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arXiv · 2402.01754

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

Hamza Ben Brahim·Fatima-Zahrae El Alaoui·Asmae Tajani·Delfim F. M. Torres
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Abstract

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.

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Hamza Ben Brahim, Fatima-Zahrae El Alaoui, Asmae Tajani, Delfim F. M. Torres. 2024-01-30. Existence and uniqueness of mild solutions for a class of psi-Caputo time-fractional systems of order from one to two. https://doi.org/10.1007/s13540-025-00434-0

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