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The existence and the uniqueness of solutions to nonlinear functional-differential and integral equations
Dariusz Bugajewski
Abstract
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 functionaldifferential 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 Fr ´echet 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.
Keywords
α-nonexpansive mapping, almost periodic function, Aronszajn property, asymptotically almost periodic function, Cauchy problem, Darboux problem of neutral type, Denjoy integral, Denjoy - Perron integral, differential equations with maxima, existence and uniqueness of solutions, fixed point theorems, function of bounded variation in the sense of Jordan, function of bounded variation in the sense of Young, functional partial differential equation, Hammerstein integral equation, Henstock - Kurzweil integral, Kuratowski measure of noncompactness, measure of weak noncompactness, nonexpansive mapping, perfect Frechet space, spectral radius, Volterra integral equation, Volterra - Hammerstein integral equation, ω-nonexpansive mapping, weakly F-contractive mapping.
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