By \(P_{K}(\mathbb{R})\) we denote the family of all nonempty, compact, and convex subsets of ℝ and define the addition and scalar multiplication in \(P_{K}(\mathbb{R})\) as usual. Denote
$$E = \bigl\{ u : \mathbb{R} \rightarrow[0,1] | u\text{ satisfies (i)-(iv) below} \bigr\} ,$$
where
-
(i)
u is normal, i.e.
\(\exists x_{0} \in\mathbb{R}\) for which \(u(x_{0})=1\),
-
(ii)
u is fuzzy convex, i.e.
$$u \bigl(\lambda x + (1-\lambda y) \bigr) \geq\min \bigl(u(x),u(y) \bigr) \quad \text{for any } x, y\in\mathbb{R}, \text{ and } \lambda\in[0,1], $$
-
(iii)
u is upper semi-continuous,
-
(iv)
\(\operatorname {supp}u = \{x\in\mathbb{R} | u(x)>0 \}\) is the support of the u, and its closure cl (suppu) is compact.
For \(0 < \alpha\leq1\), denote
$$[u ]^{\alpha} = \bigl\{ x\in\mathbb{R} | u(x) \geq\alpha \bigr\} . $$
Then, from (i)-(iv), it follows that the α-level set \([u ]^{\alpha} \in P_{K}(\mathbb{R})\) for all \(0 \leq\alpha\leq1\).
According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy-number space E as usual.
It is well known that the following properties are true for all levels:
$$[u+v ]^{\alpha} = [u ]^{\alpha} + [v ]^{\alpha} , \qquad [k u ]^{\alpha} = k [u ]^{\alpha} . $$
Let \(D : E\times E \rightarrow[0,\infty)\) be a function which is defined by the equation
$$ D(u,v) = \sup_{0\leq\alpha\leq1} d \bigl( [u]^{\alpha} , [v]^{\alpha } \bigr), $$
where d is the Hausdorff metric defined in \(P_{K}(\mathbb{R})\). Then it is easy to see that D is a metric in E and has the following properties [11]:
-
(1)
\((E,D )\) is a complete metric space;
-
(2)
\(D(u+w,v+w) = D(u,v)\) for all \(u, v, w\in E\);
-
(3)
\(D(k u,k v) = \vert k\vert D(u,v)\) for all \(u, v\in E\) and \(k\in\mathbb{R}\);
-
(4)
\(D(u+w,v+t) \leq D(u,v)+ D(w,t)\) for all \(u, v, w, t\in E\).
Definition 2.1
A fuzzy number u in parametric form is a pair \((\underline{u},\overline{u})\) of functions \(\underline{u}(r)\), \(\overline{u}(r)\), \(0\leq r\leq1\), which satisfy the following requirements:
-
(1)
\(\underline{u}(r)\) is a bounded non-decreasing left continuous function in \((0,1]\), and right continuous at 0;
-
(2)
\(\overline{u}(r)\) is a bounded non-increasing left continuous function in \((0,1]\), and right continuous at 0;
-
(3)
\(\underline{u}(r)\leq\overline{u}(r)\) for all \(0\leq r\leq1\).
A crisp number k is simply represented by \(\underline{u}(r) = \overline{u}(r) = k\), \(0\leq r\leq1\).
Let \(T=[c,d]\subset\mathbb{R}\) be a compact interval.
Definition 2.2
A mapping \(F:T\rightarrow E\) is strongly measurable if for all \(\alpha \in[0,1]\) the set-valued function \(F_{\alpha}:T\rightarrow\mathcal{P}_{K}(\mathbb{R})\) defined by \(F_{\alpha}(t)=[F(t)]^{\alpha}\) is Lebesgue measurable.
A mapping \(F:T\rightarrow E\) is called integrably bounded if there exists an integrable function k such that \(\Vert x\Vert \leq k(t)\) for all \(x\in F_{0}(t)\).
Definition 2.3
Let \(F:T\rightarrow E\), then the integral of F over T, denoted by \(\int_{T}F(t)\,dt\) or \(\int_{c}^{d}F(t)\,dt\), is defined by the equation
$$\biggl[\int_{T}F(t)\,dt \biggr]^{\alpha} = \int _{T}F_{\alpha}(t)\,dt ;\quad \alpha\in\,]0,1] $$
i.e.
$$\biggl[\int_{T}F(t)\,dt \biggr]^{\alpha} = \biggl\{ \int_{T}f(t)\,dt | f:T\rightarrow\mathbb{R} \text{ is a measurable selection for } F_{\alpha} \biggr\} .$$
Also, a strongly measurable and integrably bounded mapping \(F:T\rightarrow E\) is said to be integrable over T if \(\int_{T}F(t)\,dt\in E\).
Proposition 2.4
(Aumann [12])
If
\(F:T\rightarrow E\)
is strongly measurable and integrably bounded, then
F
is integrable.
For more measurability, integrability properties for fuzzy set-valued mappings see [2, 8, 13].
Theorem 2.5
(see [14, 15])
Let
\(f(x)\)
be a fuzzy valued-function on
\([a,\infty[\)
which is represented by
\((\underline {f}(x,r),\overline{f}(x,r))\). For any fixed
\(r\in[0,1]\), assume
\(\underline{f}(x,r)\), \(\overline {f}(x,r)\)
are Riemann integrable on
\([a,b]\)
for every
\(b\geq a\), and assume there are two positive constants
\(\underline{M}(r)\)
and
\(\overline{M}(r)\)
such that
\(\int_{a}^{b}|\underline{f}(x,r)|\,dx\leq \underline{M}(r) \)
and
\(\int_{a}^{b}|\overline{f}(x,r)|\,dx\leq\overline {M}(r) \)
for every
\(b\geq a\). Then
\(f(x)\)
is improper fuzzy Riemann integrable on
\([a,\infty[\)
and the improper fuzzy Riemann integral is a fuzzy number. Furthermore, we have
$$\int_{a}^{\infty}f(x)\,dx = \biggl(\int _{a}^{\infty}\underline{f}(x,r)\,dx, \int _{a}^{\infty}\overline{f}(x,r)\,dx \biggr) . $$
Proposition 2.6
(see [14])
If each of
\(f(x)\)
and
\(g(x)\)
is a fuzzy valued function and fuzzy Riemann integrable on
\([a,\infty[\)
then
\(f(x)+g(x)\)
is fuzzy Riemann integrable on
\([a,\infty[\). Moreover, we have
$$\int_{a}^{\infty} \bigl(f(x)+g(x) \bigr)\,dx = \int _{a}^{\infty}f(x)\,dx + \int_{a}^{\infty}g(x) \,dx . $$
For \(u,v\in E\), if there exists \(w\in E\) such that \(u=v+w\), then w is the Hukuhara difference of u and v denoted by \(u\ominus v\).
Definition 2.7
We say that a mapping \(f:(a,b)\rightarrow E\) is strongly generalized differentiable at \(x_{0}\in(a,b)\) if there exists an element \(f'(x_{0})\in E\) such that
-
(i)
for all \(h > 0\) sufficiently small, there exist \(f(x_{0}+h)\ominus f(x_{0})\), \(f(x_{0})\ominus f(x_{0}-h)\), and the limits
$$ \lim_{h\rightarrow0^{+}}\frac{f(x_{0}+h)\ominus f(x_{0})}{h} = \lim_{h\rightarrow0^{+}} \frac{f(x_{0})\ominus f(x_{0}-h)}{h}=f'(x_{0}) $$
or
-
(ii)
for all \(h > 0\) sufficiently small, there exist \(f(x_{0})\ominus f(x_{0}+h)\), \(f(x_{0}-h)\ominus f(x_{0})\), and the limits
$$ \lim_{h\rightarrow0^{+}}\frac{f(x_{0})\ominus f(x_{0}+h)}{(-h)} = \lim_{h\rightarrow0^{+}} \frac{f(x_{0}-h)\ominus f(x_{0})}{(-h)}= f'(x_{0}) $$
or
-
(iii)
for all \(h > 0\) sufficiently small, there exist \(f(x_{0} + h)\ominus f(x_{0})\), \(f(x_{0}-h)\ominus f(x_{0})\), and the limits
$$\lim_{h\rightarrow0^{+}}\frac{f(x_{0}+h)\ominus f(x_{0})}{h} = \lim_{h\rightarrow0^{+}} \frac{f(x_{0}-h)\ominus f(x_{0})}{(-h)}=f'(x_{0}) $$
or
-
(iv)
for all \(h > 0\) sufficiently small, there exist \(f(x_{0} )\ominus f(x_{0}+h)\), \(f(x_{0})\ominus f(x_{0}-h)\), and the limits
$$\lim_{h\rightarrow0^{+}}\frac{f(x_{0})\ominus f(x_{0}+h)}{(-h)} = \lim_{h\rightarrow0^{+}} \frac{f(x_{0})\ominus f(x_{0}-h)}{h}=f'(x_{0}) . $$
The following theorem (see [13]) allows us to consider case (i) or (ii) of the previous definition almost everywhere in the domain of the functions under discussion.
Theorem 2.8
Let
\(f:(a, b)\rightarrow E\)
be strongly generalized differentiable on each point
\(x\in(a,b)\)
in the sense of Definition
2.3, (iii) or (iv). Then
\(f'(x)\in\mathbb{R}\)
for all
\(x\in(a,b)\).
Theorem 2.9
(see e.g. [16])
Let
\(f : \mathbb{R}\rightarrow E\)
be a function and denote
\(f(t) = (\underline{f}(t,r),\overline{f}(t,r))\), for each
\(r\in[0,1]\). Then
-
(1)
If
f
is (i)-differentiable, then
\(\underline{f}(t,r)\)
and
\(\overline{f}(t,r)\)
are differentiable functions and
\(f'(t) = (\underline{f}'(t,r),\overline{f}'(t,r)) \).
-
(2)
If
f
is (ii)-differentiable, then
\(\underline{f}(t,r)\)
and
\(\overline{f}(t,r)\)
are differentiable functions and
\(f'(t) = (\overline{f}'(t,r),\underline{f}'(t,r)) \).
Definition 2.10
We say that a mapping \(f:(a,b)\rightarrow E\) is strongly generalized differentiable of the second-order at \(x_{0}\in(a,b)\); if there exists an element \(f''(x_{0})\in E\) such that
-
(i)
for all \(h > 0\) sufficiently small, there exist \(f'(x_{0}+h)\ominus f'(x_{0})\), \(f'(x_{0})\ominus f'(x_{0}-h)\), and the limits
$$ \lim_{h\rightarrow0^{+}}\frac{f'(x_{0}+h)\ominus f'(x_{0})}{h} = \lim_{h\rightarrow0^{+}} \frac{f'(x_{0})\ominus f'(x_{0}-h)}{h}=f''(x_{0}) $$
or
-
(ii)
for all \(h > 0\) sufficiently small, there exist \(f'(x_{0})\ominus f'(x_{0}+h)\), \(f'(x_{0}-h)\ominus f'(x_{0})\), and the limits
$$ \lim_{h\rightarrow0^{+}}\frac{f'(x_{0})\ominus f'(x_{0}+h)}{(-h)} = \lim_{h\rightarrow0^{+}} \frac{f'(x_{0}-h)\ominus f'(x_{0})}{(-h)}= f''(x_{0}) $$
or
-
(iii)
for all \(h > 0\) sufficiently small, there exist \(f'(x_{0} + h)\ominus f'(x_{0})\), \(f'(x_{0}-h)\ominus f'(x_{0})\), and the limits
$$\lim_{h\rightarrow0^{+}}\frac{f'(x_{0}+h)\ominus f'(x_{0})}{h} = \lim_{h\rightarrow0^{+}} \frac{f'(x_{0}-h)\ominus f'(x_{0})}{(-h)}=f''(x_{0}) $$
or
-
(iv)
for all \(h > 0\) sufficiently small, there exist \(f'(x_{0} )\ominus f'(x_{0}+h)\), \(f'(x_{0})\ominus f'(x_{0}-h)\), and the limits
$$\lim_{h\rightarrow0^{+}}\frac{f'(x_{0})\ominus f'(x_{0}+h)}{(-h)} = \lim_{h\rightarrow0^{+}} \frac{f'(x_{0})\ominus f'(x_{0}-h)}{h}=f''(x_{0}). $$
All the limits are taken in the metric space \((E , D )\), and at the end points of \((a,b)\) and we consider only one-sided derivatives.