Abstract
In this note, we study a new class of ordinary differential equations with non-instantaneous impulses. Both existence and generalized Ulam-Hyers-Rassias stability results are established. Finally, an example is given to illustrate our theoretical results.
Highlights
Many evolution processes studied in applied sciences are represented by differential equations
One of the mathematical models about such processes can be formulated by the following impulsive differential equations: x (t) = f (t, x(t)), t ∈ J := J \ {t, . . . , tm}, J := [, T], x(tk+) = x(tk–) + Ik(x(tk–)), k =, . . . , m, ( )
The impulsive conditions are the combination of the traditional initial value problems and the short-term perturbations whose duration can be negligible in comparison with the duration of such a process
Summary
Many evolution processes studied in applied sciences are represented by differential equations. The above situation has fallen in a new impulsive action, which starts at an arbitrary fixed point and keeps active on a finite time interval. To achieve this aim, Hernández and O’Regan [ ] introduced a new class of abstract semilinear impulsive differential equations with non-instantaneous impulses. For the recent Ulam’s stability concepts and results on ordinary differential equations (with impulses), one can see [ , ] and reference therein. Motivated by [ , , , ], we introduce a new Ulam-type stability concept for the following semilinear differential equations with non-instantaneous impulses:. In Section , we introduce a new Ulam-type stability concept for equation ( ) An example is given to illustrate our theoretical results
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