Abstract
In this paper, the stability of Ulam–Hyers and existence of solutions for semi-linear time-delay systems with linear impulsive conditions are studied. The linear parts of the impulsive systems are defined by non-permutable matrices. To obtain solution for linear impulsive delay systems with non-permutable matrices in explicit form, a new concept of impulsive delayed matrix exponential is introduced. Using the representation formula and norm estimation of the impulsive delayed matrix exponential, sufficient conditions for stability of Ulam–Hyers and existence of solutions are obtained.
Highlights
The theory of functional differential equations has been attracted by many researchers
In [2], a concept of delayed matrix exponential is introduced providing an explicit formula of solutions for linear time-delay continuous systems with commutative matrices
Motivated by the above articles, we have considered the representation of solutions of a linear time-delay impulsive differential equation of the form:
Summary
The theory of functional differential equations has been attracted by many researchers. In [2], a concept of delayed matrix exponential is introduced providing an explicit formula of solutions for linear time-delay continuous systems with commutative matrices. Authors adopted the idea of [2,3,4] obtaining the representation of solutions of linear time-delay continuous systems with impulses To do so, they introduced a concept of impulsive delayed matrix function for commutative matrices. No study exists in the literature seeking an explicit solution for linear impulsive time-delay differential equations with non-commutative matrices. We introduced a novel impulsive delayed matrix exponential function (impulsive delayed exponential) and adhered its norm estimate Using this impulsive delayed exponential and the variation of constants method, we gave an explicit representation for solutions of impulsive time-delay initial value problems with linear parts defined by non-permutable matrices.
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