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Original Articles

Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions

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Pages 1-10 | Received 20 Nov 2019, Accepted 16 Jan 2020, Published online: 05 Feb 2020
 

Abstract

Fractional Langevin equation describes the evolution of physical phenomena in fluctuating environments for the complex media systems. It is a sequential fractional differential equation with two fractional orders involving a memory kernel, which leads to non-Markovian dynamics and subdiffusion. Here by establishing a general solution of the linear fractional Langevin equations involving initial conditions with the help of well-known Mittag–Leffler functions and using the special properties of these functions, we construct a new comparison result related to linear fractional Langevin equation. Meanwhile, we investigate the existence of extremal solutions for nonlinear boundary value problems with advanced arguments. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. At last an example is presented to illustrate the main results.

2010 Mathematics Subject Classifications:

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 11972148) and Natural Science Foundation of Jiangsu Province [grant number BK20190024].

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