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

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

Authors

  • Mohamed ABBAS*

Keywords:

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

Abstract

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.

References

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.

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.

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.

T. Abdeljawad, K. Ullah, J. Ahmad, On Picard-Krasnoselskii Hybrid Iteration Process in Banach Spaces, J. Math. 2020 (2020), Article ID: 2150748, 5 p.

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.

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.

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.

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.

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, 2006.

A. A. Kilbas, Hadamard-Type Integral Equations and Fractional Calculus Operators, Operator Theory: Advances and Applications 142 (2003), 175–188.

M. A. Krasnosel’skii, Two observations about the method of successive approximations, Usp. Mat. Nauk 10 (1955), 123–127.

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.

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.

S. Micula, An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math. 339 (2018), 124–133.

G. A. Okeke, M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math. 6 (2017), 21–29.

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.

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.

A. S¸ahin, Some Results of the Picard-KrasnoselskiiHybrid Iterative Process, Filomat 33(2) (2019), 359–365.

S¸ . M. S¸oltuz, D. Otrocol, Classical results viaMann-Ishikawa iteration, Revue d’Analyse NumeriqueetdeTh

eorie de l’Approximation 36(2) (2007), 195–199.

J. Wang, Z. Lin, Ulam’s Type Stability of Hadamard Type Fractional Integral Equations, Filomat 28(7) (2014), 1323–1331.

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.

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.

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.

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Published

2023-08-01

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Articles

How to Cite

Picard and Picard-Krasnoselskii iteration methods for generalized proportional Hadamard fractional integral equations. (2023). Advances in the Theory of Nonlinear Analysis and Its Application, 6(4), 538-546. https://atnaea.org/index.php/journal/article/view/176