Existence results for hybrid integral boundary value problems of nonlinear Langevin equation with two different fractional orders

摘要

In this paper, a new type of Langevin equation with two different fractional orders is considered. By using Leray-Schauder's fixed point theorem and Banach's fixed point theorem, we obtain the existence and uniqueness results of the solution.Fractional-order models are found to be more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of prisics, chemistry, aero dynamics,electro dynamics of complex medium,polymer rheology, etc. involves derivatives of fractional order.Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details. see f3-221 and the references therein.