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Criteria for the existence of limit-periodic solutions are established to nonlinear difference as well as differential systems governed by a Stepanov-like limit-periodic forcing. The restoring nonlinearities need not be globally Lipschitzean. The obtained sufficient conditions for difference systems are significantly more effective than those for differential systems. Nevertheless, in the continuous case, our theorems also extend some known results. A possibility of more explicit formulations are discussed in concluding remarks. [ABSTRACT FROM AUTHOR]

In this paper we study and obtain the existence of almost periodic solutions to the following class of semilinear difference equations whereA, Bare singularsquare matrices () andis almost periodic in the first variable uniformly in the second one. Next, we use these results to study the existence of almost periodic to some second-order (and higher-order) singular difference equations. [ABSTRACT FROM PUBLISHER]

The relationship between Carathéodory almost-periodic (a.p.) solutions and their discretizations is clarified for differential equations and inclusions in Banach spaces. Our investigation was stimulated by an old result of Meisters [Proc. Am. Math. Soc. 10 (1959), pp. 113–119] about Bohr a.p. solutions which we generalize in several directions. Unlike for functions, Stepanov and Bohr a.p. sequences are shown to coincide. A particular attention is paid to purely (i.e. non-uniformly continuous) Stepanov a.p. solutions. Many ideas are explained in detail by means of examples illustrated. [ABSTRACT FROM PUBLISHER]

We study the superposition operators (also called Nemytskii operators) between spaces of almost periodic functions with values in a complete metric space or in a Banach space. We establish new results on their continuity and their differentiability by using several different methods. We give applications to evolution equations and to differential equations in Banach spaces. We also establish new results on the superposition operators between spaces of asymptotically almost periodic functions and spaces of almost automorphic functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result. [ABSTRACT FROM AUTHOR]

As a nontrivial application of the abstract theorem developed in our recent paper titled "Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions", the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied. [ABSTRACT FROM AUTHOR]