Interval Cauchy problem with a second type Hukuhara derivative
The paper presents some theoretical studies of an interval-valued initial value problem with the second type Hukuhara derivative of the form \[ X'(t)=F(t,X(t)),\quad X(t_0)=X_0, \] where \(F:[\alpha,\beta]\times I \to I\) and \(X_0=[X_0^-,X_0^+]\in I\) are the data of the equation, and the symbol \('\) denotes the second type Hukuhara derivative. Here, \(I\) is the family of all nonempty, compact real intervals. First, the author collects some necessary facts about the Hukuhara difference and about differentiation and integration of interval-valued functions. Further, he presents a real-world application of an interval differential equation with the second type Hukuhara derivative. The continuous dependence of the solution on the initial value and the right-hand side of the equation is shown and the existence of approximate local solutions is stated. By means of this, the author obtains the existence of at least one local solution to the problem and compactness of the set of solutions. Finally, explicit formulas of unique second type Hukuhara differentiable local solutions to linear interval differential equations are derived.
scientific article; zbMATH DE number 6107033