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Monotone iterative technique for functional differential equations with retardation and anticipation. (English) Zbl 1121.34065
Consider the functional-differential equation ${dx\over dt}= f(t, x(t), x_t, x^t)\quad\text{for }t_0< t< T\tag{$$*$$}$ under the condition
$x_t= \phi_0,\quad x^T= \psi_0,\tag{$$**$$}$ where $$x_t= x_t(\sigma)$$, $$-h_1\leq s\leq 0$$, $$x^t= x(\sigma)$$, $$0\leq \sigma\leq h_2$$. Using coupled lower and upper solutions of $$(*)$$ and $$(**)$$, the authors derive conditions such that there exist monotone sequences converging uniformly on $$[t_0-h_1, T+ h_2]$$ to coupled minimal and maximal solution of $$(*)$$, $$(**)$$. They also provide an additional condition under which $$(*)$$, $$(**)$$ has a unique solution.

##### MSC:
 34K07 Theoretical approximation of solutions to functional-differential equations 34K05 General theory of functional-differential equations
##### Keywords:
existence results
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##### References:
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