Study of time fractional order problems with proportional delay and controllability term via fixed point approach

Muhammad Sher; Kamal Shah; Zareen A. Khan; Muhammad Sher; Kamal Shah; Zareen A. Khan

doi:10.3934/math.2021317

AIMS Mathematics

2021, Volume 6, Issue 5: 5387-5396. doi: 10.3934/math.2021317

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Research article

Study of time fractional order problems with proportional delay and controllability term via fixed point approach

1.
Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan
2.
College of Science, Mathematical Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia

Received: 17 January 2021 Accepted: 08 March 2021 Published: 12 March 2021
MSC : 26A33, 34A08

In the current manuscript, we are tying to study one of the important class of differential equations known is evolution equations. Here, we considered the problem under controllability term and with proportional delay. Before going to numerical or analytical solution it is important to check the existence and uniqueness of the solution. So, we will consider our problem for qualitative theory using fixed point theorems of Banach's and Krasnoselskii's type. For numerical solution the stability is important, hence the problem is also studied for Ulam-Hyer's type stability. At the end an example is constructed to ensure the establish results.
- time fractional order problems with proportional delay,
- controllability term,
- fixed point approach
Citation: Muhammad Sher, Kamal Shah, Zareen A. Khan. Study of time fractional order problems with proportional delay and controllability term via fixed point approach[J]. AIMS Mathematics, 2021, 6(5): 5387-5396. doi: 10.3934/math.2021317

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Abstract

In the current manuscript, we are tying to study one of the important class of differential equations known is evolution equations. Here, we considered the problem under controllability term and with proportional delay. Before going to numerical or analytical solution it is important to check the existence and uniqueness of the solution. So, we will consider our problem for qualitative theory using fixed point theorems of Banach's and Krasnoselskii's type. For numerical solution the stability is important, hence the problem is also studied for Ulam-Hyer's type stability. At the end an example is constructed to ensure the establish results.

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