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Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments. (English) Zbl 1232.34091

This paper is concerned with the existence of absolutely continuous solutions of the problem

${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right),x\left(\tau \left(t,x\left(t\right),x\right)\right)\phantom{\rule{4.pt}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}t\in {I}_{+}=\left[{t}_{0},{t}_{0}+L\right],$
$x\left(t\right)={\Lambda }\left(x\right)+k\left(t\right)\phantom{\rule{4.pt}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}t\in {I}_{-}=\left[{t}_{0}-r,{t}_{0}\right],$

where ${t}_{0}\in ℝ$, $L>0$, $r\ge 0$, $k$ is a continuous function and $f$, $\tau$, ${\Lambda }$ are suitable functions not necessarily continuous. First, the authors prove an abstract result on the existence of coupled fixed points for multivalued operators. Then, they give two new theorems on the existence of coupled quasisolutions for the above initial value problem, which, in turn, yield two corresponding new theorems on the existence of unique solutions.

##### MSC:
 34K05 General theory of functional-differential equations 47N20 Applications of operator theory to differential and integral equations
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