Hybrid Fixed Point Theory and Existence of Extremal Solutions for Perturbed Neutral Functional Differential Equations

In this paper, some hybrid fixed point theorems are proved which are further applied to first and second order neutral functional differential equations for proving the existence results for the extremal solutions under the mixed Lipschitz, compactness and monotonic conditions. 1. Statement of the problems The functional differential equations (in short FDE) is a topic of great interest since long time in the theory of differential equations. These equations are modeled on a dynamical system in which the present state is determined by the past state of the related dynamical systems. During the last half century a significant efforts have been applied to study functional differential equations, i.e., equations containing derivatives of the solutions and dependencies on the solutions having non-local character (the right hand side depend not only on the solutions, but also on the “prehistory” as well). Indeed, such models based on the boundary value problems for equations with deviating arguments or integro-differential equations provide the most adequate and accurate description of different processes in physics, economics, bio-mathematics and social sciences. Therefore their study is of great importance and applications. It is well known that when there are deviations of the differentiated functions from the solutions (which characterizes neutrality), the questions of the existence of solutions are essentially more complex for the study when in the case of FDE with deviating argument only in the right hand sides. Such results for NFDE are of definite theoretical and practical importance. The exhaustive treatment of NFDEs appear in the monographs like Hale [7], Henderson [9] and a recent survey of Ntouyas [11]. The nonlinear NFDEs are generally studied for Received August 22, 2006. 2000 Mathematics Subject Classification. 47H10, 34A60.