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Abstract: We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays. [Copyright &y& Elsevier]

Abstract: We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with -boundary conditions is called -stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR2s) kind, and the Top Order Methods (TOMs), are considered in this paper. By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be -stable. [Copyright &y& Elsevier]