Abstract
A new sequence space related to the space ell _{p}, 1leq p<infty (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.
Highlights
Infinite systems of differential equations play a significant role in many subjects of nonlinear analysis
By using the measure of noncompactness technique and applying the Darbo’s fixed point theorem, we investigate the existence of solutions for the infinite system (1.1)–(1.2) in the Banach spaces p, p ≥ 1
6 Conclusion In the present research, we studied an infinite system of Langevin equations of fractional order
Summary
Infinite systems of differential equations play a significant role in many subjects of nonlinear analysis. Dαui(t) = fi t, u(t) , t ∈ (0, T), 1 < α < 2, where u(t) = {ui(t)}∞ i=0, with the initial conditions ui(0) = u0i = 0, ui(T) = aui(ξ ), i ∈ N0, aξ α–1 < T α–1 in Banach spaces, where Dα is the R–L fractional derivative of order α. The measure of noncompactness has been used extensively by many authors when studying infinite systems of differential and fractional differential equations. Motivated by the former contributions, we consider the infinite system of the generalized Langevin equations ρi c. By using the measure of noncompactness technique and applying the Darbo’s fixed point theorem, we investigate the existence of solutions for the infinite system (1.1)–(1.2) in the Banach spaces p, p ≥ 1. New results on the existence of solutions for fractional Langevin equations under variety of boundary value conditions have been published; see [20,21,22,23,24,25,26,27,28,29] and the references mentioned therein
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
