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

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.

Keywords:

• Fractional Langevin equation
• Caputo fractional derivative
• extremal solutions
• existence
• nonlinear boundary conditions

2010 Mathematics Subject Classifications:

• 26A33
• 34A08
• 34A12

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].