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Properties of intervalvalued function space under the gHdifference and their application to semilinear interval differential equations
Advances in Difference Equations volume 2016, Article number: 45 (2016)
Abstract
The conventional subtraction arithmetic on interval numbers makes studies on interval systems difficult because of irreversibility on addition, whereas the gHdifference 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 quasilinear space composed of continuous intervalvalued functions, in which some fundamental properties of continuity, differentiability, and integrability are discussed based on the gHdifference, the gHderivative, and the HausdorffPompeiu metric. Such properties are adopted to investigate semilinear 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.
1 Introduction
In realworld 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 semigroup 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. \(aa\neq0\). 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 (Hdifference, \(a\ominus b\)), 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 \(ab\) 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 \(aa=0\), and the width of the interval \(ab\) equals the absolute value of the difference of the widths of a and b. Thereafter, Stefanini [9, 27–29] extended the version of the Hdifference to the concept of a generalized Hukuhara difference (gHdifference, \(a\ominus_{g}b\)) which coincided with the nonstandard subtraction operator introduced in Markov [24], Definition 1, p.326. Such a gHdifference has been comprehensively adopted to investigate interval dynamic systems, because apart from still satisfying \(a\ominus_{g}a=0\), the gHdifference 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 gHdifference were extended to the case of intervalvalued 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 intervalvalued functions were reported [9, 16, 23, 24, 31–35]. Hukuhara [23] introduced the concept of Hdifferentiability for setvalued functions by using the concept of the Hdifference. This is a starting point to study set, fuzzy and later interval differential equations. However, the Hderivative has some shortcomings which make it difficult to study the properties of intervalvalued differential or integral equations, as the Hdifference does not always exist for any two interval numbers. This limits its wide application to interval dynamic systems. Fortunately, based on the Hdifference and the gHdifference, two recent generalized concepts of the GHderivative [33] and gHderivative [9] were introduced by Stefanini et al. Two such kinds of derivatives can be more comprehensively adopted to study IDEs by comparison with the Hderivative. 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 GHdifferential intervalvalued function is usually a gHdifferential under a few weak assumptions [9]. From the viewpoint of theoretical analysis, the gHderivative of an intervalvalued function at some point can be computed by only one formula, but the GHderivative is opposite. Therefore, in comparison with the concept of the GHderivative, the version of the gHderivative 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 intervalvalued functions, in particular continuity, differentiability, and integrability. Some reported representative achievements promote the importance of interval dynamics. ChalcoCano et al. [32, 34, 35] made hard efforts to study systematically some relationships among GH, gH, Markov, and πdifferentiability [9, 24, 33, 37]. They claimed that (i) if an intervalvalued function f was GHdifferentiable, then it was πdifferentiable, and (ii) if f was πdifferentiable, then it was gHdifferentiable. They also derived several Ostrowski inequalities capable of being used for studying IDEs’ solution estimates, relying upon the concept of the gHderivative [32]. The concepts of the GHderivative and the gHderivative 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 intervalvalued 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 GHdifferentiability, and then transformed into integral equations with the Hdifference [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 GHderivative. 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 righthand 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 GHderivative. Additionally, based on the GHderivative, Ngo et al. [12, 13, 18–20] carried out a series of studies for multiple kinds of IDEs such as intervalvalued integrodifferential equations, intervalvalued 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 gHdifference 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 gHderivative, 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, ChalcoCano et al. [22] investigated exhaustively the properties of an intervalvalued function expressed by \(Cg(t)\) with interval number C and real singlevalued 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 intervalvalued functions on time scales, while the properties of the delta generalized Hukuhra derivative and integration of intervalvalued functions on time scales were studied. An illustrative example of an IDE on time scale was also given. Lupulescu [40] also use the gHdifference to develop a theory of the fractional calculus for intervalvalued functions, and it is the foundation of intervalvalued 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 semilinear interval dynamic systems, after developing a complete normed quasilinear space. Most precisely, we first give a quasilinear space on interval number and a related continuous intervalvalued function space, and meanwhile their properties are sufficiently discussed. Second, an important and classical fixed point theorem is generalized to the intervalvalued case so as to discover IDEs’ properties. Finally, we conclude that there exists at least a strong solution for a kind of semilinear IDE as considered in this work.
2 Preliminaries and basic properties of gHdifference
Let IR denote a set composed of all closed intervals in R. For a given interval \(a=[a^{L},a^{R}]\), a is said to be a degenerate interval if \(a^{L}=a^{R}\). We say that a equals interval b only when \(a^{L}=b^{L}\) and \(a^{R}=b^{R}\), where \(b=[b^{L},b^{R}]\). Some interval arithmetic rules on IR are defined below [26]:

(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)
\(ab=a+(1)b=[a^{L}b^{R},a^{R}b^{L}]\);

(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, \(aa\) 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 \(aa\neq0\), 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 Hdifference (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])
The gHdifference of a and b is defined by
where c satisfies \(a=b+c\) if \(w(a)\geq w(b)\), or \(b=a+(1)c\) if \(w(a)< w(b)\).
The above definition indicates that any two intervals a and b have their gHdifference. In addition, we notice that the concepts of the gHdifference and Hdifference have an important relationship, namely \(a\ominus_{g}b=a\ominus b\) if \(w(a)\geq w(b)\). However, when \(w(a)< w(b)\), \(a\ominus_{g}b\) is meaningful, but \(a\ominus b\) is not. Thus, the gHdifference is an extension version of the Hdifference. Further, Stefanini [29] obtained the following basic properties of the gHdifference:

(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)
\(k(a\ominus_{g}b)=ka\ominus_{g}kb\), \(k\in R\).
In our last work [30], we also obtained some properties of the gHdifference, for example:

(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 gHdifference, we obtain some properties summed up below.
Lemma 2.2
The following properties are true:

(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 gHdifference.
Case (ii): write \((a+b)\ominus_{g}(a+c)=d\). Based on the gHdifference, 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
The following properties hold:
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 quasilinear space
3.1 Interval number space
In this section, we first develop a quasilinear space on IR, and then we analyze its properties under the gHdifference, by introducing the HausdorffPompeiu metric on interval numbers. For \(a,b,c\in IR\), and \(k,l\in R\), the addition and scalar multiplication have some wellknown 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.
In brief, IR is not a linear space under the above arithmetic rules of addition and scalar multiplication, but it can almost keep the features of linear space, provided that we replace subtraction by the gHdifference. Thus, we call IR a quasilinear space with gHdifference. In one such quasilinear space, we can easily obtain an additional property, namely for a given \(a\in IR\), there exists a unique \(d\in IR\) such that \(a\ominus_{g}d=0\). Additionally, in order to investigate the relation between elements in IR, we introduce the HausdorffPompeiu metric [26] on IR, i.e.,
Through simple induction, the triangle inequality of the HausdorffPompeiu metric on IR always holds, namely
Aubin and Cellina [41] asserted that \((IR,H)\) was a complete metric space. Further, such a metric can imply the following properties with the Hdifference [10, 42]:

(i)
\(H(a+b,a+c)=H(b,c)\);

(ii)
\(H(ka,kb)=kH(a,b)\), 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
There always exists the following inequality:
Especially, the equality holds if \(r_{ab}r_{ac}\geq0\).
Proof
Write \(d=a\ominus_{g}b\) and \(e=a\ominus_{g}c\). In the case of \(r_{ab}r_{ac}\geq0\), if \(r_{ab}\geq0\) and \(r_{ac}\geq0\), we can obtain \(d=a\ominus b\) and \(e=a\ominus c\) by the definition as in equation (2.1), and hence it follows from property (iv) of HausdorffPompeiu metric above that the equality is true; if \(r_{ab}\leq0\) and \(r_{ac}\leq0\), then \(b=a+(1)d\) and \(c=a+(1)e\). This, together with properties (i) and (ii) of HausdorffPompeiu metric above, easily shows \(H(d,e)=H(a+(1)d, a+(1)e)=H(b,c)\). Therefore, when \(r_{ab}\) and \(r_{ac}\) have the same sign, the equality in equation (3.4) holds. In the case of \(r_{ab}r_{ac}<0\), if \(r_{ab}>0\) and \(r_{ac}<0\), then \(a=b+d\) and \(c=a+(1)e\), and accordingly, we have
Similarly, when \(r_{ab}<0\) and \(r_{ac}>0\), we can also prove that equation (3.4) holds. □
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
The following inequality is always true,
Especially, the equality holds if \(r_{ab}r_{cd}\geq0\).
Proof
Write \(e=a\ominus_{g}b\) and \(h=c\ominus_{g}d\). In the case of \(r_{ab}r_{cd}\geq0\), if \(r_{ab}\geq0\) and \(r_{cd}\geq0\), we can obtain both \(e=a\ominus b\) and \(h=c\ominus d\); if \(r_{ab}\leq0\) and \(r_{cd}\leq0\), then \(b=a+(1)e\) and \(d=c+(1)h\). This easily shows that equation (3.5) is valid. In the case of \(r_{ab}r_{cd}<0\), if \(r_{ab}>0\) and \(r_{cd}<0\), then \(a=b+e\) and \(d=c+(1)h\). Therefore,
In the same way, if \(r_{ab}<0\) and \(r_{cd}>0\), then
In brief, the above conclusion holds. □
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.
Lemma 3.3
([43])
Let \(a,b,c\in IR\), then
Based on the above HausdorffPompeiu 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 quasilinear space.
Theorem 3.4
For \(a,b\in IR\), the following basic properties are true:

(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
Cases (i) and (ii) hold obviously. Case (iii): since
the conclusion is true. □
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 quasilinear 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 quasilinear 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 Intervalvalued function space
Let \(I=[t_{1},t_{2}]\) and \(t_{0}\in I\). \(f:I\rightarrow IR\) is an intervalvalued function. We say that \(a\in IR\) is the limit of f at the point \(t_{0}\) if \(\f(t)\ominus_{g}a\_{I}\rightarrow0\) as \(t\rightarrow t_{0}\). f is said to be continuous on I, if for any given \(t_{0}\in I\), \(\f(t)\ominus_{g}f(t_{0})\_{I}\rightarrow0\) as \(t\rightarrow t_{0}\). Define the following continuous intervalvalued function space,
Introduce the following wellknown arithmetic rules for \(f,g\in C(I,IR)\):

(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
For any \(t_{0}\in I\), since
kf is continuous at the point \(t_{0}\). Again, through equation (3.3), we obtain
Thus, following the definition of continuity, we derive that \(f+g\in C(I,IR)\). Further, equations (3.3) and (3.5) yield
Thus, \(f\ominus_{g}g\in C(I,IR)\). On the other hand, equations (3.2) and (3.6) imply that
and, consequently, \(fg\in C(I,IR)\). □
Through the process of the proof above, we notice that \(f\in C(I,IR)\) if and only if \(f^{L}, f^{R}\in C(I,R)\), where \(f(t)=[f^{L}(t), f^{R}(t)]\). Further, according to the above arithmetic rules, \(C(I,IR)\) is also a quasilinear space. Define
One can prove that the metric of ρ satisfies the three basic properties of a metric space, namely if \(f,g,h\in C(I,IR)\), then

(i)
\(\rho(f,g)\geq0\); \(\rho(f,g)=0\) if and only if \(f=g\);

(ii)
\(\rho(f,g)=\rho(g,f)\);

(iii)
\(\rho(f,g)\leq\rho(f,h)+\rho(h,g)\).
Thus, \((C(I,IR),\rho)\) is a metric space. In addition, in terms of the properties of the HausdorffPompeiu metric, it is easy to see that ρ has the following properties, namely if \(f,g,\varphi,\psi\in C(I,IR)\), then:

(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 intervalvalued 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 quasilinear space \((C(I,IR),\rho)\) is complete.
Proof
Let \(\{f_{n}\}_{n\geq1}\) be an arbitrary Cauchy sequence in \(C(I,IR)\). It follows that for any \(\varepsilon>0\), there exists \(N_{0}(\varepsilon)>0\) such that if \(n,m>N_{0}(\varepsilon)\), then
Therefore, for any fixed t, \(\{f_{n}(t)\}_{n\geq1}\) is a Cauchy sequence in the complete normed quasilinear space IR and, accordingly, there exists an element \(f(t)\in IR\) such that when \(n>N_{0}(\varepsilon)\), one can derive that
This way, \(\{f_{n}\}_{n\geq1}\) converges uniformly to f on I. On the other hand, for any \(t_{0}\in I\) there exists \(\delta(\varepsilon)>0\) such that if \(tt_{0}<\delta(\varepsilon)\), then \(\f_{n}(t)\ominus _{g}f_{n}(t_{0})\_{I}< \frac{\varepsilon}{3}\). According to equation (3.2) and property (i) as in Theorem 3.4, we see that
Consequently, \(f\in C(I,IR)\), and hence the proof is completed. □
Like the above normed quasilinear space IR, we can introduce the version of a norm on \(C(I,IR)\), namely \(\f\_{C}=\rho(f,0)\). By means of the HausdorffPompeiu metric on interval numbers above, one can see that \((C(I,IR), \\cdot\_{C})\) is a normed quasilinear 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
If \(f,g\in C(I,IR)\), then

(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 quasilinear space.
We next develop a fixed point theorem under the gHdifference. 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 gHdifferentiability
Since the addition arithmetic on interval number is irreversible, the concept of a derivative of an intervalvalued function has been gaining great concern among researchers. As to this point, Hukuhara [23] introduced the concept of the Hderivative related to the version of the Hdifference. However, as we mention in Section 1, this concept cannot be comprehensively adopted to investigate the properties of intervalvalued functions, as it cannot ensure that the Hdifference exists for any two interval numbers. Thereafter, Bede and Gal [33] generalized such concept of a derivative to the version of a GHderivative. However, the latter concept is relatively more useful, it still needs the same basic assumptions as the concept of Hderivative. Fortunately, Stefanini and Bede [9] proposed a more general concept of derivative (i.e., the gHderivative) by comparison with the GHderivative. The main merit consists in the fact that their concept is similar to the version of the derivative of a realvalued function.
Definition 4.1
([9])
\(f:I\rightarrow IR\) is said to be gHdifferentiable on I if f is gHdifferentiable in \(t\in I\), namely there exists an interval number \(f'(t)\in IR\) such that
f is said to be (i)differentiable in \(t\in I\) if \(f'(t)=[(f^{L})'(t), (f^{R})'(t)]\) and (ii)differentiable if \(f'(t)=[(f^{R})'(t), (f^{L})'(t)]\), where \(f(t)=[f^{L}(t), f^{R}(t)]\).
The main advantage of such a definition is that the formulation of the gHderivative is simpler than that of the GHderivative. Thus, it is easily utilized to study intervalvalued functions. According to the definition, we observe that, when f is (i)differentiable, \(w(f(t))\) is an increasing function, and conversely when f is (ii)differentiable, \(w(f(t))\) is decreasing. Further, similar to the formulation of the conventional derivative in real analysis, the formulas of left and right derivatives at the point t for f can be expressed by
Accordingly, we can give a sufficient and necessary condition on gHdifferentiability below.
Theorem 4.2
f is gHdifferentiable in \(t\in I\) if and only if \(f_{+}'(t)\) and \(f_{}'(t)\) exist and \(f_{+}'(t)=f_{}'(t)\).
Proof
If f is gHdifferentiable in \(t\in I\), then for any \(\varepsilon>0\), there exists \(\delta(\varepsilon)>0\), such that when \(h<\delta (\varepsilon)\), we have
Consequently, if taking \(0< h<\delta(\varepsilon)\) or \(\delta (\varepsilon)< h<0\), the conclusion is true. Conversely, when \(f_{+}'(t)=f_{}'(t)\), it is easy to prove that the conclusion is valid through equations (4.2) and (4.3). □
Based on the concept of the gHderivative, Stefanini et al. developed the relationship between gHderivative and conventional derivatives, in other words, the gHderivative of f can be expressed by the derivatives of its endpoint functions.
Theorem 4.3
([9])
\(f:I\rightarrow IR\) is gHdifferentiable in t if and only if \(f^{L}\) and \(f^{R}\) are both differentiable, and
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
The following property is true:
provided that f and g are simultaneously (i)differentiable or (ii)differentiable.
Proof
Assume that f and g are (i)differentiable. One can prove that both \(w(f(t))\) and \(w(g(t))\) are increasing functions. Hence, in the case of \(h>0\), since \(w(f(t+h))\geq w(f(t))\) and \(w(g(t+h))\geq w(g(t))\), through the definition of the gHdifference we obtain \(f(t+h)=f(t)+u(t,h)\) and \(g(t+h)=g(t)+v(t,h)\), and thus
Further, in the case of \(h<0\), since \(w(f(t+h))\leq w(f(t))\) and \(w(g(t+h))\leq w(g(t))\), we get \(f(t)=f(t+h)+(1)u(t,h)\) and \(g(t)=g(t+h)+(1)v(t,h)\), and accordingly
Hence,
Thus, \(f+g\) is gHdifferentiable and equation (4.6) is true. Similarly, when f and g are (ii)differentiable, one can prove that \(f+g\) is gHdifferentiable and equation (4.6) is also true. □
Theorem 4.5
The following property is true:
if one of the following conditions is satisfied:

(i)
f is (i)differentiable and g is (ii)differentiable;

(ii)
f is (ii)differentiable and g is (i)differentiable.
Proof
Case (i): by the definition of the gHderivative, \(w(f(t))\) is increasing and \(w(g(t))\) is decreasing. Consequently, in the case of \(h>0\), we obtain \(f(t+h)=f(t)+u(t,h)\) and \(g(t)=g(t+h)+(1)v(t,h)\). Hence,
Again if \(f(t)=g(t)+\omega(t)\), equation (4.10) yields
from which, together with the cancellation of addition on interval numbers, it follows that
In the same way, if \(g(t)=f(t)+(1)\omega(t)\), from equation (4.10) it follows that
Hence, equations (4.12) and (4.13) imply that
In the case of \(h<0\), since \(w(f(t+h))\leq w(f(t))\) and \(w(g(t+h))\geq w(g(t))\), the definition of the gHdifference yields \(f(t)=f(t+h)+(1)u(t,h)\) and \(g(t+h)=g(t)+v(t,h)\), and then we also see that equation (4.14) is valid. This, together with the differentiability of f and g, yields
Thus, equation (4.9) is true in case (i). Similar to the process of the proof above, one can see that equation (4.9) is also valid in case (ii). □
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 intervalvalued 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
Take \(f(t)=[e^{t}, e^{t+1}]\) and \(g(t)=[\cos t, \cos t]\) with \(0\leq t\leq\frac{\pi}{4}\). It is easy to see that \(f(t)g(t)=[e^{t+1}\cos t, e^{t+1}\cos t]\). Further, through a simple deduction, we know that
This is a hint that
In the following subsection, we degenerate the intervalvalued 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
Assume that \(f\in C^{1}(I,R)\) and g is (i)differentiable. If \(f(t)f'(t)>0\), then
Proof
In the case of \(h>0\) with \(t+h\in I\), since g is (i)differentiable, \(w(g(t))\) is increasing and, accordingly, \(g(t+h)=g(t)+v(t,h)\). Again, since \(f(t)f'(t)>0\), \(f(t)\) and \(U(t,h)\) have the same sign, which yields
Thus,
On the other hand, in the case of \(h<0\), we have \(w(g(t+h))\leq w(g(t))\), and hence \(g(t)=g(t+h)+(1)v(t,h)\). Further, it follows from \(f(t)f'(t)>0\) that \(f(t+h)\) and \((1)U(t,h)\) have the same sign. Consequently,
Hence,
Further, depending on the continuity and differentiability of f and g, equations (4.17) and (4.19) imply that
This shows that equation (4.15) is valid by Theorem 4.2. □
Theorem 4.9
Let g be (i)differentiable and \(f\in C^{1}(I,R)\). Under \(f(t)f'(t)<0\), if \(w(f(t)g(t))\) is increasing, then fg is gHdifferentiable and
if \(w(f(t)g(t))\) is decreasing, then fg is gHdifferentiable and
Proof
Let \(w(f(t)g(t))\) be increasing. Since \(f(t)f'(t)<0\), we can see that \(f(t+h)\) and \((1)U(t,h)\) have the same sign when \(h>0\), and meanwhile \(f(t)\) and \(U(t,h)\) are so if \(h<0\). In the case of \(h>0\), we know that \(g(t+h)=g(t)+v(t,h)\) by the (i)differentiability of g. Thus,
Namely,
Again since \(w(f(t+h)g(t+h))\geq w(f(t)g(t))\), we can obtain
This way, by substituting equation (4.25) into equation (4.24) and using the cancellation law on interval numbers, we see that
So,
that is,
In addition, in the case of \(h<0\), since g is (i)differentiable, we can derive that \(g(t)=g(t+h)+(1)v(t,h)\). Thus,
In other words,
Again since
by virtue of the cancellation law on interval numbers from equations (4.29) and (4.30) we infer that
Thus,
namely
Equations (4.27) and (4.32) imply that fg is gHdifferentiable, and meanwhile equation (4.21) holds. Similar to the process of the proof above, one can prove that equation (4.22) is also valid. □
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
Assume that \(f\in C^{1}(I,R)\) and g is (ii)differentiable. If \(f(t)f'(t)<0\), then
Further, under \(f(t)f'(t)>0\), no matter whether \(w(f(t)g(t))\) is increasing or decreasing, fg is gHdifferentiable. More precisely, if \(w(f(t)g(t))\) is increasing, then
if \(w(f(t)g(t))\) is decreasing, then
5 Integral of intervalvalued function
In this section, we cite the concept of an integral of an intervalvalued function originally proposed by Stefanini and Bede [9], and meanwhile some new properties are discussed. Let \(J=[t_{0},t_{f}]\) and \(f(t)=[f^{L}(t),f^{R}(t)]\) with \(t\in J\). The integral of f is defined by the integrals of the endpoints [9], namely
In such a case, f is said to be integrable on J. For an integrable intervalvalued function g, by the definition of the gHdifference we easily see that
Correspondingly, some fundamental properties have been studied.
Theorem 5.1
([41])
Let \(f,g\in C(J,IR)\). Then

(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])
Let \(f\in C(J,IR)\). Then

(i)
\(F(t)\) is gHdifferentiable, and \(F'(t)=f(t)\), where \(F(t)=\int _{t_{0}}^{t}f(t)\, dt\);

(ii)
\(G(t)\) is gHdifferentiable, 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
Let f and g be integrable on J. Then \(f\ominus_{g}g\) is integrable on J, and meanwhile
provided that \(w(f(t))\geq w(g(t))\) for \(t\in J\) or \(w(f(t))\leq w(g(t))\) for \(t\in J\).
Proof
Write \(f(t)\ominus_{g}g(t)=h(t)\). If \(w(f(t))\geq w(g(t))\) with any \(t\in J\), then \(f(t)=g(t)+h(t)\), and accordingly
If \(w(f(t))\leq w(g(t))\) with any \(t\in J\), then \(g(t)=f(t)+(1)h(t)\) and hence,
Thus, equations (5.4) and (5.5) illustrate that equation (5.3) is true. □
Theorem 5.4
If \(g_{1},g_{2}\in C(J,IR)\), the following inequality holds:
Proof
Write \(g_{1}(t)=[g_{1}^{L}(t), g_{1}^{R}(t)]\) and \(g_{2}(t)=[g_{2}^{L}(t), g_{2}^{R}(t)]\). We can prove that
This completes the proof. □
In terms of Lemma 3.3 and Theorem 5.4, one can easily gain the following conclusion.
Corollary 5.5
If \(f\in C(J,R)\) and \(g_{1},g_{2}\in C(J,IR)\), we have
6 Interval differential equation
In this section, we consider the following semilinear interval differential equation (SIDE):
where \(a:J\rightarrow R\) is an integrable real scalar function, and \(f : J\times IR\rightarrow IR\) is an intervalvalued function; \(x_{0}\) is a given initial interval number in IR. In order to analyze the properties of the solutions in SIDE, three basic concepts are introduced below.
Definition 6.1
For given \(x\in C(J,IR)\), x is continuous gHdifferentiable on J if \(x'\) is continuous.
Definition 6.2
Let x be continuous gHdifferentiable on J. x is a strong solution of SIDE if satisfying the initial condition and the above equation.
Definition 6.3
\(x\in C(J,IR)\) is the (i)solution of SIDE if
and the (ii)solution if
Equations (6.2) and (6.3) are constructed in terms of the formulation of the solutions for ordinary differentiable equations. However, (i) and the (ii)solutions are not SIDE’s strong solutions usually. Based on this consideration, we discuss the relationship between SIDE’s solutions, depending on the above properties of \(C(J,IR)\). For the convenience of representation, write
and
Theorem 6.4
Let \(a\in C(J,R)\) and \(f\in C(J\times IR,IR)\). If \(a(t)>0\) with \(t\in J\), the (i)solution x is a strong solution of SIDE; under \(a(t)<0\), if (ii)solution x satisfies
it is also SIDE’s strong solution.
Proof
Let \(a(t)>0\) with \(t\in J\) and x be a (i)solution of SIDE. Write
As related to Definition 4.1 and Theorem 5.2 it follows that \(F(t,x)\) is (i)differentiable and that
Hence, Theorem 4.4 and equation (6.5) imply that \(g(t,x)\) is (i)differentiable and
Again since \(a(t)>0\) and \(p'(t)=a(t)p(t)\), it is obvious that \(p(t)p'(t)>0\). Hence, it follows from Theorem 4.8 and equations (6.2) and (6.5)(6.7) that
Therefore, x is a strong solution of SIDE.
On the other hand, let x be a (ii)solution with \(a(t)<0\). Since \(p(t)>0\), we know that \(p(t)p'(t)<0\). This way, equation (6.3) can be rewritten by
where
As we mentioned above, \(F(t,x)\) is (i)differentiable, and accordingly, from Theorem 4.5 and equation (6.10) it follows that
Since \(w(F(t,x))\leq w(x_{0})\), equations (6.10) and (6.11) indicate that \(h(t,x)\) is (ii)differentiable. Therefore, Theorem 4.10 and equation (6.10) imply that
Hence, the conclusion is true. □
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
There exist \(K>0\) and \(\alpha>0\) such that
Lemma 6.5
Under Hypothesis 6.1, if \(f\in C(J\times IR,IR)\), there is a positive constant \(M_{\alpha}\) such that the (i) or (ii)solution x satisfies
where
Proof
By means of Theorem 3.4, Corollary 5.5, and equations (6.2) and (6.3), we can prove that
When \(\alpha=1\), the Gronwall inequality indicates that equation (6.13) is valid; when \(\alpha\neq1\), the generalized Bellman lemma [44] hints that equation (6.13) is also true. □
In addition, based on Lemma 6.5, define
We can prove that \(C_{\alpha}(J,IR)\) is a complete metric space.
Hypothesis 6.2
Assume that f satisfies the uniformly Lipschitz condition, namely there exists \(L>0\) such that
with \(\forall t\in J\) and \(\z_{1}\_{I}, \z_{2}\_{I}\leq M_{\alpha}\).
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
Let \(a\in C(J,R)\). Under Hypotheses 6.1 and 6.2, when
SIDE has a unique (i)solution in \(C_{\alpha}(J,IR)\) if \(a(t)>0\), and a unique (ii)solution in \(C_{\alpha}(J,IR)\) if \(a(t)<0\), provided that the initial value \(x_{0}\) satisfies
Proof
Under \(a(t)>0\), define a mapping \(T_{1}\) on \(C_{\alpha}(J,IR)\) given by
with \(t\in J\), namely,
where \(g(t,x)\) is decided by equation (6.5) above. For \(t, t+\Delta t\in J\) and \(x\in C_{\alpha}(J,IR)\), in terms of Lemma 2.2 and Theorem 5.1 we have
Thus, we see that \(g(t,x)\) is continuous in t, due to Hypothesis 6.1 and the prior estimate of the solution in Lemma 6.5. This way, from Theorem 3.7 it follows that \(p(t)g(t,x)\) is continuous in t, and hence \(T_{1}x\in C(J,IR)\). Additionally, by means of Lemma 3.3 and the additive property of the HausdorffPompeiu metric on interval numbers, we derive for \(x,y\in C_{\alpha}(J,IR)\) that
Thus,
Accordingly, we prove by Hypothesis 6.2 that
Further, Theorem 3.9, Hypothesis 6.1, and equations (6.16), (6.17), and (6.19) imply that
Consequently, \(T_{1}\) is a contraction mapping on \(C_{\alpha}(J,IR)\) and hence has a unique fixed point. This shows that SIDE has a unique (i)solution. We next prove that SIDE has a unique (ii)solution. If \(a(t)<0\), define a mapping \(T_{2}\) on \(C(J,IR)\) given by
with \(t\in J\), namely,
where \(h(t,x)\) is decided by equation (6.10) above. On one hand, from Lemma 2.2 it follows that
On the other hand, Lemmas 3.1 and 3.3 yield
Subsequently, through a similar deduction to above we can prove that \(T_{2}\) is a contraction mapping on \(C_{\alpha}(J,IR)\). Therefore, SIDE has a unique (ii)solution. □
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
Let \(\{t_{n}\}_{n\geq1}\) be a series of zero points of \(a(t)\). Divide the interval J into countable subintervals \(J_{n}\) with \(n\geq1\) and \(J_{n}=[t_{n1},t_{n}]\), in which \(a(t)\) always maintains the same sign on \(J_{n}\). If \(a(t)>0\) with \(t_{n1}< t< t_{n}\), Theorems 6.4 and 6.6 show that the unique (i)solution, \(x_{n1}(t)\), is a strong solution of SIDE on \(J_{n}\). In the same way, if \(a(t)<0\) with \(t_{n1}< t< t_{n}\), two such theorems ensure that SIDE has a strong solution on \(J_{n}\), i.e., the (ii)solution \(x_{n2}(t)\). Therefore, we can obtain a strong solution \(x(t)\) for the above SIDE given by
where
Especially, when \(n=1\), \(x_{1}(t)\) denotes a strong solution of SIDE on \([t_{0},t_{1}]\); when \(n=2\), \(x_{2}(t_{1})\) takes the form of \(x_{1}(t_{1})\). This way, \(x(t)\) takes the form \(x_{n1}(t_{n1})\) at the endpoint \(t_{n1}\). □
7 Illustrative examples
In this section, our experiments are implemented through MATLAB’s 7.0 standard ODE solver (ode45). Three simple intervalvalued Cauchy problems are used to examine our theoretical results.
Example 7.1
([9])
This in fact is a linear intervalvalued Cauchy problem, satisfying the conditions of Theorem 6.7 with \(a(t)=1\). Therefore, there exists a strong solution, i.e., the (ii)solution, expressed by
One such solution, \(x=[x^{L},x^{R}]\), is drawn in Figure 1, where \(x^{L}\) and \(x^{R}\) denote the lower and upper bound curves of the (ii)solution x obtained through Theorem 6.7. \(x_{1}=[x_{1}^{L},x_{1}^{R}]\) and \(x_{2}=[x_{2}^{L},x_{2}^{R}]\) represent the curves of I and IItype solutions gotten through (i)differentiability and the (ii)differentiability, respectively [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
where \(f(t,x)= \bigl \{\scriptsize{ \begin{array}{l@{\quad}l} \sin t,& 0\leq t\leq\pi, \\ x\sin t,& \pi< t\leq2\pi. \end{array}} \bigr.\)
We note that there is a zero point π for \(a(t)=\sin t\). It can be checked that f is a continuous intervalvalued function in t and x, and it satisfies the conditions as in Theorem 6.7. Consequently, there exists a strong solution composed of the (i)solution in \([0,\pi]\) and the (ii)solution in \((\pi,2\pi]\), namely
The solution x is drawn in Figure 2. In addition, \(x_{1}\) and \(x_{2}\) represent the curves of I and IItype solutions mentioned above, respectively.
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 IItype 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
It can be examined that f satisfies Hypotheses 6.1 and 6.2, where \(f(t,x)=x^{3}\varphi(t)\) and \(\varphi(t)=\frac{\sin t}{100+\sin^{2}t}\). The reason can be found below.

(i)
For any \((t,x)\in J\times IR\), since
$$\bigl\Vert f(t,x)\bigr\Vert _{I}=H\bigl(f(t,x),0\bigr)=\max\bigl\{ \bigl\vert \bigl(x^{L}\bigr)^{3}\varphi(t)\bigr\vert , \bigl\vert \bigl(x^{R}\bigr)^{3}\varphi (t)\bigr\vert \bigr\} \leq\x\_{I}^{3}. $$Therefore, Hypothesis 6.1 holds.

(ii)
For \(\x\_{C}, \y\_{C}\leq M_{\alpha}\), we have
$$\begin{aligned} H\bigl(f(t,x),f(t,y)\bigr) =& \max\bigl\{ \bigl\vert \bigl(x^{L} \bigr)^{3}\varphi(t)\bigl(y^{L}\bigr)^{3}\varphi (t)\bigr\vert ,\bigl\vert \bigl(x^{R}\bigr)^{3} \varphi(t)\bigl(y^{R}\bigr)^{3}\varphi(t)\bigr\vert \bigr\} \\ \leq& \max\bigl\{ \bigl\vert \bigl(x^{L}\bigr)^{3} \bigl(y^{L}\bigr)^{3}\bigr\vert ,\bigl\vert \bigl(x^{R}\bigr)^{3}\bigl(y^{R} \bigr)^{3}\bigr\vert \bigr\} \\ \leq& 3M_{\alpha}^{2}H(x,y). \end{aligned}$$This illustrates that Hypothesis 6.2 is true.
We note that there are two zero points \(\frac{\pi}{2}\) and \(\frac{3\pi }{2}\) for \(a(t)=\cos t\). As associated to Theorem 6.7, there exists a strong solution composed of two (i)solutions on \([0,\frac{\pi }{2}]\) and \((\frac{3\pi}{2},2\pi]\), and a (ii)solution on \((\frac{\pi }{2},\frac{3\pi}{2}]\), namely
The solution x is drawn in Figure 3. In addition, \(x_{1}\) and \(x_{2}\) represent the curves of I and IItype solutions gotten in the same fashion as above, respectively.
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 Itype solution \(x_{1}\) diverges and the IItype 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 intervalvalued functions under the gHdifference and also probing the existence of the solutions for a class of semilinear interval differential equations. As associated to the concept of the gHdifference and also the conventional arithmetic rules such as addition and scalar multiplication, we have developed a complete normed quasilinear space on interval numbers, in which some important properties of the gHdifference are found under the HausdorffPompeiu metric. Subsequently, on the basis of one such space, we introduced a continuous intervalvalued function space which has been proven to be a complete normed quasilinear space. A contracting mapping theorem on one such space, similar to the classical contracting mapping principle, has been obtained, relying upon the gHdifference. Based on these fundamental works, some arithmetic properties of the gHderivative for intervalvalued 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 semilinear equation. After some simple properties of the integral of intervalvalued 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.
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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.
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Tao, J., Zhang, Z. Properties of intervalvalued function space under the gHdifference and their application to semilinear interval differential equations. Adv Differ Equ 2016, 45 (2016). https://doi.org/10.1186/s1366201607599
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DOI: https://doi.org/10.1186/s1366201607599