A Generalization of Bochner’s Theorem and Its Applications in the Study of Impulsive Differential Equations

Factorization of Scalar Piecewise Continuous Almost Periodic Functions

“…, t ∈ R, then f ∈ P CAP (R, R) by (ii) of 3.1. We shall prove that f ∈ C(R, R), which implies the almost periodicity of f by Theorem 3.8 in [12]. A direct calculation shows that…”

Bochner-Type Property on Spaces of Generalized Almost Periodic Functions

“…The theory of almost periodic functions opened a way to study a wide class of trigonometric series of the general type and even exponential series (in this context, we can cite among others the papers [3,4,5,6,8,11]). Furthermore, it has many important applications in problems of ordinary differential equations, dynamical systems, stability theory and partial differential equations (see for example recent developments in [10,12,13]). A very important result of this theory is the approximation theorem according to which the class of almost periodic functions AP (R, C) coincides with the class of limit functions of uniformly convergent sequences of trigonometric polynomials of the type a 1 e iλ1t + .…”

Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions

“…Impulsive differential equation, which provides a natural description of observed evolution processes, is an important mathematical tool to solve some practical problems. The theory of impulsive differential equations of integer order has been widely used in practical mathematical modeling and has become an important area of research in recent years, which steadily receives attention of many authors (see [21][22][23][24][25][26][27][28][29][30][31][32][33][34]). Sun et al [35] considered a class of impulsive fractional differential equations with Riemann-Liouville fractional derivative, the existence of solution was proved by using Darbo-Sadovskii's fixed-point theorem, and the optimal control results were obtained.…”