Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

Abstract

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

Boundary Value Problem for the Langevin Equation and Inclusion with the Hilfer Fractional Derivative

“…An equation of the form md 2 x/dt 2 � λdx/dt + η(t) is called Langevin equation, introduced by Paul Langevin in 1908. e Langevin equation is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [8]. For some new developments on the fractional Langevin equation, see, for example, [16][17][18].…”

Existence Solution for Coupled System of Langevin Fractional Differential Equations of Caputo Type with Riemann–Stieltjes Integral Boundary Conditions

“…Analytical expressions of the correlation functions were obtained using the two fluctuation-dissipation theorems and fractional calculus approaches. The fractional Langevin equation has been received the attention of many scientists due to its extremely useful applications in different fields of science and has been dealt with under different conditions (see [8][9][10][11][12][13][14]).…”

A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

“…Moreover, for the multivalued maps in the equilibrium in the duopoly markets and in aquatic ecosystem there are also too many applications. In an ordinary differential equation, the application provided by J. J. Nieto and R. Rodríguez-López [10] and the system of matrix equations by Ran and Reurings [14], the fixed point for iteration to find optimal solution in statistics [15], for the stability problem in Intuitionistics Fuzzy Banach Space [16], and many more such as [17].…”