Properties of interval-valued function space under the gH-difference and their application to semi-linear interval differential equations
- Juan Tao^{1} and
- Zhuhong Zhang^{2}Email author
https://doi.org/10.1186/s13662-016-0759-9
© Tao and Zhang 2016
Received: 5 October 2015
Accepted: 18 January 2016
Published: 9 February 2016
Abstract
The conventional subtraction arithmetic on interval numbers makes studies on interval systems difficult because of irreversibility on addition, whereas the gH-difference as a popular concept can ensure interval analysis to be a valuable research branch like real analysis. However, many properties of interval numbers still remain open. This work focuses on developing a complete normed quasi-linear space composed of continuous interval-valued functions, in which some fundamental properties of continuity, differentiability, and integrability are discussed based on the gH-difference, the gH-derivative, and the Hausdorff-Pompeiu metric. Such properties are adopted to investigate semi-linear interval differential equations. While the existence and uniqueness of the (i)- or (ii)-solution are studied, a necessary condition that the (i)- and the (ii)-solutions to be strong solutions is obtained. For such a kind of equation it is demonstrated that there exists at least a strong solution under certain assumptions.
Keywords
MSC
1 Introduction
In real-world engineering fields, many dynamic problems can be formulated by dynamic models, such as motor servo systems, navigation control, and so forth. However, this kind of system involves usually multiple uncertain parameters or interval coefficients [1], and thus interval analysis, developed by Moore [2] plays an important role in studying the existence and uniqueness of the solutions for interval differential equations (IDEs). Despite being initially introduced as an attempt to handle interval uncertainty, which appears in mathematical programming problems with bounded uncertain parameters (e.g. [3–5]), one such theory has been gradually applied to IDEs [6–22], due to the development of differential dynamics. To our knowledge, a great deal of work on interval theory was done by researchers many years ago, and on studies of fundamental arithmetic properties of the interval number [2, 23–30]. Especially, after Moore established the interval arithmetic rules [2], Oppenheimer and Michel [25] claimed subsequently that interval systems with the usual addition were a commutative semi-group but failed as a group. Unfortunately, since such systems are not an Abelian group for addition, interval arithmetic cannot yield the structure of a linear space. Generally, the arithmetic of addition is irreversible, namely for any two interval numbers a and b, if \(a+b=0\), b is not equivalent to −a usually, where \(0=[0,0]\). This way, whereas the difference of a and b can be defined by such a version of addition, many properties present in real analysis are not true in the context of interval number arithmetic, e.g. \(a-a\neq0\). In order to develop a useful theoretical framework on interval number like real number theory, Hukuhara [23] introduced another concept of interval difference for a and b in 1967, namely the Hukuhara difference (H-difference, \(a\ominus b\)), where \(a\ominus b=c\) if and only if \(a=b+c\). However, although such a concept can satisfy \(a\ominus a=0\), \(a\ominus b\) is meaningful only when \(w(a)\ge w(b)\), where \(w(a)\) and \(w(b)\) denote the widths of a and b, respectively. In order to overcome one such fault, after Markov [24] pointed out that the width of \(a-b\) was equal to the sum of the widths of a and b, he gave the concept of a nonstandard subtraction expressed also by the symbol of ‘−’. Such a concept can guarantee that \(a-a=0\), and the width of the interval \(a-b\) equals the absolute value of the difference of the widths of a and b. Thereafter, Stefanini [9, 27–29] extended the version of the H-difference to the concept of a generalized Hukuhara difference (gH-difference, \(a\ominus_{g}b\)) which coincided with the nonstandard subtraction operator introduced in Markov [24], Definition 1, p.326. Such a gH-difference has been comprehensively adopted to investigate interval dynamic systems, because apart from still satisfying \(a\ominus_{g}a=0\), the gH-difference always exists for any two intervals. Thus, it is an invaluable mathematical concept in probing interval number theory. In our last work [30], some fundamental arithmetic rules on interval numbers, based on the conventional addition and gH-difference were extended to the case of interval-valued vectors, while some properties, in particular associative and distributive laws, were obtained.
After the conventional subtraction arithmetic was generalized, multiple kinds of concepts of derivatives for interval-valued functions were reported [9, 16, 23, 24, 31–35]. Hukuhara [23] introduced the concept of H-differentiability for set-valued functions by using the concept of the H-difference. This is a starting point to study set, fuzzy and later interval differential equations. However, the H-derivative has some shortcomings which make it difficult to study the properties of interval-valued differential or integral equations, as the H-difference does not always exist for any two interval numbers. This limits its wide application to interval dynamic systems. Fortunately, based on the H-difference and the gH-difference, two recent generalized concepts of the GH-derivative [33] and gH-derivative [9] were introduced by Stefanini et al. Two such kinds of derivatives can be more comprehensively adopted to study IDEs by comparison with the H-derivative. Many valuable fundamental properties have been discovered by researchers [32, 34, 36]. We also note that there exist some intrinsic relationships between these two concepts, for example a GH-differential interval-valued function is usually a gH-differential under a few weak assumptions [9]. From the viewpoint of theoretical analysis, the gH-derivative of an interval-valued function at some point can be computed by only one formula, but the GH-derivative is opposite. Therefore, in comparison with the concept of the GH-derivative, the version of the gH-derivative will become more and more focused upon in the coming theoretical research on IDEs, demonstrated by some recent results [9, 14–16, 22].
Recently, several researchers have paid great attention to studies on the properties of interval-valued functions, in particular continuity, differentiability, and integrability. Some reported representative achievements promote the importance of interval dynamics. Chalco-Cano et al. [32, 34, 35] made hard efforts to study systematically some relationships among GH-, gH-, Markov, and π-differentiability [9, 24, 33, 37]. They claimed that (i) if an interval-valued function f was GH-differentiable, then it was π-differentiable, and (ii) if f was π-differentiable, then it was gH-differentiable. They also derived several Ostrowski inequalities capable of being used for studying IDEs’ solution estimates, relying upon the concept of the gH-derivative [32]. The concepts of the GH-derivative and the gH-derivative are usually utilized to define the types of the solutions for IDEs [9–13, 16, 17, 22]. However, since many arithmetic properties of real number theory are not true in the branch of interval analysis, it is extremely difficult to probe IDEs’ theoretical foundations. Even so, some pioneering works on the existence and uniqueness of the solutions are gaining great interest among researchers. Theoretically, studies on IDEs depend greatly on the type of interval-valued derivative, as different concepts of derivatives require that IDEs satisfy different conditions so as to ensure IDEs’ solution existence and uniqueness.
More recently, several special IDEs were defined based on GH-differentiability, and then transformed into integral equations with the H-difference [10–13, 16–20, 38]. Their solution properties, including the existence, uniqueness, and continuous dependence, have been well investigated by some researchers. Malinowski [10, 11] made great contributions to analyzing a kind of IDE, depending on the second type of Hukuhara derivative included in the concept of GH-derivative. Subsequently, some important properties of the solutions were found such as the existence of local solutions, convergence, and continuous dependence of the solution on initial value and right-hand side of the equation. Skripnic [17] proved the existence of the solutions of IDEs by virtue of the Caratheodory theorem and the concept of generalized differentiability [39], in which the version of derivative was equivalent to that of GH-derivative. Additionally, based on the GH-derivative, Ngo et al. [12, 13, 18–20] carried out a series of studies for multiple kinds of IDEs such as interval-valued integro-differential equations, interval-valued functional differential equations, and so on. They obtained some significant conclusions as regards the existence of the solutions, by developing comparison theorems.
On the other hand, IDEs have also been well studied based on the concept of the gH-difference in the recent years [9, 14–17, 21, 22, 32, 40]. Stefanini and Bede [9] gave the existence and uniqueness of two types of local solutions for an initial valued IDE with a gH-derivative, and meanwhile the characteristics of the solutions were found. After that, they also carried out an experimental analysis of such a kind of IDE [14]. Especially, Chalco-Cano et al. [22] investigated exhaustively the properties of an interval-valued function expressed by \(Cg(t)\) with interval number C and real single-valued function g. They also derived out the representation of the solutions for a class of linear initial valued IDEs. In addition, Lupulescu [16] proposed the concepts of differentiability and integrability for the interval-valued functions on time scales, while the properties of the delta generalized Hukuhra derivative and integration of interval-valued functions on time scales were studied. An illustrative example of an IDE on time scale was also given. Lupulescu [40] also use the gH-difference to develop a theory of the fractional calculus for interval-valued functions, and it is the foundation of interval-valued fractional differential equations.
Summarizing, interval differential dynamic systems are a still open research topic in the context of differential dynamic systems. Three fundamental issues are under consideration: (i) how to define and analyze the space of the solutions, (ii) whether some classical conclusions such as fixed point theorems in the branch of classical functional analysis can be adapted to IDEs, and (iii) how to derive analytic solutions or numerical ones for IDEs. Thus, in this paper we probe into the existence and uniqueness of the solutions for a class of semi-linear interval dynamic systems, after developing a complete normed quasi-linear space. Most precisely, we first give a quasi-linear space on interval number and a related continuous interval-valued function space, and meanwhile their properties are sufficiently discussed. Second, an important and classical fixed point theorem is generalized to the interval-valued case so as to discover IDEs’ properties. Finally, we conclude that there exists at least a strong solution for a kind of semi-linear IDE as considered in this work.
2 Preliminaries and basic properties of gH-difference
- (i)
\(a+b=[a^{L}+b^{L},a^{R}+b^{R}]\);
- (ii)
\(ka=\bigl \{\scriptsize{ \begin{array}{l@{\quad}l} {[ka^{L},ka^{R}]}, & k \geq0, \\ {[ka^{R},ka^{L}]}, & k < 0; \end{array}} \bigr.\)
- (iii)
\(a-b=a+(-1)b=[a^{L}-b^{R},a^{R}-b^{L}]\);
- (iv)
\(ab=[\min\{u\in A\}, \max\{u\in A\}]\), where \(A=\{ a^{L}b^{L},a^{L}b^{R},a^{R}b^{L},a^{R}b^{R}\}\);
- (v)
\(|a|= \max\{|a^{L}|,|a^{R}|\}\), \(w(a)=a^{R}-a^{L}\);
- (vi)
\(a\leq b \Leftrightarrow a^{L}\leq b^{L}\), \(a^{R}\leq b^{R}\).
In general, \(a-a\) does not equal 0 except that a is a degenerate interval. This indicates that the subtraction is not the inverse of Minkowski addition above. However, the cancellation law of addition on interval numbers holds, i.e., \(a+c=b+c\) if and only if \(a=b\). Since \(a-a\neq0\), many properties of the real number theory cannot be extended to interval analysis. So, Hukuhara [23] introduced another concept of subtraction in order to overcome this drawback. He defined the H-difference (i.e., \(a\ominus b\)) of a and b as c if \(a=b+c\), namely \(a\ominus b=c\). Although such a subtraction can yield \(a\ominus a=0\), \(a\ominus b\) exists only when \(w(a)\geq w(b)\). Subsequently, Stefanini [27–29] proposed a more general concept of subtraction as below.
Definition 2.1
([29])
- (i)
\(a\ominus_{g}b=[\min\{a^{L}-b^{L},a^{R}-b^{R}\},\max\{a^{L}-b^{L},a^{R}-b^{R}\}]\);
- (ii)
\(a\ominus_{g}a=0\), \(a\ominus_{g}0=a\), \(0\ominus_{g}a=(-1)a\);
- (iii)
\((-a)\ominus_{g}b=(-b)\ominus_{g}a\);
- (iv)
\(a\ominus_{g}b=(-b)\ominus_{g}(-a)=-(b\ominus_{g}a)\);
- (v)
\((a+b)\ominus_{g}b=a\), \(a\ominus_{g}(a+b)=-b\);
- (vi)
\((a\ominus_{g}b)+b=a\), if \(w(a)\geq w(b)\); \(a+(-1)(a\ominus_{g}b)=b\), if \(w(a)< w(b)\);
- (vii)
\(k(a\ominus_{g}b)=ka\ominus_{g}kb\), \(k\in R\).
- (i)
\((a+b)\ominus_{g}c=a+b\ominus_{g}c\) if and only if \(w(c)\leq w(b)\) with \(c\in IR\);
- (ii)\(a(b\ominus_{g}c)=ab\ominus_{g}ac\), if b and c are symmetric, or one of the following conditions holds:
- (a)
\(w(b)\geq w(c)\), and \(0\leq c\leq b\), \(b\leq c\leq0\);
- (b)
\(w(b)\leq w(c)\), and \(0\leq b\leq c\), \(c\leq b\leq0\).
- (a)
In the present work, in terms of the concept of the gH-difference, we obtain some properties summed up below.
Lemma 2.2
- (i)
\(a\ominus_{g}b=0\) if and only if \(b=a\);
- (ii)
\((a+b)\ominus_{g}(a+c)=b\ominus_{g}c\);
- (iii)
\((a\ominus_{g}b)\ominus_{g}(a\ominus_{g}c)=c\ominus_{g}b\), if \(w(a)\leq \min(w(b),w(c))\) or \(w(a)\geq\max(w(b),w(c))\).
Proof
Case (i) is true by the definition of the gH-difference.
Case (ii): write \((a+b)\ominus_{g}(a+c)=d\). Based on the gH-difference, we have \(a+b=a+c+d\) or \(a+c=a+b+(-1)d\). Hence, it follows from the cancellation law of addition on interval number that \(b=c+d\) or \(c=b+(-1)d\). This illustrates that \(b\ominus_{g}c=d\).
Case (iii): write \(a\ominus_{g}b=e\), \(a\ominus_{g}c=f\), and \(e\ominus _{g}f=g\). When \(w(a)\leq\min(w(b),w(c))\), we note that \(b=a+(-1)e\) and \(c=a+(-1)f\). Thus, if \(w(e)\geq w(f)\), then \(e=f+g\), and hence \(a+(-1)e=a+(-1)f+(-1)g\). This yields \(b=c+(-1)g\). On the other hand, if \(w(e)< w(f)\), we have \(f=e+(-1)g\), and hence \(a+(-1)f=a+(-1)e+g\). Thus one derives that \(c=b+g\). In total, we obtain \(g=c\ominus_{g}b\). Similarly, when \(w(a)\geq\max(w(b),w(c))\), it follows that \(a=b+e\) and \(a=c+f\). If \(w(e)\geq w(f)\), one can derive that \(c+f=b+e=b+f+g\), that is, \(c=b+g\). On the other hand, if \(w(e)< w(f)\), we have \(b+e=c+f=c+e+(-1)g\), and then \(b = c+(-1)g\). Thus we also get \(g=c\ominus_{g}b\). □
For the convenience of notation, write \(r_{ab}=w(a)-w(b)\) and \(r_{bc}=w(b)-w(c)\). We obtain the following properties.
Lemma 2.3
Proof
Write \(a\ominus_{g}b=d\) and \(b\ominus_{g}c=d_{1}\). If \(r_{ab}\geq0\), then \(a=b+d\); if \(r_{ab}<0\), then \(b=a+(-1)d\). Similarly, if \(r_{bc}\geq0\), then \(b=c+d_{1}\); if \(r_{bc}<0\) then \(c=b+(-1)d_{1}\).
Case (i): write \(d\ominus_{g}c=e_{1}\). By definition, it implies that \(d=c+e_{1}\) if \(w(d)\geq w(c)\), and \(c=d+(-1)e_{1}\) if \(w(d)< w(c)\). In the case of \(r_{ab}\geq0\), if \(w(d)\geq w(c)\), one gets that \(a=b+d=b+c+e_{1}\); and if \(w(d)< w(c)\), then \(b+c=b+d+(-1)e_{1}=a+(-1)e_{1}\). Therefore, we have \(e_{1}=a\ominus_{g}(b+c)\). Conversely, in the case of \(r_{ab}<0\), if \(w(d)\geq w(c)\), then \(b=a+(-1)d=a+(-1)c+(-1)e_{1}\), and if \(w(d)< w(c)\), then \(a+(-1)c=a+(-1)d+e_{1}=b+e_{1}\). This indicates that \(e_{1}=(a+(-1)c)\ominus_{g}b\).
Case (ii): write \(a\ominus_{g}d_{1}=e_{2}\). We can obtain \(a=d_{1}+e_{2}\) if \(w(a)\geq w(d_{1})\), and \(d_{1}=a+(-1)e_{2}\) if \(w(a)< w(d_{1})\). In the case of \(r_{bc}\geq0\), if \(w(a)\geq w(d_{1})\), one can derive that \(a+c=c+d_{1}+e_{2}=b+e_{2}\); if \(w(a)< w(d_{1})\), then \(b=c+d_{1}=a+c+(-1)e_{2}\). These two equalities follow from \(e_{2}=(a+c)\ominus_{g}b\). On the other hand, in the case of \(r_{bc}<0\), if \(w(a)\geq w(d_{1})\), then \(b+(-1)a=b+(-1)d_{1}+(-1)e_{2}=c+(-1)e_{2}\); if \(w(a)< w(d_{1})\), then \(c=b+(-1)d_{1}=b+(-1)a+e_{2}\). Thus, we get \(e_{2}=c\ominus_{g}(b+(-1)a)\).
Case (iii): write \(a\ominus_{g}(b+c)=e_{3}\). The first equality is the same as the first one of case (i). We only need to demonstrate the second one. To this end, if \(r_{ab}<0\), it is obvious that \(w(a)\leq w(b+c)\). This means that \(b+c=a+(-1)e_{3}\). Therefore, \((a+(-1)d)+c=a+(-1)e_{3}\). Hence, we get \(e_{3}=d+(-1)c\).
Case (iv): in the case of \(r_{ab}\geq0\), we know that \(a=b+d\), which yields \(a+c=b+c+d\); again since \(w(a+c)\geq w(b)\), we get \(d+c=(a+c)\ominus_{g}b\). Conversely, in the case of \(r_{ab}<0\), we note that \(w(a)< w(b+(-1)c)\) and \(b=a+(-1)d\), which illustrates that \(b+(-1)c=a+(-1)(d+c)\). Thus, \(d+c=a\ominus_{g}(b+(-1)c)\). This completes the proof. □
3 Normed quasi-linear space
3.1 Interval number space
In this section, we first develop a quasi-linear space on IR, and then we analyze its properties under the gH-difference, by introducing the Hausdorff-Pompeiu metric on interval numbers. For \(a,b,c\in IR\), and \(k,l\in R\), the addition and scalar multiplication have some well-known properties: (i) \(a+b=b+a\), (ii) \(a+(b+c)=(a+b)+c\), (iii) \(a+0=a\), (iv) \(k(a+b)=ka+kb\), (v) \(k(la)=(kl)a\), (vi) \(1a=a\). Unfortunately, there usually does not exist \(d\in IR\) s.t. \(a+d=0\), and the equality \((k+l)a=ka+la\) is true only when \(kl\geq0\) [26]. For example, take \(a=[1,2]\), \(b=[-2,-1]\), \(k=1\), and \(l=-1\). Obviously, one can find that \((k+l)a=0\) and \(ka+la=[2, 4]\). Thus, \((k+l)a\neq ka+lb\); on the other hand, if \(a+b=0\), then \(1+b^{L}=0\) and \(2+b^{R}=0\), and hence \(b=[-1, -2]\), which yields a contradiction.
- (i)
\(H(a+b,a+c)=H(b,c)\);
- (ii)
\(H(ka,kb)=|k|H(a,b)\), where \(k\in R\);
- (iii)$$ H(a+b,c+d)\leq H(a,c)+H(b,d); $$(3.3)
- (iv)
if \(a\ominus b\), \(a\ominus c\) exist, then \(H(a\ominus b, a\ominus c)=H(b,c)\);
- (v)
if \(a\ominus b\), \(c\ominus d\) exist, then \(H(a\ominus b, c\ominus d)=H(a+d,b+c)\).
Notice that equations (iv) and (v) are true only when \(a\ominus b\), \(a\ominus c\), and \(c\ominus d\) exist. We next identify whether the two equations of (iv) and (v) above hold after replacing ⊖ by \(\ominus_{g}\). For convenience of the representation, write \(r_{ab}=w(a)-w(b)\), \(r_{ac}=w(a)-w(c)\), and \(r_{cd}=w(c)-w(d)\) with \(a,b,c,d\in IR\).
Lemma 3.1
Proof
The above lemma can be illustrated by taking \(a=[1,3]\), \(b=[-2,-1]\), and \(c=[2,6]\). Through simple inference, we obtain that \(H(a\ominus_{g}b, a\ominus_{g}c)=H([3,4],[-3,-1])=6\), and \(H(b,c)=H([-2,-1],[2,6])=7\). So, equation (3.4) is true.
Lemma 3.2
Proof
For example, take \(a=[1,2]\), \(b=[3,5]\), \(c=[2,6]\), and \(d=[-2,-1]\). We can see that \(H(a\ominus_{g}b,c\ominus_{g}d)=H([-3,-2],[4,7])=9\), and \(H(a+d,b+c)=H([-1,1],[5,11])=10\). Hence, equation (3.5) is valid.
Based on the above Hausdorff-Pompeiu metric, define \(\|a\|_{I}=H(a,0)\) with \(a\in IR\). Further, by simple inference we notice that \(\|\cdot \|_{I}\) satisfies the basic properties of the classical concept of norm. Therefore, IR can be naturally said to be a normed quasi-linear space.
Theorem 3.4
- (i)
\(\|a\ominus_{g}b\|_{I}=H(a,b)\);
- (ii)
\(\|a\|_{I}-\|b\|_{I}\leq\|a\ominus_{g}b\|_{I}\leq\|a\|_{I}+\|b\|_{I}\);
- (iii)
\(\|ab\|_{I}=\|a\|_{I}\|b\|_{I}\).
Proof
Take \(a=[-2,-1]\) and \(b=[1,3]\). Then \(\|a\ominus _{g}b\|_{I}=\|[-4,-3]\|_{I}=4\) and \(H(a,b)=4\). Further, \(\|a\|_{I}-\|b\|_{I}=-1\), \(\|a\|_{I}+\|b\|_{I}=5\), \(\|ab\|_{I}=\|[-6,-1]\|_{I}=6\), and \(\|a\|_{I}\|b\|_{I}=6\). Thus, the above conclusions in Theorem 3.4 hold.
We next discuss the completeness of the normed quasi-linear space IR, where a version of interval convergence is given.
Definition 3.5
For \(a_{n}, a\in IR\), \(n=1,2,\ldots\) , if \(\|a_{n}\ominus_{g}a\|_{I}\rightarrow 0\) as \(n\rightarrow\infty\), \(\{a_{n}\}_{n\geq1}\) is said to be convergent to a (simply written as \(\lim_{n\rightarrow\infty}a_{n}=a\)).
Similarly, we introduce the version of Cauchy convergence in IR. That is, \(\{a_{n}\}_{n\geq1}\) is convergent if and only if \(\|a_{n}\ominus _{g}a_{m}\|_{I}\rightarrow0\) as \(m,n\rightarrow\infty\). It is easy to prove that \((IR,\|\cdot\|_{I})\) is complete by means of the completeness of \((IR,H)\).
Theorem 3.6
\((IR,\|\cdot\|_{I})\) is a complete normed quasi-linear space.
Proof
Let \(\{a_{n}\}_{n\geq1}\) be an arbitrary Cauchy sequence in \((IR,\|\cdot \|_{I})\). Since \(H(a_{n},a_{m})=\|a_{n}\ominus_{g}a_{m}\|_{I}\), we obtain \(H(a_{n},a_{m})\rightarrow0\) as \(m,n\rightarrow\infty\). Therefore, \(\{ a_{n}\}_{n\geq1}\) is a Cauchy sequence in \((IR,H)\) and, accordingly, there exists \(a\in IR\) such that \(H(a_{n},a)\rightarrow0\) as \(n\rightarrow\infty\), due to the completeness of \((IR,H)\). This implies that \(\|a_{n}\ominus_{g}a\|_{I}\rightarrow0\) as \(n\rightarrow\infty\). □
3.2 Interval-valued function space
- (i)
\((f+g)(t)=f(t)+g(t)\);
- (ii)
\((kf)(t)=kf(t)\), \(k\in R\);
- (iii)
\((f\ominus_{g}g)(t)=f(t)\ominus_{g}g(t)\);
- (iv)
\((fg)(t)=f(t)g(t)\).
Under these arithmetic rules, we discuss some basic properties in \(C(I,IR)\).
Theorem 3.7
If \(f,g\in C(I,IR)\), then kf, \(f+g\), \(f\ominus_{g}g\), and fg are continuous on I.
Proof
- (i)
\(\rho(f,g)\geq0\); \(\rho(f,g)=0\) if and only if \(f=g\);
- (ii)
\(\rho(f,g)=\rho(g,f)\);
- (iii)
\(\rho(f,g)\leq\rho(f,h)+\rho(h,g)\).
- (i)
\(\rho(f+\varphi,f+\psi)=\rho(\varphi,\psi)\);
- (ii)
\(\rho(kf,kg)=|k|\rho(f,g)\), where \(k\in R\);
- (iii)
\(\rho(f\varphi,f\psi)=\rho(0,f)\rho(\varphi,\psi)\);
- (iv)
\(\rho(f+g,\varphi+\psi)\leq\rho(f,\varphi)+\rho(g,\psi)\);
- (v)
\(\rho(f\ominus_{g}\varphi,f\ominus_{g}\psi)\leq\rho(\varphi,\psi)\);
- (vi)
\(\rho(f\ominus_{g}g,\varphi\ominus_{g}\psi)\leq\rho(f+\psi,g+\varphi)\).
We further discuss some properties of \(C(I,IR)\) useful for studying the properties of IDEs. To this point, introduce the version of convergence of an interval-valued function sequence. For \(f_{n}, f\in C(I,IR)\), \(n=1, 2,\ldots\) , if \(\rho(f_{n},f)\rightarrow0\) as \(n\rightarrow\infty\), \(\{ f_{n}\}_{n\geq1}\) is said to be convergent to f. Similarly, we say that \(\{f_{n}\}_{n\geq1}\) is a Cauchy sequence if \(\rho(f_{n},f_{m})\rightarrow0\) as \(m,n\rightarrow\infty\).
Theorem 3.8
The quasi-linear space \((C(I,IR),\rho)\) is complete.
Proof
Like the above normed quasi-linear space IR, we can introduce the version of a norm on \(C(I,IR)\), namely \(\|f\|_{C}=\rho(f,0)\). By means of the Hausdorff-Pompeiu metric on interval numbers above, one can see that \((C(I,IR), \|\cdot\|_{C})\) is a normed quasi-linear space. We also notice that \(|f(t)|=H(f(t),0)\). Therefore, we can rewrite \(\|f\|_{C}\) as \(\sup_{t\in I}|f(t)|\). Additionally, by means of Theorems 3.4 and 3.8, the following basic properties are valid.
Theorem 3.9
- (i)
\(\|f\ominus_{g}g\|_{C}=\rho(f,g)\);
- (ii)
\(\|f\|_{C}-\|g\|_{C}\leq\|f\ominus_{g}g\|_{C}\leq\|f\|_{C}+\|g\|_{C}\);
- (iii)
\(\|fg\|_{C}\leq\|f\|_{C}\|g\|_{C}\);
- (iv)
\((C(I,IR),\|\cdot\|_{C})\) is a complete normed quasi-linear space.
We next develop a fixed point theorem under the gH-difference. x is said to be a fixed point of a mapping \(T:C(I,IR)\rightarrow C(I,IR)\) if \(Tx=x\). We say that T is a contraction mapping on \(C(I,IR)\), if there exists a real number α with \(0<\alpha<1\) such that \(\|Tx\ominus _{g}Ty\|_{C}\leq\alpha\|x\ominus_{g}y\|_{C}\) for any \(x,y\in C(I,IR)\). Similar to the process of the proof as in the classical principle of contraction mapping, we obtain a fixed point theorem as below.
Theorem 3.10
If \(T:C(I,IR)\rightarrow C(I,IR)\) is a contraction mapping, there exists a fixed point.
4 The properties of gH-differentiability
Since the addition arithmetic on interval number is irreversible, the concept of a derivative of an interval-valued function has been gaining great concern among researchers. As to this point, Hukuhara [23] introduced the concept of the H-derivative related to the version of the H-difference. However, as we mention in Section 1, this concept cannot be comprehensively adopted to investigate the properties of interval-valued functions, as it cannot ensure that the H-difference exists for any two interval numbers. Thereafter, Bede and Gal [33] generalized such concept of a derivative to the version of a GH-derivative. However, the latter concept is relatively more useful, it still needs the same basic assumptions as the concept of H-derivative. Fortunately, Stefanini and Bede [9] proposed a more general concept of derivative (i.e., the gH-derivative) by comparison with the GH-derivative. The main merit consists in the fact that their concept is similar to the version of the derivative of a real-valued function.
Definition 4.1
([9])
Theorem 4.2
f is gH-differentiable in \(t\in I\) if and only if \(f_{+}'(t)\) and \(f_{-}'(t)\) exist and \(f_{+}'(t)=f_{-}'(t)\).
Proof
Based on the concept of the gH-derivative, Stefanini et al. developed the relationship between gH-derivative and conventional derivatives, in other words, the gH-derivative of f can be expressed by the derivatives of its endpoint functions.
Theorem 4.3
([9])
It should be pointed out that usually one can only find that \((f+g)' \subseteq f'+g'\) when f and g are differentiable [9]. However, in some weak assumptions, the symbol of inclusion can be replaced by the symbol of equality. To this end, for convenience of notation below, we write \(\omega(t)=f(t)\ominus_{g}g(t)\), \(u(t,h)=f(t+h)\ominus_{g}f(t)\), and \(v(t,h)=g(t+h)\ominus_{g}g(t)\) with \(t+h\in I\).
Theorem 4.4
Proof
Theorem 4.5
- (i)
f is (i)-differentiable and g is (ii)-differentiable;
- (ii)
f is (ii)-differentiable and g is (i)-differentiable.
Proof
Notice that when f and g are both (i)-differentiable or both (ii)-differentiable, equation (4.9) is not true, which can be illustrated by a simple example as below.
Example 4.6
Take \(f(t)=[t,2t+1]\) and \(g(t)=[t,3t+1]\) with \(0\leq t\leq1\). Then \(f(t)\ominus_{g}g(t)=[-t,0]\). Again f and g are (i)-differentiable. One can know that \((f(t)\ominus_{g}g(t))'=[-t,0]'=[-1,0]\). However, \(f'(t)+(-1)g'(t)=[1,2]+(-1)[1,3]=[-2,1]\). Thus, \((f(t)\ominus _{g}g(t))'\neq f'(t)+(-1)g'(t)\).
As we know, if two real scalar functions are differentiable, their product function is also differentiable. However, for two given interval-valued functions, even if they are all differentiable, their product function is not differentiable usually. This can be illustrated by an example as below.
Example 4.7
In the following subsection, we degenerate the interval-valued function f into a scalar function, and we study some properties of the product function of fg. In addition to the notations presented in Theorems 4.4 and 4.5, write \(W(t,h)=f(t+h)g(t+h)\ominus _{g}f(t)g(t)\), \(U(t,h)=f(t+h)-f(t)\).
Theorem 4.8
Proof
Theorem 4.9
Proof
Similarly, when g is (ii)-differentiable, we can obtain the following properties of fg according to the sign of \(f(t)f'(t)\), for which their proofs are omitted.
Theorem 4.10
5 Integral of interval-valued function
Correspondingly, some fundamental properties have been studied.
Theorem 5.1
([41])
- (i)
\(\int_{t_{0}}^{t_{f}}(f(t)+g(t))\, dt=\int_{t_{0}}^{t_{f}}f(t)\, dt+\int _{t_{0}}^{t_{f}}g(t)\, dt\);
- (ii)
\(\int_{t_{0}}^{t_{f}}f(t)\, dt=\int_{t_{0}}^{\tau}f(t)\, dt+\int_{\tau }^{t_{f}}f(t)\, dt\), \(t_{0}<\tau<t_{f}\).
Theorem 5.2
([9])
- (i)
\(F(t)\) is gH-differentiable, and \(F'(t)=f(t)\), where \(F(t)=\int _{t_{0}}^{t}f(t)\, dt\);
- (ii)
\(G(t)\) is gH-differentiable, and \(G'(t)=-f(t)\), where \(G(t)=\int _{t}^{t_{f}}f(t)\, dt\).
We next present two important integral properties helpful for discussing the following IDE.
Theorem 5.3
Proof
Theorem 5.4
Proof
In terms of Lemma 3.3 and Theorem 5.4, one can easily gain the following conclusion.
Corollary 5.5
6 Interval differential equation
Definition 6.1
For given \(x\in C(J,IR)\), x is continuous gH-differentiable on J if \(x'\) is continuous.
Definition 6.2
Let x be continuous gH-differentiable on J. x is a strong solution of SIDE if satisfying the initial condition and the above equation.
Definition 6.3
Theorem 6.4
Proof
In the subsequent subsection, we first give a prior estimate of the solution, and then discuss the existence and uniqueness of strong solutions of SIDE.
Hypothesis 6.1
Lemma 6.5
Proof
Hypothesis 6.2
In the above SIDE problem, when \(a(t)\equiv0\), Stefanini and Bede [9] proved that there exist only two strong solutions under some limitations. We here discuss SIDE’s existence and uniqueness of strong solutions under \(a(t)\neq0\).
Theorem 6.6
Proof
In the above theoretical analysis, from Theorem 6.4 one draws the conclusion that the (i)- and the (ii)-solutions are strong solutions under certain conditions; Theorem 6.6 shows the conditions of existence and uniqueness of (i)- and the (ii)-solutions for equations (6.2) and (6.3), respectively. These hint that SIDE has at least a strong solution under certain assumptions, for which we give a conclusion to emphasize the existence of strong solutions in SIDE.
Theorem 6.7
Let \(a(t)\) be a continuous scalar function with at most countable zero points on J. Then SIDE has at least a strong solution under Theorems 6.4 and 6.6.
Proof
7 Illustrative examples
In this section, our experiments are implemented through MATLAB’s 7.0 standard ODE solver (ode45). Three simple interval-valued Cauchy problems are used to examine our theoretical results.
Example 7.1
([9])
By Figure 1, solutions x and \(x_{2}\) have the same switching point \(t_{\alpha}=1.3606\), and meanwhile are almost the same in \([0,t_{\alpha}]\). However, they present different characteristics within \(t_{\alpha}\) and 4, as \(w(x(t))\) is decreasing but \(w(x_{2}(t))\) is increasing. This indicates the uncertainty degree of x is ever smaller with time t, and thereby x is better than \(x_{2}\). In addition, whereas \(x_{2}\) is of a smaller uncertainty degree than \(x_{1}\), it will become divergent with time t. In total, \(x_{1}\) and \(x_{2}\) are not rational because of their divergence.
Example 7.2
By Figure 2, solutions x and \(x_{1}\) are almost the same in \([0,\pi]\). However, \(w(x(t))\) is decreasing but \(w(x_{1}(t))\) is increasing in \((\pi,2\pi]\). This indicates the uncertainty degree of x is smaller than that of \(x_{1}\) in \((\pi,2\pi]\), and thus x is superior to \(x_{1}\). On the other hand, \(x_{2}\) is of a smaller uncertainty degree than x and \(x_{1}\), as \(w(x_{2}(t))\) remains decreasing. It is emphasized that I- and the II-type solutions are two kinds of extreme solutions decided under (i)-differentiability and the (ii)-differentiability, whereas through Theorem 6.7, we can obtain a more rational strong solution which switches between the (i)- and the (ii)-solutions.
Example 7.3
- (i)For any \((t,x)\in J\times IR\), sinceTherefore, Hypothesis 6.1 holds.$$\bigl\Vert f(t,x)\bigr\Vert _{I}=H\bigl(f(t,x),0\bigr)=\max\bigl\{ \bigl\vert \bigl(x^{L}\bigr)^{3}\varphi(t)\bigr\vert , \bigl\vert \bigl(x^{R}\bigr)^{3}\varphi (t)\bigr\vert \bigr\} \leq\|x\|_{I}^{3}. $$
- (ii)For \(\|x\|_{C}, \|y\|_{C}\leq M_{\alpha}\), we haveThis illustrates that Hypothesis 6.2 is true.$$\begin{aligned} H\bigl(f(t,x),f(t,y)\bigr) =& \max\bigl\{ \bigl\vert \bigl(x^{L} \bigr)^{3}\varphi(t)-\bigl(y^{L}\bigr)^{3}\varphi (t)\bigr\vert ,\bigl\vert \bigl(x^{R}\bigr)^{3} \varphi(t)-\bigl(y^{R}\bigr)^{3}\varphi(t)\bigr\vert \bigr\} \\ \leq& \max\bigl\{ \bigl\vert \bigl(x^{L}\bigr)^{3}- \bigl(y^{L}\bigr)^{3}\bigr\vert ,\bigl\vert \bigl(x^{R}\bigr)^{3}-\bigl(y^{R} \bigr)^{3}\bigr\vert \bigr\} \\ \leq& 3M_{\alpha}^{2}H(x,y). \end{aligned}$$
By Figure 3, the solutions x and \(x_{1}\) are almost the same on \([0,\frac{\pi}{2}]\). \(w(x(t))\) is increasing in \([0,\frac{\pi}{2}]\), decreasing in \((\frac{\pi}{2},\frac{3\pi}{2}]\), and increasing in \((\frac{3\pi}{2},2\pi]\). On the other hand, \(w(x_{1}(t))\) always remains increasing, and \(w(x_{2}(t))\) keeps decreasing. This shows that the I-type solution \(x_{1}\) diverges and the II-type solution \(x_{2}\) is very conservative. Thus, \(x_{1}\) and \(x_{2}\) cannot effectively reflect the dynamic characteristics of the above dynamic system, but x is opposite.
8 Conclusions
This work aims at studying the properties of interval-valued functions under the gH-difference and also probing the existence of the solutions for a class of semi-linear interval differential equations. As associated to the concept of the gH-difference and also the conventional arithmetic rules such as addition and scalar multiplication, we have developed a complete normed quasi-linear space on interval numbers, in which some important properties of the gH-difference are found under the Hausdorff-Pompeiu metric. Subsequently, on the basis of one such space, we introduced a continuous interval-valued function space which has been proven to be a complete normed quasi-linear space. A contracting mapping theorem on one such space, similar to the classical contracting mapping principle, has been obtained, relying upon the gH-difference. Based on these fundamental works, some arithmetic properties of the gH-derivative for interval-valued functions were investigated exhaustively, among which some results can be adopted to study the existence and uniqueness of the solutions for such a kind of semi-linear equation. After some simple properties of the integral of interval-valued functions were discussed, we have obtained a necessary condition that the (i)- and the (ii)-solutions are strong solutions, including the conditions of the existence and uniqueness of the solutions.
Declarations
Acknowledgements
This work is supported by the Doctoral Fund of Ministry of Education of China (20125201110003), National Natural Science Foundation (61563009). The authors would like to thank the editor in chief and the anonymous referee, for their comments and suggestions that greatly improved the paper.
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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