- Research
- Open Access
Existence, uniqueness and stability of fuzzy fractional differential equations with local Lipschitz and linear growth conditions
- SS Mansouri^{1},
- M Gachpazan^{1}Email author and
- O Solaymani Fard^{1}
https://doi.org/10.1186/s13662-017-1293-0
© The Author(s) 2017
Received: 24 March 2017
Accepted: 24 July 2017
Published: 16 August 2017
Abstract
Fuzzy fractional differential equations (FFDEs) driven by Liu’s process are a type of fractional differential equations. In this paper, we intend to provide and prove a novel existence and uniqueness theorem for the solutions of FFDEs under local Lipschitz and linear growth conditions. We also investigate the stability of solutions to FFDEs by a theorem. Finally, some examples are provided.
Keywords
- credibility theory
- fractional derivative
- fractional differential equation
- fuzzy differential equations
- fuzzy Liu’s process
- existence and uniqueness
1 Introduction
A large number of physical processes such as real-life phenomena appear to display fractional-order demeanor that may vary with space or time. The fractional calculus has authorized the operations of differentiation and integration to all fractional order [1–3]. The order may take on all real or imaginary values. Multitude systems modeled with the support of fractional calculus also demonstrate fractional dynamical conduct such as viscoelastic systems [4, 5], colored noise [6], boundary layer effects in ducts and electromagnetic waves [7].
In recent decades, the theory of fractional differential equations (FDEs) has enticed many researchers, such as [8–14], who have applied FDEs for acoustic models, thermal systems, signal processing, system identification, robotics and control, etc.
A FDE is a differential equation which includes fractional derivatives. The results of various studies have clearly declared that fractional derivatives seem to arise universally and generally from major mathematical reasons. There are different kinds of fractional derivatives, like Riemann-Liouville and Caputo. For details, one can refer to [12] and [4].
Stochastic fractional differential equations were used to model dynamical systems affected by random noises [15–23].
As already known, the concepts of fuzzy sets have been first investigated by Zadeh via the membership function [24]. The literature concerning the applications of nonlinear FDEs has grown rapidly in the recent years [25–28].
In this paper, we investigate the theory of fuzzy FDEs (FFDEs) in the sense of Liu’s process. Liu introduced credibility theory for the first time and presented the concept of credibility measure to facilitate the measurement of fuzzy events [29]. It is worthy to note that this measurement is a powerful tool for dealing with fuzzy phenomena and is based on normality, monotonicity, self-duality, and maximality axioms.
Also, Liu has proposed the concept of fuzzy process [30–32]. Liu’s process is a particular fuzzy process with stationary and independent increment, which is just like a stochastic process described by Brownian motion.
A large number of researches have been recently published on the Liu process and its applications in other sciences such as economics and optimal control [33–38]. Many researchers were inspired by stochastic notions and the Ito process introduced fuzzy differential equations which were driven by Liu’s process for better understanding of the fuzzy phenomenon [37, 39–43].
Qin and Li applied fuzzy differential equations, driven by Liu’s process, to solve European pricing problems in a fuzzy environment [44].
Regarding the importance of the existence and uniqueness of the solution to fuzzy differential equations driven by Liu’s process, You investigated the existence and uniqueness of the solution to the fuzzy differential equations by employing Lipschitz and linear growth conditions [45]. Afterwards, Fei studied the uniqueness of the solution to the fuzzy differential equations driven by Liu’s process with non-Lipschitz coefficients [46], but unfortunately, the existence and uniqueness of the solution to the FFDEs driven by Liu’s process have not been adequately investigated. Moreover, stability, which is one of the most important issues in differential equations, has not been studied with regard to such equations.
In this paper, weaker conditions are provided in order to guarantee the existence and uniqueness of the solution to the FFDEs, which makes it possible for more functions to be verified in such conditions. Furthermore, stability can also be investigated by considering such conditions.
The basic concepts of credibility theory are first discussed in this paper since they are needed in the later sections. FFDEs driven by Liu’s process are taken into consideration in the second section. The existence and uniqueness theorem of the solution with regard to weaker conditions such as the local Lipschitz and Linear growth conditions is proved in the third section, and finally in the last and the most important section, the stability of FFDEs in weaker conditions is elaborated.
2 Preliminaries
The emphasis in this section is mainly on introducing some concepts such as credibility measure, credibility space, fuzzy variables, independence of fuzzy variables, expected value, variance, fuzzy process, Liu process, and stopping time.
- 1.
Axiom (normality) \(\operatorname{\mathbf{Cr}}\{\Theta\}= 1\).
- 2.
Axiom (monotonicity) \(\operatorname{\mathbf{Cr}}\{\kappa\} \leq\operatorname{\mathbf{Cr}}\{ \beta\}\) whenever \(\kappa\subset\beta\).
- 3.
Axiom (self-duality) \(\operatorname{\mathbf{Cr}}\{\kappa\} + \operatorname{\mathbf{Cr}}\{ \kappa^{c}\} = 1\) for any event κ.
- 4.Axiom (maximality) \(\operatorname{\mathbf{Cr}}\{\mathbf{U}_{i}\kappa_{i}\} = \sup_{i} \operatorname{\mathbf{Cr}}\{\kappa_{i}\}\) for any events \(\{\kappa_{i}\}\) with$$ \sup_{i} \operatorname{\mathbf{Cr}}\{\kappa_{i}\} < 0.5. $$
Definition 2.1
[32]
The set function Cr is called a credibility measure if it satisfies the normality, monotonicity, self-duality, and maximality axioms.
A family \(\mathcal{P}\) with the above mentioned four properties is called a σ-algebra. The pair \((\Theta,\mathcal{P})\) is called a measurable space, and each element of \(\mathcal{P}\) is consequently called \(\mathcal{P}\)-measurable sets instead of events.
Definition 2.2
[47]
The triple \((\Theta,\mathcal{P},\operatorname{\mathbf{Cr}})\) is a credibility space if Θ is a non-empty set, \(\mathcal{P}\) the power set of Θ, and Cr a credibility measure.
Let \((\Theta,\mathcal{P},\operatorname{\mathbf{Cr}})\) be a credibility space. A filtration is a family \(\{\mathcal{P}_{t}\}_{t\geq0}\) of increasing sub-σ-algebras of \(\mathcal{P}\) (i.e. \(\mathcal {P}_{t}\subset\mathcal{P}_{s}\subset\mathcal{P}\) for all \(0 \leq t < s < \infty\)). The filtration is said to be right continuous if \(\mathcal{P}_{t} = \bigcap_{s>t}\mathcal{P}_{s}\) for all \(t\leq0\). When the credibility space is complete, the filtration is said to satisfy the usual conditions if it is right continuous and \(\mathcal{P}_{0}\) contains all Cr-null sets.
We also define \(\mathcal{P_{\infty}}=\sigma(U_{t\geq0}\mathcal {P}_{t})\) (i.e. a σ-algebra generated by \(U_{t\geq 0}\mathcal{P}_{t}\)). \(\mathcal{P}\)-measurable fuzzy variables are denoted by \(\mathbf{L}^{p}(\Theta,\mathbf{R^{d}})\), which will be defined later. A process is called \(\mathcal{P}\)-adapted if for all \(t\in[0,t]\) the fuzzy variable \(x(t)\) is \(\mathcal{P}\)-measurable.
Definition 2.3
[47]
A fuzzy variable is defined as a (measurable) function \(\xi: (\Theta,\mathcal{P}, \operatorname{\mathbf{Cr}})\longrightarrow\mathbf{R}\).
Definition 2.4
[47]
Definition 2.5
[47]
Definition 2.6
[31]
Definition 2.7
[47]
Considering T to be an index set and \((\Theta,\mathcal {P},\operatorname{\mathbf{Cr}})\) to be a credibility space, a fuzzy process can be described as a function from \(\mathbf{T} \times(\Theta,\mathcal {P},\operatorname{\mathbf{Cr}})\) to the set of real numbers.
A fuzzy process is basically a sequence of fuzzy variables indexed by time or space.
Definition 2.8
[47]
- 1.
\(\mathbf{C}_{0} = 0\),
- 2.
\(\mathbf{C}_{t}\) has stationary and independent increments,
- 3.every increment \(\mathbf{C}_{t+s} - \mathbf{C}_{s}\) is a normally distributed fuzzy variable with expected value et and variance \(\sigma ^{2}t^{2}\) whose membership function is$$ \eta(w)=2\biggl(1+\exp\biggl(\frac{\pi \vert w-et \vert }{\sqrt {6}\sigma t}\biggr)\biggr)^{-1}, \quad -\infty< w< +\infty. $$
Based on the Liu process, a Liu integral is defined as a fuzzy counterpart of an Ito integral as follows:
Theorem 2.1
[32]
Let us define a sequence of credibilistic stopping times.
Definition 2.9
[35]
Definition 2.10
[47]
If \(W = \{W_{t}\}_{t\geq0}\) is a measurable process and τ is a stopping time, then \(\{W_{\tau\wedge t}\}_{t\geq0}\) is called a stopped process of W.
There are some useful inequalities for fuzzy variables such as the Hölder inequality and the Chebyshev inequality. In this sequence, we introduce generalized inequalities for fuzzy variables.
Theorem 2.2
Hölder’s inequality, [47]
Theorem 2.3
Chebychev’s inequality
Proof
Before ending this section it is essential to introduce some symbols that are used in the next sections.
Notation 1
\(\mathbf{L}^{\mathbf{n}}(\theta,\mathbf{R}^{\mathbf{d}})\) the family of \(\mathbf{R}^{\mathbf{d}}\)-valued fuzzy variables ς with \(\mathbf {E} \vert \xi \vert ^{\mathbf{p}}<\infty\).
Notation 2
\(\ell^{\mathbf{p}}([a,b],\mathbf{R}^{\mathbf{d}})\) the family of \(\mathbf {R}^{\mathbf{d}}\)-valued \(\mathcal{P}_{t}\)-adapted processes \(\{h(t)\}_{ a \leq t \leq b}\) such that \(\int^{b}_{a} \vert h(t) \vert ^{\mathbf{n}}\,\mathrm{d}t<\infty\) almost surely.
Notation 3
\(M^{\mathbf{n}}([a,b],\mathbf{R}^{\mathbf{d}}) \) the family of processes \(\{h(t)\}_{ a \leq t \leq b}\) in \(\ell^{\mathbf{n}}([a,b],\mathbf {R}^{\mathbf{d}})\) such that \(\int^{b}_{a} \vert h(t) \vert ^{\mathbf{n}}\,dt<\infty\).
Notation 4
\(\ell^{\mathbf{n}}(\mathbf{R}_{+},\mathbf{R}^{\mathbf{d}})\) the family of processes \(\{h(t)\}_{t >0}\) such that, for every \(T > 0\), \(\{h(t)\}_{ a \leq t \leq T} \in\ell^{\mathbf{n}}([0,T],\mathbf {R}^{\mathbf{d}}) \).
Remark 2.1
[29]
Lemma 2.1
[31]
3 Fuzzy fractional differential equations
In this section, some basic definitions, notations and lemmas which will be used throughout the paper are provided in order to establish our main results.
Two common notations for the fractional differential operator also need to be introduced, which are Riemann-Liouville and Caputo. See [12] and [4] for more details.
Definition 3.1
[12]
Definition 3.2
[4]
Remark 3.1
Lemma 3.1
3.1 Existence and uniqueness of the solution for FFDEs
- 1.
We expect initial conditions of the mathematical model to have a solution.
- 2.
According to initial conditions, we want any mathematical model to have a unique solution.
- (I)Local Lipschitz condition: There exists a positive constant number L such thatfor those \(w(t),y(t)\in\mathbf{R}^{n} \).$$ \bigl\vert p \bigl(w(t), t\bigr)-p \bigl(y(t), t\bigr) \bigr\vert + \bigl\vert q\bigl(w(t), t\bigr)-q\bigl(y(t), t\bigr) \bigr\vert \leq\mathbf{L} \bigl\vert w(t)-y(t) \bigr\vert , $$
- (II)Linear growth condition: There exists a positive number L such that$$ \bigl\vert p \bigl(w(t), t\bigr) \bigr\vert + \bigl\vert q\bigl(w(t), t \bigr) \bigr\vert \leq\mathbf{L}\bigl( 1+ \bigl\vert w(t) \bigr\vert \bigr). $$
Lemma 3.2
For any \(\tau\in\Theta\), and \(w(t,\tau) \in\mathbf{C}([-\tau,\mathbf {R}],\mathbf{R})\) we have \(A(w(t,\tau))\in\mathbf{C}([-\tau,\mathbf {R}],\mathbf{R})\).
Proof
Theorem 3.1
Existence and uniqueness
The FFDE (3.4) has a unique solution \(w(t)\) in \([0,+\infty)\) if the coefficients \(f(w(t),t)\) and \(g(w(t),t)\) satisfy (I) and (II).
Proof
3.2 Stability theorem for FFDEs
Regarding the fact that if a physical examination is repeated twice with the same conditions and how much measurements are conducted carefully, the initial imposed conditions will not be exactly the same. It is expected that the results of two physical examinations have a little difference with each other. This means that very small changes in initial conditions are expected to cause only small changes in the answer. In other words, the answer to the mathematical model is stable. In this section, the concept of stability for a FFDE in the sense of fuzzy measure is proposed. A sufficient condition will also be derived for a FFDE to be stable.
Corollary 3.1
Proof
Corollary 3.2
Proof
Definition 3.3
Theorem 3.2
The FFDE (3.10) is stable if the coefficients \(f(w,t)\) and \(g(w,t)\) satisfy (I) and (II).
Proof
4 Conclusion
In this paper, weaker conditions are provided in order to guarantee the existence and uniqueness of the solution to the fuzzy fractional deferential equations driven by Liu’s process, which makes it possible for more functions to be verified in such conditions. Furthermore, stability can also be investigated by considering such conditions.
Declarations
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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