A study of nonlocal fractional delay differential equations with hemivariational inequality

Ebrahem A. Algehyne; Abdur Raheem; Mohd Adnan; Asma Afreen; Ahmed Alamer; Ebrahem A. Algehyne; Abdur Raheem; Mohd Adnan; Asma Afreen; Ahmed Alamer

doi:10.3934/math.2023659

AIMS Mathematics

2023, Volume 8, Issue 6: 13073-13087. doi: 10.3934/math.2023659

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Research article

A study of nonlocal fractional delay differential equations with hemivariational inequality

1.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk-71491, Saudi Arabia
2.
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

Received: 19 January 2023 Accepted: 21 March 2023 Published: 03 April 2023
MSC : 34A08, 34G25, 34K37, 49J40

In this paper, we study an abstract system of fractional delay differential equations of order $ 1 < q < 2 $ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.
- Caputo fractional derivative,
- hemivariational inequality,
- Rothe method,
- Clarke subdifferential,
- pseudomonotone
Citation: Ebrahem A. Algehyne, Abdur Raheem, Mohd Adnan, Asma Afreen, Ahmed Alamer. A study of nonlocal fractional delay differential equations with hemivariational inequality[J]. AIMS Mathematics, 2023, 8(6): 13073-13087. doi: 10.3934/math.2023659

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Abstract

In this paper, we study an abstract system of fractional delay differential equations of order $ 1 < q < 2 $ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.

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