The existence and the uniqueness of solutions to nonlinear functional-differential and integral equations

The aim of this paper is to present a collection of recent results, especially concerning the problems of the existence and the uniqueness of solutions to some classes of functional-differential and integral equations. In Section 1 we present results concerning differential equations (especially concerning the equation of n-th order) formulated in terms of approximate derivatives. This type of a derivative seems to be one of the most important generalizations of the classical one. The second section contains the results concerning the existence and the uniqueness of solutions to nonlinear Hammerstein as well as to nonlinear Volterra - Hammerstein integral equation in the classes of functions of bounded variation in the sense of Jordan and Young. In Section 3 we present the results concerning the existence and the uniqueness of solutions to some equations of neutral type, especially concerning the Darboux problem. In the next section we deal with almost periodic solutions to differential and integral equations in Frechet spaces. Moreover, we present there Bohr - Neugebauer type theorems for almost periodic solutions to linear equations in such spaces. Finally, in the Appendix we present fixed point theorems for locally convex spaces as well as for Banach spaces, which can be tools in investigations concerning the existence of solutions to nonlinear differential and integral equations.