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A general rule of thumb is that vehicles with even model years are required to pass an emissions inspection in even years, and vehicles with odd model years in odd years. For complete information, however, see .
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t \multimap F(t,x) is measurable, for every x \in \mathbb{R}^{n} ,
x \multimap F(t,x) is u.s.c., for almost all (a.a.) t\in J ,
|y| \leq r(t) (1+|x|) , for every (t,x) \in J \times \mathbb{R}^{n} , and every y\in F(t,x) , where r\colon J \rightarrow [0, \infty) is a Lebesgue integrable function.
Let F \colon X \multimap Y be a multivalued mapping and f \colon X \rightarrow Y be a single-valued mapping. We say that f is a selection of F (written f \subset F ) if f(x) \in F(x) , for every x \in X .
We will employ the following definitions and statements.
(Castaing representation, cf. Theorem III.7 in [ 35 ])
(, , Lemma7.1 in [ 36 ])
F\colon J\times\mathbb{R}^{n} \multimap\mathbb{R}^{n} -. F(t, q(t)) , q \in C(J, \mathbb{R}^{n}) , -.
If X \subset Y and \varphi\colon X \multimap Y , then a point x \in X is called a fixed point of φ if x \in\varphi (x) . We set \operatorname{Fix} (\varphi) := \{ x \in X \mid x \in\varphi(x) \} .
Let and be subsets of normed linear spaces and \varphi\colon X \multimap Y be a multivalued mapping. If is convex, then is called a , provided is u.s.c. with (nonempty) compact, convex values.
Let and be metric spaces. A multivalued mapping \varphi\colon X \multimap Y is called if its image \varphi(X)= \bigcup\{\varphi(x) \mid x \in X \} is contained in a compact subset of .
The following statement, which we state in the form of proposition, is usually called the Kakutani-Ky Fan-type fixed point theorem.
(, , Theorem II.8.4 in [
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() , \varphi\colon C \multimap C . .
Observe that Φ is a bounded u.s.c. mapping with (nonempty) compact and convex values.
( cf. , e.g. , Theorem3.2.1 in [ 38 ])