Abstract
This article studies the existence of solutions for a three-point inclusion problem of Langevin equation with two fractional orders. Our main tools of study include a nonlinear alternative of Leray-Schauder type, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps, and a fixed point theorem for multivalued map due to Covitz and Nadler. An illustrative example is also presented. Mathematical Subject Classification 2000: 26A33; 34A12; 34A40.
Highlights
The study of fractional calculus has recently gained a great momentum and has emerged as an interesting and important field of research
Some results concerning the initial and boundary value problems of fractional equations and inclusions can be found in a series of articles [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and the references therein
Motivated by recent work on Langevin equation of fractional order, we study the following inclusion problem of Langevin equation of two fractional orders in different intervals with three-point boundary conditions cDβx(t) ∈ F(t, x(t)), 0 < t < 1, x(0) = 0, x(η) = 0, x(1) = 0, 0 < α ≤ 1, 0 < η < 1, 1 < β ≤ 2, (1:1)
Summary
The study of fractional calculus has recently gained a great momentum and has emerged as an interesting and important field of research. A Dirichlet boundary value problem for Langevin equation involving two fractional orders has been studied in [42]. In a more recent article [47], the authors studied a nonlinear Langevin equation involving two fractional orders a Î
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