On memo-viability of fractional equations with the Caputo derivative
- Ewa Girejko^{1},
- Dorota Mozyrska^{1} and
- Małgorzata Wyrwas^{1}Email author
https://doi.org/10.1186/s13662-015-0403-0
© Girejko et al.; licensee Springer. 2015
Received: 21 October 2014
Accepted: 3 February 2015
Published: 24 February 2015
Abstract
In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give a necessary condition for fractional viability of a locally closed set with respect to a nonlinear function. A specific sufficient condition is also provided.
Keywords
1 Introduction
Fractional calculus deals with fractional derivatives and integrals of any order, and it is a field of mathematics that grows out of the traditional definitions of calculus integral and derivative operators. Hence, fractional differential equations are generalizations of ordinary differential equations to equations with an arbitrary order.
In recent years, fractional differential equations have been investigated by many authors [1–6]. However, the problem of viability of fractional differential equations, which consists in finding at least one solution to the equation starting and staying in a constrained set, has not been well developed so far. The classical viability theory has its origin in the Nagumo theorem [7] and is widely exploited starting from ordinary differential equations and reaching differential inclusions based on set-valued maps with a wide range of applications [8–10]. In this paper we continue the subject of the Nagumo theorem for a fractional differential equation with the Caputo derivative. In [11] we showed a sufficient condition for solutions to be viable with respect to a constrained set. In the present paper a necessary condition of viability of a fractional differential equation with the Caputo derivative is proved. It is not trivial to show a necessary condition of viability of a fractional differential equation with the Caputo derivative. Thus we implement the idea that we used in [12] for a fractional differential equation with the Riemann-Liouville derivative, i.e., the initialization problem that leads to a modification of the problem, namely we consider viability of solutions in the memory domain. Then we employ the formula that joins these two types of fractional derivatives. This idea allows us to prove the necessity of the Nagumo theorem going through the classical tools as Bouligand cone, contingent vectors, etc.
The paper is organized as follows. In Section 2 we gather preliminary definitions, notations and some results. Section 3 includes the initialization problem. We formulate the inner value problem with the Caputo derivative on the basis of a similar problem with the Riemann-Liouville derivative. The key result is given in Proposition 7. The last section concerns the viability problem. Theorem 11 and corollaries give necessary conditions of viability of a fractional differential equation with the Caputo derivative. Then an illustrative example is provided. Finally Theorem 17 gives a base to formulate a sufficient condition of viability for an equation involving the Caputo derivative that is, however, slightly different from the one that we formulated in [11]. We finish the paper with a block scheme that shows relations among viability conditions presented in the paper.
2 Preliminaries
In this section we make a review of notations, definitions, and some preliminary facts, which are useful for the paper. We recall definitions of fractional integrals of arbitrary order, the Caputo and Riemann-Liouville derivatives of order \(q\in(0,1)\), and a description of special functions in the fractional calculus.
Definition 1
Remark 2
([15])
Note that \(I_{0+}^{q}f(t)=(f*\varphi_{q})(t)\), where \(\varphi_{q}(t)=\frac{t^{q-1}}{\Gamma(q)}\) for \(t>0\), \(\varphi_{q}(t)=0\) for \(t\leq0\), and \(\varphi_{q}\rightarrow\delta(t)\) as \(q\rightarrow0\), with δ the delta Dirac pseudo function.
The best known fractional derivatives are the Riemann-Liouville and the Caputo ones.
Definition 3
Remark 4
From [13], Lemmas 2.4 and 2.5, we have the following properties.
Proposition 5
3 The inner value problem
Remark 6
Since for \(x(0^{+})=0\) we have \(f\equiv g\), we look for the solutions of (6) in the set of functions \(x(\cdot)\) such that \(x(0^{+})\neq0\).
Calculating values of \(f(s,x(s))\) for \(s\in(0,t_{0}]\) one finds some difficulties in predicting values of the memory in the light of viability, thus we consider the modification of the problem that leads to the main results.
Proposition 7
Proof
4 Viability problem
By definitions of m, \(\widetilde{m}\) and Proposition 7 the following proposition is obvious.
Proposition 8
Similarly as for the ordinary differential equations (see [16]), one can define the viability of a subset with respect to the fractional differential equation (3).
Let us denote by I an open interval in ℝ.
Definition 9
Let \(K\subset\mathbb{R}^{n}\) be a nonempty locally closed set and \(f: I\times K\to\mathbb{R}^{n}\). The subset K is fractionally memo-viable with respect to f if for any \((t_{0},x_{0})\in I\times K\) equation (3) has at least one solution \(x:[t_{0},T]\to \mathbb{R}^{n}\) satisfying \(m(t_{0},t)\in K\) for \(t\in[t_{0},T]\), where \(t_{0}>0\).
The idea of viability of fractional differential equations can be expressed using the concept of tangent cone. There are many notions of tangency of a vector to a set (see, for example, [16], Section 2.3). We will follow the concept of contingent vectors (see [9]).
Let us recall that for \(K \subset\mathbb{R}^{n}\) and \(x_{0}\in K\) one can define the vector tangent to the set K as follows.
Definition 10
The set of all vectors that are contingent to the set K at point \(x_{0}\) is a closed cone, see [16], Proposition 2.3.1. This cone, denoted by \(\mathcal{T}_{K} (x_{0})\), is called contingent cone (Bouligand cone) to the set K at \(x_{0}\in K\). From [16], Proposition 2.3.2, we know that \(\eta\in\mathcal{T}_{K} (x_{0})\) if and only if for every \(\varepsilon>0\) there exist \(h\in(0, \varepsilon)\) and \(p_{h} \in B(0, \varepsilon)\) such that \(x_{0} +{h}(\eta+p_{h})\in K\), where \(B(0, \varepsilon)\) denotes the closed ball in \(\mathbb{R}^{n}\) centered at 0 and of radius \(\varepsilon>0\).
Theorem 11
Let \(K\subset\mathbb{R}^{n}\) be nonempty and \(f: I\times K \to\mathbb {R}^{n}\). If the subset K is fractionally memo-viable with respect to f, then \(g(t_{0},x_{0})\in\mathcal{T}_{K}(m_{0})\), where \(x_{0}=x(t_{0})\) and \(m_{0}=m(t_{0},t_{0})= (I^{1-q}_{0+}x )(t_{0})\).
Proof
Corollary 12
Let \(K\subset K^{\varepsilon}\subset\mathbb{R}^{n}\) and \(f: I\times K\to \mathbb{R}^{n}\). If the subset K is fractionally memo-viable with respect to f, then \(g(t_{0},x_{0})\in\mathcal{T}_{K^{\varepsilon}}(\widetilde{m}_{0})\).
From Theorem 11 we get the following weaker result.
Corollary 13
Let \(K\subset\mathbb{R}^{n}\) be nonempty and \(f: I\times K\to\mathbb {R}^{n}\). If the subset K is fractionally memo-viable with respect to f, then \(f(t_{0},x_{0})\in\mathcal{T}_{K}(m_{0})\) or \(x(0^{+})\in\mathcal {T}_{K}(m_{0})\), where \(x_{0}=x(t_{0})\) and \(m_{0}=m(t_{0},t_{0})= (I^{1-q}_{0+}x )(t_{0})\).
Example 14
Let \(m_{0}=0\), then \(g(t_{0},x_{0})<0\), i.e., \(c+\frac {t_{0}^{-q}}{\Gamma(1-q)}x(0^{+})<0\). Let us notice that the term \(\frac {t_{0}^{-q}}{\Gamma(1-q)}\) is positive.
Finally, in order to prove the sufficient condition of viability, we need the following definition and proposition.
Definition 15
([11])
Let \(K\subset\mathbb{R}^{n}\) be nonempty and \(f: I\times K\to\mathbb {R}^{n}\). The subset K is fractionally viable with respect to f if for any \((t_{0},x_{0})\in I\times K\) equation (3) has at least one solution \(x:[t_{0},T]\to K\) satisfying \(x(t_{0})=x_{0}\). Such a solution we call viable with respect to f.
Proposition 16
([11])
Let \(K\subset \mathbb {R}^{n}\) be a nonempty and locally closed set, and let \(f:I\times K\rightarrow \mathbb {R}^{n}\) be a vector-valued continuous function. If \(f(t_{0},x_{0})\in\mathcal{T}_{K}(x_{0})\) for every \((t_{0}, x_{0})\in I\times K\), then K is fractionally viable with respect to f.
Theorem 17
Let \(K\subset K^{\varepsilon}\subset\mathbb{R}^{n}\) be nonempty and \(f: I\times K \to\mathbb{R}^{n}\). If the subset K is fractionally memo-viable with respect to f, then \(g(t_{0},x_{0})\in\mathcal{T}_{\tilde {K}^{\varepsilon}}(x_{0})\), where \(x_{0}=x(t_{0})\).
Proof
The next corollary gives, in fact, a sufficient condition of viability for an equation involving the Caputo derivative that is, however, slightly different from the one that we formulated in [11].
Corollary 18
Let \(K\subset K^{\varepsilon}\subset\mathbb{R}^{n}\) be nonempty and \(f: I\times\mathbb{R}^{n} \to\mathbb{R}^{n}\). Let \(\tilde{K}^{\varepsilon}=\frac {t_{0}^{q-1}}{\Gamma(q)}K^{\varepsilon}+\frac{1-q}{\Gamma(q+1)}t_{0}^{q} f(t_{0},x_{0})+\frac{1-q}{\Gamma(q+1)\Gamma(1-q)}x(0^{+})\). If \(g(t_{0},x_{0})\in \mathcal{T}_{\tilde{K}^{\varepsilon}}(x_{0})\), where \(x_{0}=x(t_{0})\), then \(\tilde{K}^{\varepsilon}\) is viable with respect to g.
5 Conclusions
Declarations
Acknowledgements
The authors are very grateful to anonymous reviewers for valuable suggestions and comments, which improved the quality of the paper. The work was supported by Bialystok University of Technology grant S/WI/2/2011.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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