[c^{-},c^{+}]\in\mathcal{K}\) and let \(\alpha\in(0,1]\). Define the interval-valued function \(G(t):=\frac{(t-a)^{\alpha-1}}{\Gamma(\alpha)}C\) on \((a,b]\). Then \(w(G_{1-\alpha}(t))=c^{+}-c^{-}\) is a constant function on \([a,b]\).
Proof
3 The Cauchy problem for interval-valued fractional differential equations
This section is devoted to deriving the relationship between the solutions to the Cauchy type problem for interval-valued differential equations of fractional order and the solutions to the corresponding interval-valued integral equation.
Theorem 3.1
Let G be an open set in \(\mathcal{K}\) and let \(F:[a,b]\times G\rightarrow\mathcal{K}\) be an interval-valued function such that \(F(t,Y(t))\in L([a,b],\mathcal{K})\) for any \(Y\in{G}\). If \(Y(t)\in L([a,b],\mathcal{K})\) satisfies a.e. the relations (1) and (2) (i.e., \(Y(t)\) is a solution of the problem (1)-(2)), and it satisfies either \(\frac{d}{dt}w(Y_{1-\alpha}(t))\geq0\) for a.e. \(t\in[a,b]\) or \(\frac{d}{dt}w(Y_{1-\alpha}(t))\leq0\) for a.e. \(t\in[a,b]\), then \(Y(t)\) is also a solution of the integral equation (4).
Proof
Theorem 3.2
Let G be an open set in \(\mathcal{K}\) and let \(F:[a,b]\times G\rightarrow\mathcal{K}\) be an interval-valued function such that \(F(t,Y(t))\in L([a,b],\mathcal{K})\) for any \(Y\in{G}\). Assume that \(Y(t)\in L([a,b],\mathcal{K})\) is w-monotonic and satisfies a.e. the interval-valued integral equation (4) with \(w(Y(t))-\frac{(t-a)^{\alpha-1}}{\Gamma(\alpha)}w(B)\) has a constant sign on \([a,b]\). If \(Y_{1-\alpha}(t)\) is w-monotonic, then \(Y(t)\) is also a solution of the problem (1)-(2).
Proof
Remark 1
Theorems 3.1 and 3.2 show that, in general, the Cauchy type problem (1)-(2) for interval-valued fractional differential equations and the corresponding interval-valued integral equation (4) are not equivalent in the sense that, if \(Y(t)\in L([a,b],\mathcal{K})\) satisfies one of these relations, then it also satisfies the other, unless the solutions satisfy some strict conditions.
Next we shall establish an important result related to the existence of a solution to the interval-valued integral equation (4), and then we can obtain the existence of a solution to the Cauchy type problem (1)-(2) under certain conditions.
Theorem 3.3
Proof
- (i)
if \(Y\in L([a,t^{\ast}],\mathcal{K})\), then \((PY)(t),(QY)(t)\in L([a,t^{\ast}],\mathcal{K})\);
- (ii)for any \(Y_{1},Y_{2}\in L([a,t^{\ast}],\mathcal{K})\), the following estimates hold:$$\begin{aligned} & \mathcal{H}_{1}(PY_{1},PY_{2})\leq\omega \mathcal {H}_{1}(Y_{1},Y_{2}), \end{aligned}$$(15)where \(\omega=L\frac{(t^{*}-a)^{\alpha}}{\Gamma(\alpha+1)}\).$$\begin{aligned} & \mathcal{H}_{1}(QY_{1},QY_{2})\leq\omega \mathcal{H}_{1}(Y_{1},Y_{2}), \end{aligned}$$(16)
Obviously, \(Y_{0}(t)\in L([a,t^{\ast}],\mathcal{K})\). By Lemma 2.2, we know that the interval-valued integrals in the right-hand side of (13) and (14) belong to \(L([a,t^{\ast}],\mathcal{K})\), since \(F(t,Y(t))\in L([a,t^{\ast}],\mathcal{K})\).
Remark 2
Combining Theorem 3.2 with Theorem 3.3, we can formulate the following result associated with the existence of the solutions to the Cauchy type problem (1)-(2).
Theorem 3.4
Let G be an open set in \(\mathcal{K}\) and let \(F:[a,b]\times G\rightarrow\mathcal{K}\) be an interval-valued function such that \(F(t,Y(t))\in L([a,b],\mathcal{K})\) for any \(Y\in L([a,b],G)\). Let \(M>0\) such that \(w(F(t,Y(t)))\leq M\) for any \(t\in[a,b]\). Assume that F satisfies the Lipschitz condition (7) for all \(t\in[a,b]\) and all \(Y_{1},Y_{2}\in L([a,b],\mathcal{K})\). Then there exist two unique solutions Ỹ, Ŷ to the interval-valued integral equation (4) in the space \(\mathcal {L}^{\alpha}([a,t^{\ast}],\mathcal{K})\).
Furthermore, if \(Y(t)\in L([a,b],\mathcal{K})\) is w-monotonic and \(w(\widetilde{Y}(t))-w(B)\) and \(w(\widehat{Y}(t))-w(B)\) has a constant sign on \([a,t^{\ast}]\), \(\widetilde{Y}_{1-\alpha}(t)\) and \(\widehat{Y}_{1-\alpha}(t)\) are w-monotonic, then \(\widehat{Y}(t)\) and \(\widehat{Y}(t)\) are also two unique solutions to the Cauchy type problem (1)-(2), where \(t^{\ast}\) is given as in Theorem 3.3.
4 Conclusions
Usually, the existence of the solutions to the Cauchy problem (or initial value problem) for a differential equation is characterized by the existence of the solutions to the equivalent integral equation. Accordingly, it also becomes a fundamental way to construct the successive approximation sequence by means of the integral equation. In this paper, we note that, in general, the Cauchy type problem for interval-valued fractional differential equations and the corresponding integral equation are not equivalent. However, under certain conditions, we have derived the relationship between the solutions to the Cauchy type problem and the ones to the interval-valued integral equation. Therefore, these results provide the possibility for us to solve the Cauchy type problem for interval-valued fractional equations by the corresponding integral equation.
Declarations
Acknowledgements
This work was supported by the ‘Qing Lan’ Talent Engineering Funds by Tianshui Normal University and the Research Project of Higher Learning of Gansu Province (No. 2014B-080).
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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