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Boundary value problems for ordinary differential equations with deviated arguments. (English) Zbl 1140.34027

Consider the boundary value problem

$\begin{array}{c}{x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right),x\left(\alpha \left(t\right)\right),x\left(t\right),x\left(\beta \left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],\\ 0=g\left(x\left(0\right),x\left(T\right)\right),\end{array}\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $f$, $g$, $\alpha$, $\beta$ are continuous functions. The authors give sufficient conditions such that

(i) $\left(*\right)$ has quasisolutions;

(ii) $\left(*\right)$ has a unique solution.

The proofs are based on the monotone iterative method. They present two examples satisfying the required assumptions.

##### MSC:
 34K10 Boundary value problems for functional-differential equations 34K07 Theoretical approximation of solutions of functional-differential equations
##### References:
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