Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions

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Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions Weera Yukunthorn1 , Sotiris K Ntouyas2,3 and Jessada Tariboon1* * Correspondence: [email protected] 1 Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand Full list of author information is available at the end of the article

Abstract In this paper, we study the existence and uniqueness of solution for a problem consisting of a sequential nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. A variety of fixed point theorems, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory, are used. Examples illustrating the obtained results are also presented. MSC: 26A33; 34A08; 34B10 Keywords: fractional differential equations; nonlocal boundary conditions; fixed point theorems

1 Introduction In this paper, we concentrate on the study of existence and uniqueness of solution for the following nonlinear fractional Caputo-Langevin equation with nonlocal RiemannLiouville fractional integral conditions:     Dp Dq + λ x(t) = f t, x(t) , m 

t ∈ [, T],

μi I αi x(ηi ) = σ ,

(.)

i= n 

νj I βj x(ξj ) = σ ,

j=

where  < p, q ≤ ,  < p + q ≤ , Dq and Dp are the Caputo fractional derivatives of order q and p, respectively, I φ is the Riemann-Liouville fractional integral of order φ, where φ = αi , βj > , ηi , ξj ∈ (, T) are given points, μi , νj , λ, σ , σ ∈ R, i = , , . . . , m, j = , , . . . , n, and f : [, T] × R → R is a continuous function. The significance of studying problem (.) is that the nonlocal conditions are very general and include many conditions as special cases. In particular, if αi = βj = , for all i = , , . . . , m, j = , , . . . , n, then the nonlocal condition of (.) reduces to 

η η η μ   x(s) ds + μ   x(s) ds + · · · + μm  m x(s) ds = σ ,  ξ  ξ  ξn ν  x(s) ds + ν  x(s) ds + · · · + νn  x(s) ds = σ ,

(.)

©2014 Yukunthorn et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Yukunthorn et al. Advances in Difference Equations 2014, 2014:315 http://www.advancesindifferenceequations.com/content/2014/1/315

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and if σ = σ = , m = n = , μ , ν = , then (.) is reduced to 



η

x(s) ds =

ε



η

x(s) ds,







ξ

ξ

x(s) ds =

ε 

x(s) ds,

(.)



where ε = –(μ /μ ) and ε = –(ν /ν ). Note that the nonlocal conditions (.) and (.) do not contain values of an unknown function x on the left-hand side and the right-hand side of boundary points t =  and t = T, respectively.

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