Picard and Picard-Krasnoselskii iteration methods for generalized proportional Hadamard fractional integral equations

Yıl 2022, , 538 - 546, 30.12.2022

Mohamed Abbas

https://doi.org/10.31197/atnaa.1070142

Öz

In the current paper, some existence and uniqueness results for a generalized proportional Hadamard fractional integral equation are established via Picard and Picard-Krasnoselskii iteration methods together with the Banach contraction principle. A simulative example was provided to verify the applicability of the theoretical findings.

Anahtar Kelimeler

Picard method, Picard-Krasnoselskii method, Proportional Hadamard fractional integral

Kaynakça

• [1] M. I. Abbas, M. A. Ragusa, On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function, Symmetry 13(2) (2021), Article ID:264.
• [2] M. I. Abbas, Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative Miskolc Mathematical Notes 22(2) (2021), 1–12.
• [3] M. I. Abbas, Non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another function, Math. Meth. Appl. Sci. 44(13) (2021), 10432–10447.
• [4] T. Abdeljawad, K. Ullah, J. Ahmad, On Picard-Krasnoselskii Hybrid Iteration Process in Banach Spaces, J. Math. 2020 (2020), Article ID: 2150748, 5 p.
• [5] D. Boucenna, D. Baleanu, A. Makhlouf, A. M. Nagy, Analysis and numerical solution of the generalized proportional fractional Cauchy problem, Appl. Numer. Math. 167 (2021), 173–186.
• [6] A. M. A. El-Sayed, H. H. G. Hashem, E. A. A. Ziada, Picard and Adomian decomposition methods for a quadratic integral equation of fractional order, Comp. Appl. Math. 33 (2014), 95–109.
• [7] S. Hristova, M. I. Abbas, Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses, Symmetry 13(6) (2021), Article ID:996.
• [8] F. Jarad, T.Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Special topics 226 (2017), 3457–3471.
• [9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, 2006.
• [10] A. A. Kilbas, Hadamard-Type Integral Equations and Fractional Calculus Operators, Operator Theory: Advances and Applications 142 (2003), 175–188.
• [11] M. A. Krasnosel’skii, Two observations about the method of successive approximations, Usp. Mat. Nauk 10 (1955), 123–127.
• [12] Ch. Li, Uniqueness of the Hadamard-type integral equations, Advances in Difference Equations 2021 (2021), Article ID:40, doi:10.1186/s13662-020-03205-8.
• [13] H. R. Marasi, A. S. Joujehi, H. Aydi, An extension of the Picard theorem to fractional differential equations with a Caputo-Fabrizio derivative, J. Funct. Spaces 2021 (2021), Article ID:6624861.
• [14] S. Micula, An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math. 339 (2018), 124–133.
• [15] G. A. Okeke, M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math. 6 (2017), 21–29.
• [16] E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl. 6 (1890), 145–210.
• [17] G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral, Advances in Difference Equations 2019 (2019), Article ID:454, doi:10.1186/s13662-019-2381-0.
• [18] A. S¸ahin, Some Results of the Picard-KrasnoselskiiHybrid Iterative Process, Filomat 33(2) (2019), 359–365.
• [19] S¸ . M. S¸oltuz, D. Otrocol, Classical results viaMann-Ishikawa iteration, Revue d’Analyse Num`erique et de Th`eorie de l’Approximation 36(2) (2007), 195–199.
• [20] J. Wang, Z. Lin, Ulam’s Type Stability of Hadamard Type Fractional Integral Equations, Filomat 28(7) (2014), 1323–1331.
• [21] J. Wang, M. Fe˘ckan, Y. Zhou, Weakly Picard operators method for modified fractional iterative functional differential equations, Fixed Point Theory 15(1) (2014), 297–310.
• [22] J.Wang, Y. Zhou, M. Medve˘d, Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces, J. Math. Anal. Appl. 389 (2012), 261–274.
• [23] Sh. Zhou, S. Rashid, E. Set, A. G. Ahmad, Y. S. Hamed, On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications, AIMS Math. 6(9) (2021), 9154–9176.