Monotone iterative technique for functional differential equations with retardation and anticipation (Q876928)

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.