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To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/math.2024443","type":"journal-article","created":{"date-parts":[[2024,3,6]],"date-time":"2024-03-06T06:58:00Z","timestamp":1709708280000},"page":"9107-9127","source":"Crossref","is-referenced-by-count":10,"title":["Fractional tempered differential equations depending on arbitrary kernels"],"prefix":"10.3934","volume":"9","author":[{"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal"}]},{"given":"Nat\u00e1lia","family":"Martins","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal"}]},{"given":"J. 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