# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Nonordered discontinuous upper and lower solutions for first-order ordinary differential equations. (English) Zbl 1001.34001

The author studies the first-order equations

${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right)\right),\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{a.}\phantom{\rule{4.pt}{0ex}}\text{e.}\phantom{\rule{4.pt}{0ex}}t\in I=\left[0,1\right],\phantom{\rule{1.em}{0ex}}x\left(0\right)={x}_{0},$

and

${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right)\right),\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{a.}\phantom{\rule{4.pt}{0ex}}\text{e.}\phantom{\rule{4.pt}{0ex}}t\in I=\left[0,1\right],\phantom{\rule{1.em}{0ex}}x\left(1\right)={y}_{0},$

where $f:I×ℝ\to ℝ$ is a Carathéodory function.

He proves that if there exist $\alpha \in B{V}^{-}\left(I\right)$ and $\beta \in B{V}^{+}\left(I\right)$ such that ${\alpha }^{\text{'}}\left(t\right)\le f\left(t,\alpha \left(t\right)\right)$ and ${\beta }^{\text{'}}\left(t\right)\ge f\left(t,\beta \left(t\right)\right)$ for a. e. $t\in I$ then such problems have extremal solutions in $\left[m,M\right]$ for all ${x}_{0}\in \left[\alpha \left(0\right),\beta \left(0\right)\right]$ and ${y}_{0}\in \left[\beta \left(1\right),\alpha \left(1\right)\right]$. Here, $m\left(t\right)=min\left\{\alpha \left(t\right),\beta \left(t\right)\right\}$, $M\left(t\right)=max\left\{\alpha \left(t\right),\beta \left(t\right)\right\}$ for all $t\in I$, and $B{V}^{+}\left(I\right)$ ($B{V}^{-}\left(I\right)$) denotes the set of functions of bounded variation in $I$ with nondecreasing (nonincreasing) singular part.

Moreover, if $\alpha \left(0\right)\le \beta \left(0\right)$ and $\alpha \left(1\right)\ge \beta \left(1\right)$ then, for every ${x}_{0}\in \left[\alpha \left(0\right),\beta \left(0\right)\right]$ and ${y}_{0}\in \left[\beta \left(1\right),\alpha \left(1\right)\right]$, the problem ${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right)\right)$ for a.e. $t\in I=\left[0,1\right]$, $x\left(0\right)={x}_{0}$, $x\left(1\right)={y}_{0}$, has extremal solutions in $\left[m,M\right]$.

These results are extended to the discontinuous problem ${x}^{\text{'}}\left(t\right)=q\left(x\left(t\right)\right)f\left(t,x\left(t\right)\right)$, with $f$ a Carathéodory function and $q:ℝ\to \left(0,+\infty \right)$ such that $q$, $1/q\in {L}_{\text{loc}}^{\infty }\left(ℝ\right)$.

Finally, some examples of functions $f$ that do not satisfy the Carathéodory conditions and for which there is no solution in $\left[m,M\right]$ are presented in the paper.

##### MSC:
 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 34A36 Discontinuous equations
##### Keywords:
upper and lower solutions; Carathéodory conditions