# The Ulam's types stability of non-linear Volterra integro-delay dynamic system with simple non-instantaneous impulses on time scales

Volume 4, Issue 1, pp 13--25 Publication Date: August 07, 2019       Article History
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### Authors

Syed Omar Shah - Department of Mathematics, University of Peshawar, Peshawar 25000. Akbar Zada - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan. Cemil Tunc - Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey.

### Abstract

This manuscript presents Hyers-Ulam stability and Hyers-Ulam-Rassias stability results of non-linear Volterra integro-delay dynamic system on time scales with non-instantaneous impulses. Picard fixed point theorem is used for obtaining existence and uniqueness of solutions. By means of abstract Gronwall lemma, Gronwall's inequality on time scales and applications of Gronwall's inequality on time scales, we establish Hyers-Ulam stability and Hyers-Ulam-Rassias stability results. There are some primary lemmas, inequalities and relevant assumptions that helps in our stability results.

### Keywords

• Hyers-Ulam stability
• time scale
• impulses
• delay dynamic equation
• Gronwall's inequality
• abstract Gronwall lemma
• Banach fixed point theorem

•  34N05
•  34G20
•  45J05
•  45M10

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