Mild solutions and controllability of $ (k, \varphi) $-Hilfer fractional delay differential equations with history-dependent operators

Doha A. Kattan; Hasanen A. Hammad; Najat Almutairi; Doha A. Kattan; Hasanen A. Hammad; Najat Almutairi

doi:10.3934/math.2026637

AIMS Mathematics

2026, Volume 11, Issue 6: 15485-15512. doi: 10.3934/math.2026637

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Research article

Mild solutions and controllability of $ (k, \varphi) $-Hilfer fractional delay differential equations with history-dependent operators

1.
Department of Mathematics, College of Sciences and Art, King Abdulaziz University, Rabigh, Saudi Arabia
2.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received: 21 March 2026 Revised: 24 April 2026 Accepted: 20 May 2026 Published: 02 June 2026
MSC : 26A33, 35K58, 47H10, 49J15

The objective of this work was to investigate a class of Hilfer-type fractional differential equations governed by the interaction of semigroup operators, history-dependent mechanisms, and probability density functions, with particular attention to their analytical and control properties. First, the existence of mild solutions was established through semigroup theory and fixed-point arguments tailored to fractional dynamics. We then addressed the controllability problem for the corresponding $ (k, \varphi) $-Hilfer fractional delay differential equation, taking into account the influence of memory and delay effects induced by history-dependent operators. By combining Mönch's fixed-point theorem with the measure of noncompactness, a set of sufficient conditions for controllability was obtained. This approach not only captures the complexity of the system but also deepens the understanding of how fractional-order behavior and past-state dependence affect controllability.
- fractional derivative,
- Hilfer operator,
- fixed-point technique,
- history-dependent operator,
- measure of noncompactness,
- delay term
Citation: Doha A. Kattan, Hasanen A. Hammad, Najat Almutairi. Mild solutions and controllability of $ (k, \varphi) $-Hilfer fractional delay differential equations with history-dependent operators[J]. AIMS Mathematics, 2026, 11(6): 15485-15512. doi: 10.3934/math.2026637

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Abstract

The objective of this work was to investigate a class of Hilfer-type fractional differential equations governed by the interaction of semigroup operators, history-dependent mechanisms, and probability density functions, with particular attention to their analytical and control properties. First, the existence of mild solutions was established through semigroup theory and fixed-point arguments tailored to fractional dynamics. We then addressed the controllability problem for the corresponding $ (k, \varphi) $-Hilfer fractional delay differential equation, taking into account the influence of memory and delay effects induced by history-dependent operators. By combining Mönch's fixed-point theorem with the measure of noncompactness, a set of sufficient conditions for controllability was obtained. This approach not only captures the complexity of the system but also deepens the understanding of how fractional-order behavior and past-state dependence affect controllability.

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