Existence and uniqueness of mild solutions for a class of psi-Caputo time-fractional systems of order from one to two

We prove the existence and uniqueness of mild solutions for a specific class of time-fractional \(\psi\)-Caputo evolution systems with a derivative order ranging from 1 to 2 in Banach spaces. By using the properties of cosine and sine family operators, along with the generalized Laplace transform, w...

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Bibliographic Details
Published in:arXiv.org 2024-01
Main Authors: Hamza Ben Brahim, Fatima-Zahrae El Alaoui, Tajani, Asmae, Torres, Delfim F M
Format: Article
Language:English
Subjects:
Banach spaces
Fixed points (mathematics)
Operators (mathematics)
Trigonometric functions
Uniqueness
Online Access:Get full text
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Summary:We prove the existence and uniqueness of mild solutions for a specific class of time-fractional \(\psi\)-Caputo evolution systems with a derivative order ranging from 1 to 2 in Banach spaces. By using the properties of cosine and sine family operators, along with the generalized Laplace transform, we derive a more concise expression for the mild solution. This expression is formulated as an integral, incorporating Mainardi's Wright-type function. Furthermore, we provide various valuable properties associated with the operators present in the mild solution. Additionally, employing the fixed-point technique and Gr\"{o}nwall's inequality, we establish the existence and uniqueness of the mild solution. To illustrate our results, we conclude with an example of a time-fractional equation, presenting the expression for its corresponding mild solution.
ISSN:2331-8422