International Monthly Seminar on Time Scales Analysis #4

In this work, we introduce a perturbed non-convex sweeping process with a class of subsmooth moving sets depending on the time and the state. In the first result we study the existence of solution and we present some topological properties of the attainable set, the perturbation considered here is an upper semi-continuous set-valued mapping with nonempty closed convex values unnecessarily bounded. In the second result we prove the existence to the minimal time problem and we give a description to the attainable set of control systems.

In this conversation, examines the long-term existence of solutions for a system of weakly coupled equations involving fractional evolution and various nonlinearities. The main focus is on analyzing the relationship between the regularity of initial data, memory terms, and the allowable range of exponents in a specific equation. Using L^p–L^q estimates for solutions of associated linear fractional $σ$-evolution equations with vanishing right-hand sides, along with a fixed-point method, the study establishes the existence of small-data solutions with in certain admissible exponent ranges.