Ulam’s-Type Stability of First-Order Impulsive Differential Equations with Variable Delay in Quasi–Banach Spaces

In this paper, Ulam’s-type stabilities are studied for a class of first-order impulsive differential equations with bounded variable delays on compact interval with finite number of impulses. Results of stability are proved via newly established integral inequality of Bellman–Grönwall–Bihari type with delay for discontinuous functions. Using this inequality for the first time and assumption of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> α -H <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>o</m:mi> <m:mo>¨</m:mo> </m:mover> </m:math> ö lder’s condition instead of common Lipschitz condition is novelty of this paper. Moreover, solution is obtained in quasi–Banach spaces which is best suited for obtaining results under the assumptions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> α -H <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>o</m:mi> <m:mo>¨</m:mo> </m:mover> </m:math> ö lder’s condition.