Abstract
Abstract In this paper, we study a new type of a Langevin equation involving two different fractional orders and impulses. Sufficient conditions are formulated for the existence and uniqueness of solutions of the given problems. MSC:34A08, 34B10, 34B37, 46N10.
Highlights
1 Introduction Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes
For the systems in complex media, an integer-order Langevin equation does not provide the correct description of the dynamics
Nonlocal conditions were initiated by Byszewski [ ] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems
Summary
Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This gives rise to a fractional Langevin equation, see [ – ] and the references therein. In , Lim, Li and Teo [ ] firstly introduced a new type of a Langevin equation with two different fractional orders. In , by using the contraction mapping principle and Krasnoselskii’s fixed point theorem, Ahmad and Nieto [ ] studied a Langevin equation involving two fractional orders with Dirichlet boundary conditions.
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