The exponential stability property of an evolutionary process is characterized in terms of the existence of some functionals on certain function spaces. Thus are generalized some well-known results obtained by Datko, Rolewicz, Littman and Van Neerven.
We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.
We introduce the notion of uniform quietness at zero for a real-valued function and we study one-parameter nonlocal boundary value problems for second order differential equations involving such functions. By using the Krasnosel'skiĭ fixed point theorem in a cone, we give values of the parameter for which the problems have at least two positive solutions.
Suppose that the function in the differential equation (1) is decreasing on where . We give conditions on which ensure that (1) has a pair of solutions such that the -th derivative () of the function has the sign for sufficiently large and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.
We revisit the concept of Stepanov--Orlicz almost periodic functions introduced by Hillmann in terms of Bochner transform. Some structural properties of these functions are investigated. A particular attention is paid to the Nemytskii operator between spaces of Stepanov--Orlicz almost periodic functions. Finally, we establish an existence and uniqueness result of Bohr almost periodic mild solution to a class of semilinear evolution equations with Stepanov--Orlicz almost periodic forcing term.