EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAUCHY-TYPE PROBLEMS INVOLVING mth-LEVEL FRACTIONAL DERIVATIVES

Bany-Ahmad R., Ibrahim A., Noorani M. S. M., ABDELJAWAD T.

Fractals, 2025 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1142/s0218348x25402790
  • Dergi Adı: Fractals
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, INSPEC, zbMATH
  • Anahtar Kelimeler: Existence and Uniqueness, Hilfer Fractional Derivative, mth-level Fractional Derivatives, Riemann–Liouville Fractional Derivative, Second-level Fractional Derivative
  • İstanbul Gelişim Üniversitesi Adresli: Evet

Özet

This paper investigates the existence and uniqueness of global solutions in the space of weighted continuous functions for Cauchy-type problems involving fractional differential equations with mth-level fractional derivatives. Using the Banach fixed-point theorem and the step method, we establish our results within an appropriate functional space, demonstrating the equivalence between the given problem and a corresponding Volterra integral equation. As a particular case, we examine the Cauchy-type problem for fractional differential equations with second-level fractional derivatives. Key properties and fundamental results related to this type of fractional calculus are discussed. Additionally, we derive significant Cauchy-type problems from both mth- and second-level fractional derivatives, which have been extensively studied in Riemann–Liouville, Caputo, and Hilfer fractional derivatives. Finally, we analyze the stability of the solutions to the weighted Cauchy-type problem.