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.


34K07 Theoretical approximation of solutions to functional-differential equations
34K05 General theory of functional-differential equations
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