An analysis on the boundary control for nonlinear integro-differential evolution systems with impulsive effects and time delays via fixed point theorems

Kamalendra Kumar; Rohit Patel; Mohammad Sajid; Rakesh Kumar; Kamalendra Kumar; Rohit Patel; Mohammad Sajid; Rakesh Kumar

doi:10.3934/math.2025646

AIMS Mathematics

2025, Volume 10, Issue 6: 14347-14371. doi: 10.3934/math.2025646

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Research article

An analysis on the boundary control for nonlinear integro-differential evolution systems with impulsive effects and time delays via fixed point theorems

1.
Department of Basic Science, Shri Ram Murti Smarak College of Engineering and Technology, Bareilly, India
2.
Department of Mathematics, Government P.G. College Bisalpur, Pilibhit, India
3.
Department of Mechanical Engineering, College of Engineering, Qasim University, Saudi Arabia
4.
Department of Mathematics, Hindu College, Moradabad (Affiliated to MJP Rohilkhand University, Bareilly), India

Received: 19 February 2025 Revised: 27 May 2025 Accepted: 10 June 2025 Published: 23 June 2025
MSC : 34A37, 47D06, 93B05

This study investigated the boundary controllability of nonlinear impulsive integro-differential evolution systems (NIIESs) with time-varying delays within Banach spaces. Two classes of NIIESs were considered, and sufficient conditions for their controllability were established using fixed point theorems and semigroup theory. For the first class, Schaefer's fixed point theorem was employed in combination with compact semigroup theory, whereas for the second class, Schauder's fixed point theorem was utilized. The research defined essential hypotheses and mathematical structures to ensure the robustness and applicability of the results. Illustrative examples were provided to confirm the applicability and effectiveness of the developed theoretical framework. This work significantly contributes to the study of partial functional integro-differential equations in nonlinear systems, particularly systems influenced by impulsive effects and time delays, addressing gaps in the existing literature.
- boundary controllability,
- impulsive systems,
- nonlinear systems,
- integro-differential equations,
- fixed point theory,
- time-delay systems
Citation: Kamalendra Kumar, Rohit Patel, Mohammad Sajid, Rakesh Kumar. An analysis on the boundary control for nonlinear integro-differential evolution systems with impulsive effects and time delays via fixed point theorems[J]. AIMS Mathematics, 2025, 10(6): 14347-14371. doi: 10.3934/math.2025646

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Abstract

This study investigated the boundary controllability of nonlinear impulsive integro-differential evolution systems (NIIESs) with time-varying delays within Banach spaces. Two classes of NIIESs were considered, and sufficient conditions for their controllability were established using fixed point theorems and semigroup theory. For the first class, Schaefer's fixed point theorem was employed in combination with compact semigroup theory, whereas for the second class, Schauder's fixed point theorem was utilized. The research defined essential hypotheses and mathematical structures to ensure the robustness and applicability of the results. Illustrative examples were provided to confirm the applicability and effectiveness of the developed theoretical framework. This work significantly contributes to the study of partial functional integro-differential equations in nonlinear systems, particularly systems influenced by impulsive effects and time delays, addressing gaps in the existing literature.

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