Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives
type: Item
instance of: scholarly article
title: Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives
zbMATH Open document ID: 1482.34062
author: Zohre Kiyamehr
author: Hamid Baghani
published in: Journal of Applied Analysis
publication date: 2021-08-02T00:00:00Z
review text: This article investigates the existence and uniqueness of solutions for the following boundary value problem of the Langevin equation with two different fractional orders: \[ D^\beta(D^\alpha + \lambda)x(t) = f(t, x(t)),\,\, 0 < t < 1,\, 0 < a \leq 1, \,1 < \beta \leq 2, \] subject to the multi-point boundary conditions \[ x(0) = 0, \;D^{2\alpha}x(1) + \lambda D^\alpha x(1) = 0,\; x(1) =\int_0^\eta x(\tau)d\tau,\quad \text{for some } 0<\eta<1, \] where \(D^a\) is the Caputo fractional derivative of order \(a\), \(f : [0, 1] \times \mathbb{R} \rightarrow \mathbb{R}\) is a given continuous function, and \(\lambda\) is a real number. The corresponding nonlinear integral equation is considered and a fixed point theorem isn applied to show the existence and uniqueness results. At last two examples are considered to prove the applicability of the results.
reviewed by: Renu Chaudhary
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7377250
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: Caputo fractional derivative
zbMATH Keywords: fractional Langevin equation
zbMATH Keywords: three-point boundary conditions
MaRDI profile type: Publication
full work available at URL: jaa-2020-2029
OpenAlex ID: W3114967263
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DOI: 10.1515/JAA-2020-2029
description: scientific article; zbMATH DE number 7377250