Extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses

Abstract

We study the existence of extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses by employing the monotone iterative technique coupled with the method of lower and upper solutions. Our existence results are established by appealing to two different theories. In our first result, the semigroup generated by the linear operator associated with the problem is assumed to be compact. On the other hand, in our second result, the semigroup need not be compact and our conclusions are obtained using the theory of measures of noncompactness. In order to illustrate our results, we also discuss a concrete problem regarding heat conduction in a metal bar with predefined impulsive points.

Keywords:

• Fractional differential equations
• measure of noncompactness
• monotone iterative technique
• non-instantaneous impulses
• semigroup theory

2020 Mathematics Subject classifications:

• 34A08
• 34G20
• 34K30
• 34K45
• 47D06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. Both authors are grateful to two anonymous referees for providing them with pertinent comments and useful suggestions. They are also grateful to Dr. Chinedu Izuchukwu for his help with Section 4.