Open AccessExplicit iteration and unique solution for $ \phi $-Hilfer type fractional Langevin equations
Author(s) -
Abdulkafi Mohammed Saeed,
AUTHOR_ID,
Mohammed A. Almalahi,
Mohammed S. Abdo,
AUTHOR_ID,
AUTHOR_ID,
AUTHOR_ID
Publication year - 2021
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022192
Subject(s) - mathematics , monotone polygon , iterative method , fractional calculus , fixed point , type (biology) , fixed point theorem , function (biology) , mathematical analysis , mathematical optimization , geometry , evolutionary biology , biology , ecology
This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving $ \phi $-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the Mittag-Leffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.